An Equivariant Lefschetz Fixed-Point Formula

for Correspondences

Ivo Dell’Ambrogio, Heath Emerson,^{1} and Ralf Meyer^{2}

Received: June 26, 2013

Communicated by Joachim Cuntz

Abstract. We compute the trace of an endomorphism in equivariant bivariant K-theory for a compact group G in several ways: geometrically using geometric correspondences, algebraically using localisation, and as a Hattori–Stallings trace. This results in an equivariant version of the classical Lefschetz fixed-point theorem, which applies to arbitrary equivariant correspondences, not just maps.

We dedicate this article to Tamaz Kandelaki, who was a coauthor in an earlier version of this article, and passed away in 2012. We will remember him for his warm character and his perseverance in doing mathematics in difficult circumstances.

2010 Mathematics Subject Classification: 19K99, 19K35, 19D55

1 Introduction

Here we continue a series of articles by the last two authors about Euler characteristics and Lefschetz invariants in equivariant bivariant K-theory. These invariants were introduced in [11, 13–16]. The goal is to compute Lefschetz invariants explicitly in a way that generalises the Lefschetz–Hopf fixed-point formula. Let X be a smooth compact manifold and f: X → X a self-map with simple isolated fixed points. The Lefschetz–Hopf fixed-point formula identifies 1. the sum over the fixed points of f, where each fixed point contributes ±1

1supported by a National Science and Engineering Research Council of Canada Discovery Grant

2supported by the Volkswagen Foundation (Georgian–German non-commutative partnership) and the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG))

through the Institutional Strategy of the University of Göttingen.

Documenta Mathematica 19 (2014) 141–193

depending on its index;

2. the supertrace of the ℚ-linear, grading-preserving map on K^{∗}(X) ⊗ ℚ induced by f.

It makes no difference in (2) whether we use rational cohomology or K-theory because the Chern character is an isomorphism between them. We will generalise this result in two ways. First, we allow a compact group G to act on X and get elements of the representation ring R(G) instead of numbers. Secondly, we replace self-maps by self-correspondences in the sense of [15]. Sections 2 and 3 generalise the invariants (1) and (2) respectively to this setting. The invariant of Section 2 is local and geometric and generalises (1) above; the formulas in Sections 3 and 4 are global and homological and generalise (2) (in two different ways.) The equality of the geometric and homological invariants is our generalisation of the Lefschetz fixed-point theorem. A first step is to interpret the invariants (1) or (2) in a category-theoretic way in terms of the trace of an endomorphism of a dualisable object in a symmetric monoidal category. Let C be a symmetric monoidal category with tensor product ⊗ and tensor unit 𝟙. An object A of C is called dualisable if there is an object A^{∗}, called its dual, and a natural isomorphism

C(A ⊗ B, C) ≅ C(B, A^{∗} ⊗ C)

for all objects B and C of C. Such duality isomorphisms exist if and only if there are two morphisms η: 𝟙 → A ⊗ A^{∗} and ε: A^{∗} ⊗ A → 𝟙, called unit and counit of the duality, that satisfy two appropriate conditions. Let f: A → A be an endomorphism in C. Then the trace of f is the composite endomorphism

𝟙 ⟶^{η} A ⊗ A^{∗} −−→^{sym} A^{∗} ⊗ A −−−⟶^{id_{A^{∗}} ⊗f} A^{∗} ⊗ A ⟶^{ε} 𝟙,

where symdenotes the symmetry (or braiding) isomorphism. In this article we also call the trace the Lefschetz index of the morphism. This is justified by the following example. Let C be the Kasparov category KK with its usual tensor product structure, A = C(X) for a smooth compact manifold X, and f̂ ∈ KK_{0}(A, A) for some morphism. We may construct a dual A^{∗} from the tangent bundle or the stable normal bundle of X. In the case of a smooth self-map of X, and assuming a certain transversality condition, the trace of the morphism f̂ induced by the self-map equals the invariant (1), that is, equals the number of fixed-points of the map, counted with appropriate signs. This is checked by direct computation in Kasparov theory, see [13] for more general results. This paper springs in part from the reference [13]. A similar invariant to the Lefschetz index was introduced there, called the Lefschetz class (of the morphism). The Lefschetz class for an equivariant Kasparov endomorphism of X was defined as an equivariant K-homology class for X. The Lefschetz index, that is, the categorical trace, discussed above, is the Atiyah–Singer index of the Lefschetz class of [13]. The main goal of this article is to give a global, homological formula for the Lefschetz index generalising the invariant (2) for a non-equivariant self-map. The formulation and proof of our homological formula works best for Hodgkin Lie groups. A more complicated form applies to all compact groups. The article [13] also provides two formulas for the equivariant Lefschetz class whose equality generalises that of the invariants (1) and (2), but the methods there are completely different. The other main contribution of this article is to compute the geometric expression for the Lefschetz index in the category k̂k^{G} of geometric correspondences introduced in [15]. This simplifies the computation in Kasparov’s analytic theory in [13] and also gives a more general result, since we can work with general smooth correspondences rather than just maps. Furthermore, using an idea of Baum and Block in [4], we give a recipe for composing two smooth equivariant correspondences under a weakening of the usual transversality assumption (of [6]). This technique is important for computing the Lefschetz index in the case of continuous group actions, where transversality is sometimes difficult to achieve, and in particular, aids in describing equivariant Euler characteristics in our framework. Section 2 contains our geometric formula for the Lefschetz index of an equivariant self-correspondence. Why is there a nice geometric formula for the Lefschetz index of a self-map in Kasparov theory? A good explanation is that Connes and Skandalis [6] describe KK-theory for commutative C^{∗}-algebras geometrically, including the Kasparov product; furthermore, the unit and counit of the KK-duality for smooth manifolds have a simple form in this geometric variant of KK. An equivariant version of the theory in [6] is developed in [15]. In Section 2, we also recall some basic results about the geometric KK-theory introduced in [15]. If X is a smooth compact G-manifold for a compact group G, then KK_{∗}^{G}(C(X), C(X)) is isomorphic to the geometrically defined group k̂k _{∗}^{G}(X, X). Its elements are smooth correspondences

X ⟵^{b} (M, ξ) ⟶^{f} X  (1.1)

consisting of a smooth G-map b, a K_{G}-oriented smooth G-map f, and ξ ∈ K_{G}^{∗}(M). Theorem 2.18 computes the categorical trace, or Lefschetz index, of such a correspondence under suitable assumptions on b and f. Assume first that X has no boundary and that b and f are transverse; equivalently, for all m ∈ M with f(m) = b(m) the linear map Db − Df: T_{m}M → T_{f(m)} X is surjective. Then

Q := {m ∈ M | b(m) = f(m)}  (1.2)

is naturally a K_{G}-oriented smooth manifold. We show that the Lefschetz index is the G-index of the Dirac operator on Q twisted by ξ|_{Q} ∈ K_{G}^{∗}(Q) (Theorem 2.18). More generally, suppose that the coincidence space Q as defined above is merely assumed to be a smooth submanifold of M, and that x ∈ TX and Df(ξ) = Db(ξ) implies that ξ ∈ TQ. Then we say that f and b intersect smoothly. For example, the identity correspondence, where f and b are the identity maps on X, does not satisfy the above transversality hypothesis, but f and b clearly intersect smoothly. In the case of a smooth intersection, the cokernels of the map Df − Db form a vector bundle on Q which we call the excess intersection bundle η. This bundle measures the failure of transversality of f and b. Let η be K_{G}-oriented. Then TQ also inherits a canonical K_{G}-orientation. The restriction of the Thom class of η to the zero section gives a class e(η) ∈ K_{G}^{∗}(Q). Then Theorem 2.18 asserts that the Lefschetz index of the correspondence (1.1) with smoothly intersecting f and b is the index of the Dirac operator on the coincidence manifold Q twisted by ξ ⊗e(η). This is the main result of Section 2. In Section 3 we generalise the global homological formula involved in the classical Lefschetz fixed-point theorem, to the equivalent situation. This involves completely different ideas. The basic idea to use Künneth and Universal Coefficient theorems for such a formula already appears in [9]. In the equivariant case, these theorems become much more complicated, however. The new idea that we need here is to first localise KK^{G} and compute the Lefschetz index in the localisation. In the introduction, we only state our result in the simpler case of a Hodgkin Lie group G. Then R(G) is an integral domain and thus embeds into its field of fractions F. For any G-C^{∗}-algebra A, K_{∗}^{G}(A) is a ℤ/2-graded R(G)- module. Thus K_{∗}^{G}(A; F) := K_{∗}^{G}(A) ⊗_{R(G)} F becomes a ℤ/2-graded F -vector space. Assume that A is dualisable and belongs to the bootstrap class in KK^{G}. Then K_{∗}^{G}(A; F) is finite-dimensional, so that the map on K_{∗}^{G}(A; F) induced by an endomorphism φ ∈ KK_{0}^{G}(A, A) has a well-defined (super)trace in F. Theorem 3.4 asserts that this supertrace belongs to R(G) ⊆ F and is equal to the Lefschetz index of φ. In particular, this applies if A = C(X) for a compact G-manifold. The results of Sections 2 and 3 together thus prove the following:

Theorem 1.1. Let G be a Hodgkin Lie group, let F be the field of fractions of R(G). Let X be a closed G-manifold. Let X ⟵^{b} (M, ξ) ⟶^{f} X be a smooth G-equivariant correspondence from X to X with ξ ∈ K_{G}^{dim M−dim X} (X); it represents a class φ ∈ k̂k _{0}^{G}(X, X). Assume that b and f intersect smoothly with K_{G}-oriented excess intersection bundle η. Equip Q := {m ∈ M | b(m) = f(m)} with its induced K_{G}-orientation. Then the R(G)-valued index of the Dirac operator on Q twisted by ξ|_{Q} ⊗ e(η) is equal to the supertrace of the F -linear map on K_{G}^{∗}(X) ⊗_{R(G)} F induced by φ.

If G is a connected Lie group, then there is a finite covering Ĝ ↠ G that is a Hodgkin Lie group. We may turn G-actions into Ĝ-actions using the projection map, and get a symmetric monoidal functor KK^{G} → KKĜ. Since the map R(G) → R(Ĝ) is clearly injective, we may compute the Lefschetz index of φ ∈ KK_{0}^{G}(A, A) by computing instead the Lefschetz index of the image of φ in KKĜ ; 0(A, A). By the result mentioned above, this uses the induced map on KĜ ; ∗(A) ⊗_{R(Ĝ)} F̂, where F̂ is the field of fractions of R(Ĝ). Thus we get a satisfactory trace formula for all connected Lie groups. But the result may be quite different from the trace of the induced map on K_{∗}^{G}(A) ⊗_{R(G)} F. If G is not connected, then the total ring of fractions of G is a product of finitely many fields. Its factors correspond to conjugacy classes of Cartan subgroups in G. Each Cartan subgroup H ⊆ G corresponds to a minimal prime ideal 𝔭_{H} in R(G). The quotient R(G)/𝔭_{H} is an integral domain and embeds into a field of fractions F_{H}. We show that the map R(G) → F_{H} maps the Lefschetz index of φ to the supertrace of K_{∗}^{H} (φ; F_{H}) (Theorem 3.23). It is crucial to use H-equivariant K-theory here. The very simple counterexample 3.7 shows that there may be two elements φ_{1}, φ_{2} ∈ KK_{0}^{G}(A, A) with different Lefschetz index but inducing the same map on K_{∗}^{G}(A). Thus the generalisation of Theorem 1.1 to disconnected G identifies the image of the index of the Dirac operator on Q twisted by ξ|_{Q}⊗e(η) under the canonical map R(G) → F_{H} with the supertrace of the F_{H} -linear map on K_{G}^{∗}(X)⊗_{R(G)} F_{H} induced by φ, for each Cartan subgroup H. The trace formulas in Section 3 require the algebra A on which we compute the trace to be dualisable and to belong to an appropriate bootstrap class, namely, the class of all G-C^{∗}-algebras that are KK^{G}-equivalent to a type I G-C^{∗}-algebra. This is strictly larger than the class of G-C^{∗}-algebras that are KK^{G}-equivalent to a commutative one, already if G is the circle group (see [10]). We describe the bootstrap class as being generated by so-called elementary G-C^{∗}-algebras in Section 3.1. This list of generators is rather long, but for the purpose of the trace computations, we may localise KK^{G} at the multiplicatively closed subset of non-zero divisors in R(G). The image of the bootstrap class in this localisation has a very simple structure, which is described in Section 3.2. The homological formula for the Lefschetz index follows easily from this description of the localised bootstrap category. In Section 4, we give a variant of the global homological formula for the trace for a Hodgkin Lie group G. Given a commutative ring R and an R-module M with a projective resolution of finite type, we may define a Hattori–Stallings trace for endomorphisms of M by lifting the endomorphism to a finite type projective resolution and using the standard trace for endomorphisms of finitely generated projective resolutions. This defines the trace of the R(G)-module homomorphism K_{∗}^{G}(φ): K_{∗}^{G}(A) → K_{∗}^{G}(A) in R(G) without passing through a field of fractions.

2 Lefschetz indices in geometric bivariant K-theory

The category k̂k^{G} introduced in [15] provides a geometric analogue of Kasparov theory. We first recall some basic facts about this category and duality in bivariant K-theory from [14–16] and then compute Lefschetz indices in it as intersection products. Later we are going to compare this with other formulas for Lefschetz indices. We also prove an excess intersection formula to compute the composition of geometric correspondences under a weaker assumption than transversality. This formula goes back to Baum and Block [4]. All results in this section extend to the case where G is a proper Lie groupoid with enough G-vector bundles in the sense of [14, Definition 3.1] because the theory in [14–16] is already developed in this generality. For the sake of concreteness, we limit our treatment here to compact Lie groups acting on smooth manifolds. The results in this section work both for real and complex K-theory. For concreteness, we assume in our notation that we are dealing with the complex case. In the real case, K must be replaced by KO throughout. In particular, K_{G}-orientations (that is, G-equivariant Spin^{c}-structures) must be replaced by KO^{G}-orientations (that is, G-equivariant Spin structures). In some examples, we use the isomorphisms k̂k _{2n}^{G} (pt, pt) = R(G) and k̂k _{2n+1}^{G} (pt, pt) = 0 for all n ∈ ℤ. Here R(G) denotes the representation ring of G. Of course, this is true only in complex K-theory.

2.1 Geometric bivariant K-theory

Like Kasparov theory, geometric bivariant K-theory yields a category k̂k^{G}. Its objects are (Hausdorff) locally compact G-spaces; arrows from X to Y are geometric correspondences from X to Y in the sense of [15, Definition 2.3]. These consist of

• M: a G-space;

• b: a G-map (that is, a continuous G-equivariant map) b: M → X;

• ξ: a G-equivariant K-theory class on M with X-compact support (where we view M as a space over X via the map b); we write ξ ∈ RK_{G,X}^{∗} (M);

• f: a K_{G}-oriented normally non-singular G-map f: M → Y.

Equivariant K-theory with X-compact support and equivariant vector bundles are defined in [12, Definitions 2.5 and 2.6]. If b is a proper map, in particular if M is compact, then RK_{G,X}^{∗} (M) is the ordinary G-equivariant (compactly supported) K-theory K_{G}^{∗}(M) of M. A K_{G}-oriented normally non-singular map from M to Y consists of

• V: a K_{G}-oriented G-vector bundle on M,

• E: a K_{G}-oriented finite-dimensional linear G-representation, giving rise to a trivial K_{G}-oriented G-vector bundle Y × E on Y,

• f̂: a G-equivariant homeomorphism from the total space of V to an open subset in the total space of Y × E, f̂: V ↪ Y × E.

We will not distinguish between a vector bundle and its total space in our notation. A normally non-singular map f = (V, E, f̂) has an underlying map

M ↣ V ⟶^{f̂} Y × E ↠ Y,

where the first map is the zero section of the vector bundle V and the third map is the coordinate projection. This map is called its “trace” in [14], but we avoid this name here because we use “trace” in a different sense. The degree of f is d = dimV − dimE. A wrong-way element f_{!} ∈ KK_{d}^{G}(C_{0}(M), C_{0}(Y)) induced by f is defined in [14, Section 5.3]). Our geometric correspondences are variants of those introduced by Alain Connes and Georges Skandalis in [6]. The changes in the definition avoid technical problems with the usual definition in the equivariant case. The (ℤ/2-graded) geometric KK-group k̂k _{∗}^{G}(X, Y) is defined as the quotient of the set of geometric correspondences from X to Y by an appropriate equivalence relation, generated by bordism, Thom modification, and equivalence of normally non-singular maps. Bordism includes homotopies for the maps b and f by [15, Lemma 2.12]. We will use this several times below. The Thom modification allows to replace the space M by the total space of a K_{G}-oriented vector bundle on M. In particular, we could take the K_{G}-oriented vector bundle from the normally non-singular map f. This results in an equivalent normally non-singular map where f: M → Y is a special submersion, that is, an open embedding followed by a coordinate projection Y × E ↠ Y for some linear G-representation E. Correspondences with this property are called special. The composition in k̂k^{G} is defined as an intersection product (see Section 2.2) if the map f: M → Y is such a special submersion. This turns k̂k^{G} into a category; the identity map on X is the correspondence with f = b = id_{X} and ξ = 1. The product of G-spaces provides a symmetric monoidal structure in k̂k^{G} (see [15, Theorem 2.27]). There is an additive, grading-preserving, symmetric monoidal functor

k̂k _{∗}^{G}(X, Y) → KK_{∗}^{G}(C_{0}(X), C_{0}(Y)).

This is an isomorphism if X is normally non-singular by [15, Corollary 4.3], that is, if there is a normally non-singular map X → pt. This means that there is a G-vector bundle V over X whose total space is G-equivariantly homeomorphic to an open G-invariant subset of some linear G-space. In particular, by Mostow’s Embedding Theorem smooth G-manifolds of finite orbit type are normally non-singular (see [14, Theorem 3.22]). Stable K_{G}-orientations play an important technical role in our trace formulas and should therefore be treated with care. A K_{G}-orientation on a G-vector bundle V is, by definition, a G-equivariant complex spinor bundle for V. (This is equivalent to a reduction of the structure group to Spin^{c}.) Given such K_{G}-orientations on V_{1} and V_{2}, we get an induced K_{G}-orientation on V_{1} ⊕ V_{2}; conversely, K_{G}-orientations on V_{1} ⊕ V_{2} and V_{1} induce one on V_{2}. Let ξ ∈ RK_{G}^{0}(M) be represented by the formal difference [V_{1}] − [V_{2}] of two G-vector bundles. A stable K_{G}-orientation on ξ means that we are given another G-vector bundle V_{3} and K_{G}-orientations on both V_{1} ⊕ V_{3} and V_{2} ⊕ V_{3}. Since ξ = [V_{1} ⊕ V_{3}] − [V_{2} ⊕ V_{3}], this implies that ξ is a formal difference of two K_{G}-oriented G-vector bundles. Conversely, assume that ξ = [W_{1}] − [W_{2}] with two K_{G}-oriented G-vector bundles; then there are G-vector bundles V_{3} and W_{3} such that V_{i} ⊕ V_{3} ≅ W_{i} ⊕ W_{3} for i = 1, 2; since W_{3} is a direct summand in a K_{G}-oriented G-vector bundle, we may enlarge V_{3} and W_{3} so that W_{3} itself is K_{G}-oriented. Then V_{i} ⊕ V_{3} ≅ W_{i} ⊕ W_{3} for i = 1, 2 inherit K_{G}-orientations. Roughly speaking, stably K_{G}-oriented K-theory classes are equivalent to formal differences of K_{G}-oriented G-vector bundles. A K_{G}-orientation on a normally non-singular map f = (V, E, f̂) from M to Y means that both V and E are K_{G}-oriented. Since “lifting” allows us to replace E by E ⊕ E′ and V by V ⊕ (M × E′), we may assume without loss of generality that E is already K_{G}-oriented. Thus a K_{G}-orientation on f becomes equivalent to one on V. But the chosen K_{G}-orientation on E remains part of the data: changing it changes the K_{G}-orientation on f. By [14, Lemma 5.13], all essential information is contained in a K_{G}-orientation on the formal difference [V] − [M × E] ∈ RK_{G}^{0}(M), which we call the stable normal bundle of the normally non-singular map f. If [V]−[M ×E] is K_{G}-oriented, then we may find a G-vector bundle V_{3} such that V ⊕V_{3} and (M ×E)⊕V_{3} are K_{G}-oriented. Since (M ×E)⊕V_{3} is a direct summand in a K_{G}-oriented trivial G-vector bundle, we may assume without loss of generality that V_{3} itself is trivial, V_{3} = M ×E′, and that already E ⊕ E′ is K_{G}-oriented. Lifting f along E′ then gives a normally non-singular map (V ⊕ (M × E′), E ⊕ E′, f̂× id_{E′}), where both V ⊕ (M × E′) and E ⊕ E′ are K_{G}-oriented. Thus a K_{G}-orientation on f is equivalent to a stable K_{G}-orientation on the stable normal bundle of f.

Lemma 2.1. If f = (V, E, f̂) is a smooth normally non-singular map with underlying map f̄: M → Y, then its stable normal bundle is equal to f̄^{∗} [TY] − [TM] ∈ RK_{G}^{0}(M).

Proof. The tangent bundles of the total spaces of V and Y ×E are TM ⊕V and TY ⊕ E, respectively. Since f̂ is an open embedding, f̂^{∗} (TY ⊕ E) ≅ TM ⊕ V. This implies f̄^{∗} (TY) ⊕ (M × E) ≅ TM ⊕ V. Thus [V] − [M × E] = f̄^{∗} [TY] − [TM].

This lemma also shows that the stable normal bundle of f and hence the orientability assumption depend only on the equivalence class of f. Another equivalent way to describe stable K_{G}-orientations is the following. Suppose we are already given a G-vector bundle V on Y such that TY ⊕ V is K_{G}-oriented. Then a stable K_{G}-orientation on f is equivalent to one on

[f̄^{∗} V ⊕ TM] = f̄^{∗} [TY ⊕ V] − (f̄^{∗} [TY] − [TM]), which is equivalent to a K_{G}-orientation on f̄^{∗} V ⊕ TM in the usual sense. If X and Y are smooth G-manifolds (without boundary), we may require the maps b and f̂ and the vector bundles V and E to be smooth. This leads to a smooth variant of k̂k^{G}. This variant is isomorphic to the one defined above by [15, Theorem 4.8] provided X is of finite orbit type and hence normally non-singular. Working in the smooth setting has two advantages. First, assuming M to be of finite orbit type, [14, Theorem 3.22] shows that any smooth G-map f: M → Y lifts to a smooth normally non-singular map that is unique up to equivalence. Thus we may replace normally non-singular maps by smooth maps in the usual sense in the definition of a geometric correspondence. Moreover, Nf = f^{∗}[TY] − [TM], so f is K_{G}-oriented if and only if there are K_{G}-oriented G-vector bundles V_{1} and V_{2} over M with f^{∗}[TY] ⊕ V_{1} ≅ TM ⊕ V_{2} (compare [14, Corollary 5.15]). Secondly, in the smooth setting there is a particularly elegant way of composing correspondences when they satisfy a suitable transversality condition, see [15, Corollary 2.39]. This description of the composition is due to Connes and Skandalis [6].

2.2 Composition of geometric correspondences

By [15, Theorem 2.38], a smooth normally non-singular map lifting f: M_{1} → Y and a smooth map b: M_{2} → Y are transverse if

D_{m_{1}} f(T_{m_{1}} M_{1}) + D_{m_{2}} b(T_{m_{2}} M_{2}) = T_{y}Y

for all m_{1} ∈ M_{1}, m_{2} ∈ M_{2} with y := f(m_{1}) = b(m_{2}). Equivalently, the map

Df − Db: pr_{1}^{∗}(TM_{1}) ⊕ pr_{2}^{∗}(TM_{2}) → (f ∘ pr_{1})^{∗}(TY)

is surjective; this is a bundle map of vector bundles over

M_{1} ×_{Y} M_{2} := {(m_{1}, m_{2}) | f(m_{1}) = b(m_{2})},

where pr_{1} : M_{1} ×_{Y} M_{2} → M_{1} and pr_{2} : M_{1} ×_{Y} M_{2} → M_{2} denote the restrictions to M_{1} ×_{Y} M_{2} of the coordinate projections. (We shall always use this notation for restrictions of coordinate projections.) A commuting square diagram of smooth manifolds is called Cartesian if it is isomorphic (as a diagram) to a square

\begin{matrix}M_{1} ×_{Y} M_{2} & M_{2} \\ M_{1} & Y \end{matrix} pr_{2} pr_{1} f b

where f and b are transverse smooth maps in the sense above; then M_{1} ×_{Y} M_{2} is again a smooth manifold and pr_{1} and pr_{2} are smooth maps.

The tangent bundles of these four manifolds are related by an exact sequence

0 → T(M_{1} ×_{Y} M_{2}) −−−−−−⟶^{(Dpr_{1} ,Dpr_{2} )} pr_{1}^{∗}(TM_{1}) ⊕ pr_{2}^{∗}(TM_{2})

−−−⟶^{Df−Db} (f ∘ pr_{1})^{∗}TY → 0. (2.1)

That is, T(M_{1} ×_{Y} M_{2}) is the sub-bundle of pr_{1}^{∗}(TM_{1}) ⊕ pr_{2}^{∗}(TM_{2}) consisting of those vectors (m_{1}, ξ, m_{2}, η) ∈ TM_{1} ⊕ TM_{2} (where f(m_{1}) = b(m_{2})) with D_{m_{1}} f(ξ) = D_{m_{2}} b(η). We may denote this bundle briefly by TM_{1} ⊕_{TY} TM_{2}. Furthermore, from (2.1),

T(M_{1} ×_{Y} M_{2}) − pr_{2}^{∗}(TM_{2}) = pr_{1}^{∗}(TM_{1} − f^{∗}(TY))  (2.2)

as stable G-vector bundles. Thus a stable K_{G}-orientation for TM_{1} − f^{∗}(TY) may be pulled back to one for T(M_{1} ×_{Y} M_{2}) − pr_{2}^{∗}(TM_{2}). More succinctly, a K_{G}-orientation for the map f induces one for pr_{2}. Now consider two composable smooth correspondences

\begin{aligned}& M_{1} & & M_{2} \\ X & & Y & & Z, \end{aligned} ; b_{1} ; f_{1} ; b_{2} ; f_{2} (2.3)

with K-theory classes ξ_{1} ∈ RK_{∗,X}^{G} (M_{1}) and ξ_{2} ∈ RK_{∗,Y}^{G} (M_{2}). We assume that the pair of smooth maps (f_{1}, b_{2}) is transverse. Then there is an essentially unique commuting diagram

\begin{aligned}& & M_{1} ×_{Y} M_{2} \\ & M_{1} & & M_{2} \\ X & & Y & & Z, \end{aligned} ; pr_{1} ; pr_{2} ; b_{1} ; f_{1} ; b_{2} ; f_{2} (2.4)

where the square is Cartesian. We briefly call such a diagram an intersection diagram for the two given correspondences. By the discussion above, the map pr_{2} inherits a K_{G}-orientation from f_{1}, so that the map f := f_{2} ∘ pr_{2} is also K_{G}-oriented. Let M := M_{1} ×_{Y} M_{2} and b := b_{1} ∘ pr_{1}. The product ξ := pr_{1}^{∗}(ξ_{1}) ⊗ pr_{2}^{∗}(ξ_{2}) belongs to RK_{∗,X}^{G} (M), that is, it has X-compact support with respect to the map b: M → X. Thus we get a G-equivariant correspondence (M, b, f, ξ) from X to Y. The assertion of [15, Corollary 2.39] – following [6] – is that this represents the composition of the two given correspondences. It is called their intersection product.

Example 2.2. Consider the diagonal embedding δ: X → X × X and the graph embedding f̄: X → X × X, x ↦ (x, f(x)), for a smooth map f: X → X. These two maps are transverse if and only if f has simple fixed points. If this is the case, then the intersection space is the set of fixed points of f. If, say, f = id_{X}, then δ and f̄ are not transverse. To define the composition also in the non-transverse case, a Thom modification is used in [15] to achieve transversality (see [15, Theorem 2.32]). Take two composable (smooth) correspondences as in (2.3), and let f_{1} = (V_{1}, E_{1}, f̂_{1} ) as a normally non-singular map. By a Thom modification, the geometric correspondence X ⟵^{b_{1}} (M_{1}, ξ) ⟶^{f_{1}} Y is equivalent to

X ⟵−−−^{b_{1} ∘π_{V_{1}}} (V_{1}, τ_{V_{1}} ⊗ π_{V_{1}}^{∗}ξ) −−−⟶^{π_{E_{1}} ∘f̂_{1}} Y,  (2.5)

where π_{V_{1}} : V_{1} → M_{1} and π_{E_{1}} : Y × E → Y are the bundle projections, and τ_{V_{1}} ∈ RK_{G,M_{1}}^{∗} (V_{1}) is the Thom class of V_{1}. We write ⊗ for the multiplication of K-theory classes. The support of such a product is the intersection of the supports of the factors. Hence the support of τ_{V_{1}} ⊗π_{V_{1}}^{∗}ξ is an X-compact subset of V_{1}. The forward map V_{1} → Y in (2.5) is a special submersion and, in particular, a submersion. As such it is transverse to any other map b_{2} : M_{2} → Y. Hence after the Thom modification we may compute the composition of correspondences as an intersection product of the correspondence (2.5) with the correspondence Y ⟵^{b_{2}} M_{2} ⟶^{f_{2}} Y. This yields

X ⟵−−−−−−^{b_{1} ∘π_{V_{1}} ∘pr_{1}} (V_{1} ×_{Y} M_{2}, pr_{V_{1}}^{∗}(τ_{V_{1}} ⊗ π_{V_{1}}^{∗}(ξ))) −−−−→^{f_{2} ∘pr_{2}} Z,  (2.6)

where

V_{1} ×_{Y} M_{2} := {(x, v, m_{2}) ∈ V_{1} × M_{2} | (π_{E_{1}} ∘ f̂_{1} )(x, v) = b_{2}(m_{2})}

and pr_{1} : V_{1} ×_{Y} M_{2} → V_{1} and pr_{2} : V_{1} ×_{Y} M_{2} → M_{2} are the coordinate projections. The intersection space V_{1} ×_{Y} M_{2} is a smooth manifold with tangent bundle

TV_{1} ⊕_{TY} TM_{2} := pr_{1}^{∗}(TV_{1}) ⊕_{(π_{E_{1}} ∘f̂_{1} )^{∗} (TY)} pr_{2}^{∗}(TM_{2}),

and the map pr_{2} is a submersion with fibres tangent to E_{1}. Thus it is K_{G}-oriented. This recipe to define the composition product for all geometric correspondences is introduced in [15]. It is shown there that it is equivalent to the intersection product if f_{1} and b_{2} are transverse. But the space V_{1} ×_{Y} M_{2} has high dimension, making it inefficient to compute with this formula. And we are usually given only the underlying map f_{1} : M_{1} → Y, not its factorisation as a normally non-singular map – and the latter is difficult to compute. We will weaken the transversality requirement in Section 2.5. The more general condition still applies, say, if f_{1} = b_{2}. This is particularly useful for computing Euler characteristics.

2.3 Duality and the Lefschetz index

Duality plays a crucial role in [15] in order to compare the geometric and analytic models of equivariant Kasparov theory. Duality is also used in [16, Definition 4.26] to construct a Lefschetz map

L: KK_{∗}^{G}(C(X), C(X)) → KK_{∗}^{G}(C(X), ℂ),  (2.7)

for a compact smooth G-manifold X. We may compose L with the index map KK_{∗}^{G}(C(X), ℂ) → KK_{∗}^{G}(ℂ, ℂ) ≅ R(G) to get a Lefschetz index L-ind(f) ∈ R(G) for any f ∈ KK_{∗}^{G}(C(X), C(X)). This is the invariant we will be studying in this paper. This Lefschetz map L is a special case of a very general construction. Let C be a symmetric monoidal category. Let A be a dualisable object of C with a dual A^{∗}. Let η: 𝟙 → A ⊗ A^{∗} and ε: A^{∗} ⊗ A → 𝟙 be the unit and counit of the duality. Being unit and counit of a duality means that they satisfy the zigzag equations: the composition

A −−⟶^{η⊗id_{A}} A ⊗ A^{∗} ⊗ A −−⟶^{id_{A} ⊗ε} A  (2.8)

is equal to the identity id_{A}: A → A, and similarly for the composition

A^{∗} −−−⟶^{id_{A^{∗}} ⊗η} A^{∗} ⊗ A ⊗ A^{∗} −−−⟶^{ε⊗id_{A^{∗}}} A^{∗}.  (2.9)

If C is ℤ-graded, then we may allow dualities to shift degrees. Then some signs are necessary in the zigzag equations, see [16, Theorem 5.5]. Given a multiplication map m: A ⊗ A → A, we define the Lefschetz map

L: C(A, A) → C(A, 𝟙)

by sending an endomorphism f: A → A to the composite morphism

A ≅ A⊗𝟙 −−⟶^{id_{A} ⊗η} A⊗A⊗A^{∗} −−−−−→^{m⊗id_{A^{∗}}} A⊗A^{∗} −−−⟶^{f⊗id_{A^{∗}}} A⊗A^{∗} −−→^{sym} A^{∗}⊗A ⟶^{ε} 𝟙.

This depends only on m and f, not on the choices of the dual, unit and counit. For f = id_{A} we get the higher Euler characteristic of A in C(A, 𝟙). While the geometric computations below give the Lefschetz map as defined above, the global homological computations in Sections 3 and 4 only apply to the following coarser invariant:

Definition 2.3. The Lefschetz index L-ind(f) (or trace tr(f)) of an endomorphism f: A → A is the composite

𝟙 ⟶^{η} A ⊗ A^{∗} −−→^{sym} A^{∗} ⊗ A −−−⟶^{id_{A^{∗}} ⊗f} A^{∗} ⊗ A ⟶^{ε} 𝟙,  (2.10)

where sym denotes the symmetry isomorphism. The Lefschetz index of id_{A} is called the Euler characteristic of A.

If A is a unital algebra object in C with multiplication m: A ⊗ A → A and unit u: 𝟙 → A, then L-ind(f) = L(f) ∘ u. In particular, the Euler characteristic is the composite of the higher Euler characteristic with u. In this section, we work in C = k̂k^{G} for a compact group G with 𝟙 = pt and ⊗ = ×. In Section 3, we work in the related analytic category C = KK^{G} with 𝟙 = ℂ and the usual tensor product. We will show below that any compact smooth G-manifold X is dualisable in k̂k^{G}. The multiplication m: X × X → X and unit u: pt → X are given by the geometric correspondences

X × X ⟵^{Δ} X −−→_{=}^{id_{X}} X, pt ← X −−→_{=}^{id_{X}} X

with Δ(x) = (x, x); these induce the multiplication ^{∗}-homomorphism

m: C(X × X) ≅ C(X) ⊗ C(X) → C(X)

and the embedding ℂ → C(X) of constant functions. Composing with u corresponds to taking the index of a K-homology class.

Remark 2.4. In [11, 13, 16] Lefschetz maps are also studied for non-compact spaces X, equipped with group actions of possibly non-compact groups. A non-compact G-manifold X is usually not dualisable in k̂k^{G}, and even if it were, the Lefschetz map that we would get from this duality would not be the one studied in [11, 13, 16].

2.4 Duality for smooth compact manifolds

We are going to show that compact smooth G-manifolds are dualisable in the equivariant correspondence theory k̂k^{G}. This was already proved in [15], but since we need to know the unit and counit to compute Lefschetz indices, we recall the proof in detail. It is of some interest to treat duality for smooth manifolds with boundary because any finite CW-complex is homotopy equivalent to a manifold with boundary. In case X has a boundary ∂X, let X̊ := X \ ∂X denote its interior and let ι: X̊ → X denote the inclusion map. The boundary ∂X ⊆ X admits a G-equivariant collar, that is, the embedding ∂X → X extends to a G-equivariant diffeomorphism from ∂X × [0, 1) onto an open neighbourhood of ∂X in X (see also [16, Lemma 7.6] for this standard result). This collar neighbourhood together with a smooth map [0, 1) → (0, 1) that is the identity near 1 provides a smooth G-equivariant map ρ: X → X̊ that is inverse to ι up to smooth G-homotopy. Furthermore, we may assume that ρ is a diffeomorphism onto its image. If X has no boundary, then X̊ = X, ι = id, and ρ = id. The results about smooth normally non-singular maps in [14] extend to smooth manifolds with boundary if we add suitable assumptions about the behaviour near the boundary. We mention one result of this type and a counterexample.

Proposition 2.5. Let X and Y be smooth G-manifolds with X of finite orbit type and let f: X → Y be a smooth map with f(∂X) ⊆ ∂Y and f transverse to ∂Y. Then f lifts to a normally non-singular map, and any two such normally non-singular liftings of f are equivalent.

Proof. Since X has finite orbit type, we may smoothly embed X into a finitedimensional linear G-representation E. Our assumptions ensure that the resulting map X → Y × E is a smooth embedding between G-manifolds with boundary in the sense of [14, Definition 3.17] and hence has a tubular neighbourhood by [14, Theorem 3.18]. This provides a normally non-singular map X → Y lifting f. The uniqueness up to equivalence is proved as in the proof of [14, Theorem 4.36].

Example 2.6. The inclusion map {0} → [0, 1) is a smooth map between manifolds with boundary, but it does not lift to a smooth normally non-singular map.

Let X be a smooth compact G-manifold. Since X has finite orbit type, it embeds into some linear G-representation E. We may choose this G-representation to be K_{G}-oriented and even-dimensional by a further stabilisation. Let NX ↠ X be the normal bundle for such an embedding X → E. Thus TX ⊕ NX ≅ X × E is G-equivariantly isomorphic to a K_{G}-oriented trivial G-vector bundle.

Theorem 2.7. Let X be a smooth compact G-manifold, possibly with boundary. Then X is dualisable in k̂k _{∗}^{G} with dual NX̊, and the unit and counit for the duality are the geometric correspondences

pt ← X −−−−→^{(id,ζρ)} X × NX̊, NX̊ × X ⟵−−−^{(id,ιπ)} NX̊ → pt,

where ζ: X̊ → NX̊ is the zero section, ρ: X → X̊ is some G-equivariant collar retraction, π: NX̊ → X̊ is the bundle projection, and ι: X̊ → X the identical inclusion. The K-theory classes on the space in the middle are the trivial rankone vector bundles for both correspondences.

Proof. First we must check that the purported unit and counit above are indeed geometric correspondences; this contains describing the K_{G}-orientations on the forward maps, which is part of the data of the geometric correspondences. The maps X → pt and NX̊ → NX̊ × X above are proper. Hence there is no support restriction for the K-theory class on the middle space, and the trivial rank-one vector bundle is allowed. By the Tubular Neighbourhood Theorem, the normal bundle NX̊ of the embedding X̊ → E is diffeomorphic to an open subset of E. This gives a canonical isomorphism between the tangent bundle of NX̊ and E. We choose this isomorphism and the given K_{G}-orientation on the linear G-representation E to K_{G}-orient NX̊ and thus the projection NX̊ → pt. With this K_{G}-orientation, the counit NX̊ × X ⟵−−−^{(id,ιπ)} NX̊ → pt is a G-equivariant geometric correspondence – even a special one in the sense of [15].

We identify the tangent bundle of X × NX̊ with TX × TX̊ ⊕ NX̊ in the obvious way. The normal bundle of the embedding (id, ζρ): X → X × NX̊ is isomorphic to the quotient of TX ⊕ ρ^{∗}(TX̊) ⊕ ρ^{∗}(NX̊) by the relation (ξ, Dρ(ξ), 0) ∼ 0 for ξ ∈ TX. We identify this with TX ⊕ NX ≅ X × E by (ξ_{1}, ξ_{2}, η) ↦ (Dρ^{−1} (ξ_{2}) − ξ_{1}, Dρ^{−1} (η)) for ξ_{1} ∈ T_{x}X, ξ_{2} ∈ T_{ρ(x)} X, η ∈ ρ^{∗}(NX̊)_{x} = N_{ρ(x)} X. With this K_{G}-orientation on (id, ζρ), the unit above is a G-equivariant geometric correspondence. A boundary of X, if present, causes no problems here. The same goes for the computations below: although the results in [15] are formulated for smooth manifolds without boundary, they continue to hold in the cases we need.

We establish the duality isomorphism by checking the zigzag equations as in [16, Theorem 5.5]. This amounts to composing geometric correspondences. In the case at hand, the correspondences we want to compose are transverse, so that they may be composed by intersections as in Section 2.2. Actually, we are dealing with manifolds with boundary, but the argument goes through nevertheless. We write down the diagrams together with the relevant Cartesian square.

The intersection diagram for the first zigzag equation is

X

X × X  X × NX̊

X  X × NX̊ × X  X.

(id, ιρ)

(id, ζρ)

pr_{2}

(id, ζρ) × id

id × (id, ιπ)

pr_{1}

(2.11)

The square is Cartesian because (x, y, z, (w, ν)) ∈ X^{3} × NX̊ satisfies

(x, (ρ(x), 0), y) = (z, (w, ν), w)

if and only if y = ρ(x), z = x, w = ρ(x), and ν = 0 for some x ∈ X. The K_{G}-orientation on the map (id, ζρ) described above is chosen such that the composite map f := pr_{1} ∘ (id, ζρ) = id carries the standard K_{G}-orientation. The map b := pr_{2} ∘ (id, ιρ) = ιρ is properly homotopic to the identity map. Hence the composition above gives the identity map on X as required.

The intersection diagram for the second zigzag equation is

NX̊

NX̊ × X  NX̊ × NX̊

NX̊  NX̊ × X × NX̊  NX̊

(id, ιπ)

(id, ζρπ)

pr_{1}

id × (id, ζρ)

(id, ιπ) × id

pr_{2}

(2.12)

because ((x, ν), y, (w, μ), (z, κ)) ∈ NX̊ × X × (NX̊)^{2} satisfy

((x, ν), y, (ρ(y), 0)) = ((w, μ), w, (z, κ))

if and only if (w, μ) = (x, ν), y = x, z = ρ(x), κ = 0 for some (x, ν) ∈ NX̊. The map (id, ζρπ) is smoothly homotopic to the diagonal embedding δ: NX̊ → NX̊ × NX̊. Replacing (id, ζρπ) by δ gives an equivalent geometric correspondence. The K_{G}-orientation on the normal bundle of (id, ζρπ) that comes with the composition product is transformed by this homotopy to the K_{G}-orientation on the normal bundle of the diagonal embedding that we get by identifying the latter with the pull-back of E by mapping

(ξ_{1}, η_{1}, ξ_{2}, η_{2}) ∈ T_{(x,ζ,x,ζ)} (NX̊ × NX̊) ≅ T_{x} X̊ ⊕ N_{x} X̊ × T_{x} X̊ × N_{x} X̊ ≅ E_{x} × E_{x}

to (ξ_{2} − ξ_{1}, η_{2} − η_{1}) ∈ E_{x}. Since E has even dimension, changing this to (ξ_{1} − ξ_{2}, η_{1} − η_{2}) does not change the K_{G}-orientation. Hence the induced K_{G}-orientation on the fibres of Dpr_{2} is the same one that we used to K_{G}-orient pr_{2}. The induced K_{G}-orientation on pr_{2} ∘ δ = id is the standard one. Thus the composition in (2.12) is the identity on NX̊.

Corollary 2.8. Let X be a compact smooth G-manifold and let Y be any locally compact G-space. Then every element of k̂k _{∗}^{G}(X, Y) is represented by a geometric correspondence of the form

X ⟵−−−^{ι∘π∘pr_{1}} NX̊ × Y ⟶^{pr_{2}} Y, ξ ∈ K_{G}^{∗}(NX̊ × Y),

and two such correspondences for ξ_{1}, ξ_{2} ∈ K_{G}^{∗}(NX̊ × Y) give the same element of k̂k _{∗}^{G}(X, Y) if and only if ξ_{1} = ξ_{2}. Here pr_{1} : NX̊ × Y → NX̊ and pr_{2} : NX̊ × Y → Y are the coordinate projections and ι ∘ π: NX̊ → X̊ ⊆ X is as above.

Proof. Duality provides a canonical isomorphism

K_{G}^{∗}(NX̊ × Y) ≅ k̂k _{∗}^{G}(pt, NX̊ × Y) ≅ k̂k _{∗}^{G}(X, Y).

It maps ξ ∈ K_{G}^{∗}(NX̊ × Y) to the composition of correspondences described by the following intersection diagram:

NX̊ × Y

X × NX̊ × Y  NX̊ × Y

X  X × NX̊ × Y  Y,

(ιπ, id) × id id

pr_{1}

id (ιπ, id) × id

pr_{2}

with the K-theory class ξ on NX̊×Y. Hence it involves the maps ιπ: NX̊×Y → X and pr_{2} : NX̊ × Y → Y.

If X is, in addition, K_{G}-oriented, then the Thom isomorphism provides an isomorphism NX̊ ≅ X̊ in k̂k _{∗}^{G} (which has odd parity if the dimension of X is odd). A variant of Corollary 2.8 yields a duality isomorphism

K_{G}^{∗+dim (X)} (X̊ × Y) ≅ k̂k _{∗}^{G}(X, Y),

which maps ξ ∈ K_{G}^{∗}(X̊ × Y) to the geometric correspondence

X ⟵−−^{ι∘pr_{1}} X̊ × Y ⟶^{pr_{2}} Y, ξ ∈ K_{G}^{∗}(X̊ × Y).

Hence any element of k̂k _{∗}^{G}(X, Y) is represented by a correspondence of this form. If X is K_{G}-oriented and has no boundary, this becomes

X ⟵^{pr_{1}} X × Y ⟶^{pr_{2}} Y, ξ ∈ K_{G}^{∗}(X × Y).

These standard forms for correspondences are less useful than one may hope at first because their intersection products are no longer in this standard form.

2.5 More on composition of geometric correspondences

With our geometric formulas for the unit and counit of the duality, we could now compute Lefschetz indices geometrically, assuming the necessary intersections are transverse. While this works well, say, for self-maps with regular nondegenerate fixed points, it fails badly for the identity correspondence, whose Lefschetz index is the Euler characteristic. Building on work of Baum and Block [4], we now describe the composition as a modified intersection product under a much weaker assumption than transversality that still covers the computation of Euler characteristics.

Definition 2.9. We say that the smooth maps f_{1} : M_{1} → Y and b_{2} : M_{2} → Y intersect smoothly if

M := M_{1} ×_{Y} M_{2}

is a smooth submanifold of M_{1} × M_{2} and any (ξ_{1}, ξ_{2}) ∈ TM_{1} × TM_{2} with Df_{1}(ξ_{1}) = Db_{2}(ξ_{2}) ∈ TY is tangent to M. If f_{1} and b_{2} intersect smoothly, then we define the excess intersection bundle η(f_{1}, b_{2}) on M as the cokernel of the vector bundle map

(Df_{1}, −Db_{2}): pr_{1}^{∗}(TM_{1}) ⊕ pr_{2}^{∗}(TM_{2}) → f^{∗}(TY),  (2.13)

where f := f_{1} ∘ pr_{1} = b_{2} ∘ pr_{2} : M → Y. If the maps f_{1} and b_{2} are G-equivariant with respect to a compact group G, then the excess intersection bundle is a G-vector bundle. We call the square

\begin{matrix}M & M_{2} \\ M_{1} & Y \end{matrix} pr_{2} pr_{1} f_{1} b_{2}

η-Cartesian if f_{1} and b_{2} intersect smoothly with excess intersection bundle η.

If M is a smooth submanifold of M_{1} × M_{2}, then TM ⊆ T(M_{1} × M_{2}); and if (ξ_{1}, ξ_{2}) ∈ T(M_{1} × M_{2}) is tangent to M, then Df_{1}(ξ_{1}) = Db_{2}(ξ_{2}) in TY. These pairs (ξ_{1}, ξ_{2}) form a subspace of T(M_{1} × M_{2})|_{M} = pr_{1}^{∗}TM_{1} ⊕ pr_{2}^{∗}TM_{2}, which in general need not be a vector bundle, that is, its rank need not be locally constant. The smooth intersection assumption forces it to be a subbundle: the kernel of the map in (2.13). Hence the excess intersection bundle is a vector bundle over M, and there is the following exact sequence of vector bundles over M:

0 → TM → pr_{1}^{∗}(TM_{1}) ⊕ pr_{2}^{∗}(TM_{2}) −−−−−−⟶^{(Df_{1} ,−Db_{2} )} (f_{1} ∘ pr_{1})^{∗}(TY) → η → 0.

(2.14)

Example 2.10. Let M_{1} = M_{2} = X and let f_{1} = b_{2} = i: X → Y be an injective immersion. Then M_{1} ×_{Y} M_{2} ≅ X is the diagonal in M_{1} × M_{2} = X^{2}, which is a smooth submanifold. Furthermore, if (ξ_{1}, ξ_{2}) ∈ TM_{1} × TM_{2} satisfy Df_{1}(ξ_{1}) = Db_{2}(ξ_{2}), then ξ_{1} = ξ_{2} because Di: TM → TY is assumed injective. Hence M_{1} and M_{2} intersect smoothly, and the excess intersection bundle is the normal bundle of the immersion i. Example 2.11. Let M_{1} and M_{2} be two circles embedded in Y = ℝ^{2}. The four possible configurations are illustrated in Figure 1.

1. The circles meet in two points. Then M = {p_{1}, p_{2}} and the intersection is transverse.

2. The two circles are disjoint. Then M = ∅ and the intersection is transverse.

3. The two circles are identical. Then M = M_{1} = M_{2}. The intersection is not transverse, but smooth by Example 2.10; the excess intersection bundle is the normal bundle of the circle, which is trivial of rank 1.

\begin{matrix}p_{1} \\ p_{2} \end{matrix} ; M_{1} = M_{2} ; p

Figure 1: Four possible configurations of two circles in the plane

4. The two circles touch in one point. Then M := M_{1} ×_{Y} M_{2} = {p}, so that the tangent bundle of M is zero-dimensional. But T_{p}M_{1} ∩ T_{p}M_{2} is one-dimensional because T_{p}M_{1} = T_{p}M_{2}. Hence the embeddings do not intersect smoothly.

Remark 2.12. The maps f: M_{1} → Y and b: M_{2} → Y intersect smoothly if and only if f × b: M_{1} × M_{2} → Y × Y and the diagonal embedding Y → Y × Y intersect smoothly; both pairs of maps have the same excess intersection bundle. Thus we may always normalise intersections to the case where one map is a diagonal embedding and thus an embedding.

Example 2.13. Let η be a K_{G}-oriented vector bundle over X. Let M_{1} = M_{2} = X, Y = η, and let f_{1} = b_{2} = ζ: Y → η be the zero section of η. This is a special case of Example 2.10. The maps f_{1} and b_{2} intersect smoothly with excess intersection bundle η. In this example it is easy to compose the geometric correspondences X = X → η and η ← X = X. A Thom modification of the first one along the K_{G}-oriented vector bundle η gives the special correspondence

X ← (η, τ_{η}) = η,

where τ_{η} ∈ RK_{G,X}^{∗} (η) is the Thom class of η. The intersection product of this

with η ← X = X is X = (X, ζ^{∗}(τ_{η})) = X, that is, it is the class in k̂k _{∗}^{G}(X, X) of ζ^{∗}(τ_{η}) ∈ RK_{G}^{∗}(X). This K-theory class is the restriction of τ_{η} to the zero section of η. By the construction of the Thom class, it is the K-theory class of the spinor bundle of η.

Definition 2.14. Let η be a K_{G}-oriented G-vector bundle over a G-space X. Let ζ: X → η be the zero section and let τ_{η} ∈ RK_{G,X}^{∗} (η) be the Thom class. The Euler class of η is ζ^{∗}(τ_{η}), the restriction of τ_{η} to the zero section.

By definition, the Euler class is the composition of the correspondences pt ← X → η and η ← X = X involving the zero section ζ: X → η in both cases.

Example 2.15. Assume that there is a G-equivariant section s: X → η of η with isolated simple zeros; that is, s and ζ are transverse. The linear homotopy connects s to the zero section and hence gives an equivalent correspondence η ⟵^{s} X = X. Since s and ζ are transverse by assumption, the composition is X ← Z → X, where Z is the zero set of s and the maps Z → X are the inclusion map, suitably K_{G}-oriented.

Example 2.16. Let M_{1} = S^{1}, M_{2} = S^{2}, Y = ℝ^{3}, b_{2} : M_{2} → ℝ^{3} be the standard embedding of the 2-sphere in ℝ^{3}, and let f_{1} : M_{1} → M_{2} → ℝ^{3} be the embedding corresponding to the equator of the circle. Then M_{1}×_{Y}M_{2} = M_{1}×_{M_{2}} M_{2} = M_{1}, embedded diagonally into M_{1} × M_{1} ⊂ M_{1} × M_{2}. This is a case of smooth intersection. The excess intersection bundle is the restriction to the equator of the normal bundle of the embedding b_{2}. This is isomorphic to the rank-one trivial bundle on S^{2}. Hence the Euler class e(η) is zero in this case.

Theorem 2.17. Let

X ⟵^{b_{1}} (M_{1}, ξ_{1}) ⟶^{f_{1}} Y ⟵^{b_{2}} (M_{2}, ξ_{2}) ⟶^{f_{2}} Z  (2.15)

be a pair of G-equivariant correspondences as in (2.3). Assume that b_{2} and f_{1} intersect smoothly and with a K_{G}-oriented excess intersection bundle η. Then the composition of (2.15) is represented by the G-equivariant correspondence

X ⟵−−^{b_{1} ∘pr_{1}} (M_{1} ×_{Y} M_{2}, e(η) ⊗ pr_{1}^{∗}(ξ_{1}) ⊗ pr_{2}^{∗}(ξ_{2})) −−−−→^{f_{2} ∘pr_{2}} Z,  (2.16)

where e(η) is the Euler class and the projection pr_{2} : M_{1} ×_{Y} M_{2} → M_{2} carries the K_{G}-orientation induced by the K_{G}-orientations on f_{1} and η (explained below).

In the above situation of smooth intersection, we call the diagram (2.4) an η-intersection diagram. It still computes the composition, but we need the Euler class of the excess intersection bundle η to compensate the lack of transversality. We describe the canonical K_{G}-orientation of pr_{2} : M_{1} ×_{Y} M_{2} → M_{2}. The excess intersection bundle η is defined so as to give an exact sequence of vector bundles (2.14). From this it follows that

[η] = (f_{1} ∘ pr_{1})^{∗}[TY] + TM − pr_{1}^{∗}[TM_{1}] − pr_{2}^{∗}[TM_{2}].

On the other hand, the stable normal bundle Npr_{2} of pr_{2} is equal to pr_{2}^{∗}[TM_{2}]− [TM]. Hence

[η] = pr_{1}^{∗}(f_{1}^{∗}[TY] − [TM_{1}]) − Npr_{2}.

A K_{G}-orientation on f_{1} means a stable K_{G}-orientation on Nf_{1} = f_{1}^{∗}[TY] − [TM_{1}]. If such an orientation is given, it pulls back to one on pr_{1}^{∗}(f_{1}^{∗}[TY] − [TM_{1}]), and then (stable) K_{G}-orientations on [η] and on Npr_{2} are in 1-to-1-correspondence. In particular, a K_{G}-orientation on the bundle η induces one on the normal bundle of pr_{2}. This induced K_{G}-orientation on pr_{2} is used in (2.16). ([14, Lemma 5.13] justifies working with K_{G}-orientations on stable normal bundles.)

Proof of Theorem 2.17. Lift f_{1} to a G-equivariant smooth normally nonsingular map (V_{1}, E_{1}, f̂_{1} ). The composition of (2.15) is defined in [15, Section 2.5] as the intersection product

X ⟵−−−−−−^{b_{1} ∘π_{V_{1}} ∘pr_{V_{1}}} V_{1} ×_{Y} M_{2} −−−−→^{f_{2} ∘pr_{2}} Z  (2.17)

with K-theory datum pr_{V_{1}}^{∗}(τ_{V_{1}}) ⊗ π_{V_{1}}^{∗}(ξ_{1}) ⊗ pr_{2}^{∗}(ξ_{2}) ∈ RK_{G,X}^{∗} (V_{1} ×_{Y} M_{2}). We define the manifold V_{1}×_{Y} M_{2} using the (transverse) maps π_{E_{1}} ∘f̂_{1} : V_{1} → Y and b_{2} : M_{2} → Y. We must compare this with the correspondence in the statement of the theorem. We have a commuting square of embeddings of smooth manifolds

\begin{matrix}M_{1} ×_{Y} M_{2} & M_{1} × M_{2} \\ V_{1} ×_{Y} M_{2} & V_{1} × M_{2} \end{matrix} ; ι_{0} ; ζ_{0} ; ι_{1} ; ζ_{1} (2.18)

where the vertical maps are induced by the zero section M_{1} → V_{1} and the horizontal ones are the obvious inclusion maps. The map ζ_{0} is a smooth embedding because the other three maps in the square are so. Let Nι_{0} and ν := Nζ_{0} denote the normal bundles of the maps ι_{0} and ζ_{0} in (2.18). The normal bundle of ι_{1} is isomorphic to the pull-back of TY because V_{1} → Y is submersive. Since M_{1} × M_{2} → V_{1} × M_{2} is the zero section of the pull back of the vector bundle V_{1} to M_{1} × M_{2}, the normal bundle of ζ_{1} is isomorphic to pr_{1}^{∗}(V_{1}). Recall that M := M_{1} ×_{Y} M_{2}. We get a diagram of vector bundles over M:

0  0  0

0  TM  T(M_{1} × M_{2})|_{M}  Nι_{0}  0

0  T(V_{1} ×_{Y} M_{2})|_{M}  T(V_{1} × M_{2})|_{M}  f^{∗}(TY) 0

0  ν  pr_{1}^{∗}(V_{1})  η  0

0  0  0 Dζ_{0} Dζ_{1} Dι_{0} Dι_{1}

The first two rows and the first two columns are exact by definition or by our description of the normal bundles of ζ_{1} and ι_{1}. The third row is exact with the excess intersection bundle η by (2.14). Hence the dotted arrow exists and makes the third row exact. Since extensions of G-vector bundles always split, we get

ν ⊕ η ≅ pr_{1}^{∗}(V_{1}).

Since η and V_{1} are K_{G}-oriented, the bundle ν inherits a K_{G}-orientation. We apply Thom modification with the K_{G}-oriented G-vector bundle ν to the correspondence in (2.16). This gives the geometric correspondence

X ⟵−−−−−^{b_{1} ∘pr_{1} ∘π_{ν}} ν −−−−−−→^{f_{2} ∘pr_{2} ∘π_{ν}} Z  (2.19)

with K-theory datum

ξ := τ_{ν} ⊗ π_{ν}^{∗}(e(η) ⊗ pr_{1}^{∗}(ξ_{1}) ⊗ pr_{2}^{∗}(ξ_{2})) ∈ RK_{G,X}^{∗} (ν).

The Tubular Neighbourhood Theorem gives a G-equivariant open embedding ζ̂_{0}: ν → V_{1} ×_{Y} M_{2} onto some G-invariant open neighbourhood of M (see [14, Theorem 3.18]). We may find an open G-invariant neighbourhood U of the zero section in V_{1} such that U ×_{Y} M_{2} ⊆ V_{1} ×_{Y} M_{2} is contained in the image of ζ̂_{0} and relatively M-compact. We may choose the Thom class τ_{V_{1}} ∈ K_{G}^{dim V_{1}}(V_{1}) to be supported in U. Hence we may assume that pr_{1}^{∗}(τ_{V_{1}}), the pull-back of τ_{V_{1}} along the coordinate projection pr_{1}^{∗}: V_{1} ×_{Y} M_{2} → V_{1}, is supported inside a relatively M-compact subset of ζ̂_{0}(ν). Then [15, Example 2.14] provides a bordism between the cycle in (2.17) and

X ⟵−−−−−−−−^{b_{1} ∘π_{V_{1}} ∘pr_{1} ∘ζ̂_{0}} ν −−−−−−→^{f_{2} ∘pr_{2} ∘ζ̂_{0}} Z,  (2.20)

with K-theory class pr_{1}^{∗}(τ_{V_{1}}) ⊗ ζ̂_{0}^{∗}pr_{1}^{∗}π_{V_{1}}^{∗}(ξ_{1}) ⊗ ζ̂_{0}^{∗}pr_{2}^{∗}(ξ_{2}). Let s_{t}: ν → ν be the scalar multiplication by t ∈ [0, 1]. Composition with s_{t} is a G-equivariant homotopy

π_{V_{1}}pr_{1} ζ̂_{0} ∼ pr_{1}π_{ν} : ν → M_{1}, pr_{2} ζ̂_{0} ∼ pr_{2}π_{ν} : ν → M_{2}.

Hence s_{t}^{∗}(ζ̂_{0}^{∗}pr_{1}^{∗}π_{V_{1}}^{∗}(ξ_{1}) ⊗ ζ̂_{0}^{∗}pr_{2}^{∗}(ξ_{2})) is a G-equivariant homotopy

ζ̂_{0}^{∗}pr_{1}^{∗}π_{V_{1}}^{∗}(ξ_{1}) ⊗ ζ̂_{0}^{∗}pr_{2}^{∗}(ξ_{2}) ∼ π_{ν}^{∗}(pr_{1}^{∗}(ξ_{1}) ⊗ pr_{2}^{∗}(ξ_{2})).

When we tensor with pr_{1}^{∗}(τ_{V_{1}}), this homotopy has X-compact support because the support of pr_{1}^{∗}(τ_{V_{1}}) is relatively M-compact. This gives a homotopy of geometric correspondences between (2.17) and the variant of (2.19) with K-theory datum

pr_{1}^{∗}(τ_{V_{1}}) ⊗ π_{ν}^{∗}pr_{1}^{∗}(ξ_{1}) ⊗ π_{ν}^{∗}pr_{2}^{∗}(ξ_{2});

the relative M-compactness of the support of pr_{V_{1}}^{∗}(τ_{V_{1}}) ensures that the homotopy of K_{G}-cycles implicit here has X-compact support. (We use [15, Lemma 2.12] here, but the statement of the lemma is unclear about the necessary compatibility between the homotopy and the support of ξ.) The K-theory class pr_{1}^{∗}(τ_{V_{1}}) in this formula is the restriction of the Thom class for the vector bundle pr_{1}^{∗}(V_{1}) over M to ν. Since pr_{1}^{∗}(V_{1}) ≅ ν ⊕ η and the Thom isomorphism for a direct sum bundle is the composition of the Thom isomorphisms for the factors, the Thom class of pr_{1}^{∗}(V_{1}) is pr_{1}^{∗}(τ_{V_{1}}) = τ_{ν} ⊗ τ_{η}. Restricting this to the subbundle ν gives τ_{ν} ⊗ π_{ν}^{∗}(e(η)). Hence the K-theory classes that come from (2.17) and (2.19) are equal. This finishes the proof.

2.6 The geometric Lefschetz index formula

In this section we compute Lefschetz indices in the symmetric monoidal category k̂k^{G} for smooth G-manifolds with boundary. Our computation is geometric and uses the intersection theory of equivariant correspondences discussed in Sections 2.2 and 2.5. Let X be a smooth compact G-manifold, possibly with boundary. Let X̊ be its interior. Let

X ⟵^{b} M ⟶^{f} X,  ξ ∈ RK_{G,X}^{∗} (M)  (2.21)

be a K_{G}-oriented smooth geometric correspondence from X to itself, with M of finite orbit type to ensure that f: M → X lifts to an essentially unique normally non-singular map. Since X is compact, RK_{G,X}^{∗} (M) = K_{G}^{∗}(M) is the usual K-theory with compact support. The K_{G}-orientation for (2.21) means a K_{G}-orientation on the stable normal bundle of f. This is equivalent to giving a G-vector bundle V over X and K_{G}-orientations on TM ⊕ f^{∗}(V) and TX ⊕ V. If X has a boundary, then the requirements for a smooth correspondence are that M be a smooth manifold with boundary of finite orbit type, such that f(∂M) ⊆ ∂X and f is transverse to ∂X. This ensures that f has an essentially unique lift to a normally non-singular map from M to X by Proposition 2.5. Recall the map ρ: X → X̊, which is shrinking the collar around ∂X.

Theorem 2.18. Let α ∈ k̂k _{i}^{G}(X, X) be represented by a K_{G}-oriented smooth geometric correspondence as in (2.21). Assume that (ρb, f): M → X × X and the diagonal embedding X → X × X intersect smoothly with a K_{G}-oriented excess intersection bundle η. Then Q_{ρb,f} := {m ∈ M | ρb(m) = f(m)} is a smooth manifold without boundary. For a certain canonical K_{G}-orientation on Q_{ρb,f} , L(α) ∈ k̂k _{i}^{G}(X, pt) is represented by the geometric correspondence X ← Q_{ρb,f} → pt with K-theory class ξ|_{Q_{ρb,f}} ⊗ e(η) on Q_{ρb,f} ; here the map Q_{ρb,f} → X is given by m ↦ ρb(m) = f(m). The Lefschetz index of α in k̂k _{i}^{G}(pt, pt) is represented by the geometric correspondence pt ← Q_{ρb,f} → pt with K_{G}-theory class ξ|_{Q_{ρb,f}} ⊗ e(η) on Q_{ρb,f} . The Lefschetz index of α is the index of the Dirac operator on Q_{ρb,f} with coefficients in ξ|_{Q_{ρb,f}} ⊗ e(η).

Proof. We abbreviate Q := Q_{ρb,f} throughout the proof. We have Q ⊆ M̊ because ρb(M) ⊆ ρ(X) ⊆ X̊ and f(∂M) ⊆ ∂X. The intersection M̊ ×_{X̊ ×X̊} X̊ 164 Ivo Dell’Ambrogio, Heath Emerson, and Ralf Meyer

Q_{ρb,f}

M

X × X  M × NX̊  NX̊

X  X × X × NX̊  X × NX̊ ≅ NX̊ × X pt j ζfj Δb (id, ζρb) pr_{1} id × (id, ζρ) (Δb) × id f × id (id, ιπ)

Figure 2: The intersection diagram for the computation of L(α) in the proof of Theorem 2.18. Here j: Q_{ρb,f} → M denotes the inclusion map; ζ the zero section X → NX or X̊ → NX̊; π: NX̊ → X̊ the bundle projection; ι: X̊ → X the inclusion; Δ: X → X × X the diagonal embedding; pr_{1} : X × X → X the projection onto the first factor.

is Q and hence a smooth submanifold of M̊. We compute L(α) using the dual of X constructed in Theorem 2.7. This involves a G-vector bundle NX such that TX ⊕NX ≅ X ×E for a K_{G}-oriented G-vector space E. With the unit and counit from Theorem 2.7, L(α) becomes the composition of the three geometric correspondences in the bottom zigzag in Figure 2; here we already composed α with the multiplication correspondence, which simply composes b with Δ. We first consider the small left square. Computing its intersection space naively gives M, which is a manifold with boundary. We would hope that this square is Cartesian. But X × X is only a manifold with corners if X has a boundary, and we we did not discuss smooth correspondences in this generality. Hence we check directly that the composition of the correspondences from X to X × X × NX̊ and on to M × NX̊ is represented by X ← M → M × NX̊. The manifold NX̊ is an open subset of E by construction. Hence the map

id × (id, ζρb): X × X → X × X × NX̊

extends to an open embedding

ψ: X ×X ×E → X ×X ×NX̊, (x_{1}, x_{2}, e) ↦ (x_{1}, x_{2}, ζρ(x_{2})+h_{x_{2}} (∥e∥^{2})·e),

where h_{x_{2}} : ℝ_{+} → ℝ_{+} is a diffeomorphism onto a bounded interval [0, t) depending smoothly and G-invariantly on x_{2}, such that the t-ball in E around ζρ(x_{2}) ∈ NX̊ is contained in NX̊. The map ψ gives a special correspondence

X ⟵−−−^{pr_{1} ∘π_{E}} X × X × E ⟶^{ψ} X × X × NX̊

with K-theory class the pull-back of the Thom class of E. This is equivalent to the given correspondence from X to X × X × NX̊ because of a Thom modification for the trivial vector bundle E and a homotopy. In particular, the K_{G}-orientation of id× (id, ζρ) that is implicit here is the one that we get from the K_{G}-orientation in the proof of Theorem 2.7. For a special correspondence, the intersection always gives the composition product. Here we get the space

{(x_{1}, x_{2}, e, m, y, μ) ∈ X × X × E × M × NX̊ |

(x_{1}, x_{2}, ζρ(x_{2}) + h_{x_{2}} (∥e∥^{2}) · e)) = (b(m), b(m), y, μ)}.

That is, x_{1} = x_{2} = b(m), (y, μ) = ρb(m) + h_{b(m)} (∥e∥^{2}) · e). Since m ∈ M and e ∈ E may be arbitrary and determine the other variables, we may identify this space with M × E. In the same way, we may replace

X ⟵^{b} M −−−⟶^{(id,ζρb)} M × NX̊  (2.22)

by an equivalent special correspondence with space M × E in the middle. This gives exactly the composition computed above. Hence (2.22) also represents the composition of the correspondences from X to M × NX̊ in Figure 2. Composing further with f × id simply composes K_{G}-oriented normally nonsingular maps. Since we are now in the world of manifolds with boundary, we may identify smooth maps and smooth normally non-singular maps. The large right square contains the G-maps

(f, ζρb) = (f × id) ∘ (id, ζρb): M → X × NX̊,
(ιπ, id): NX̊ → X × NX̊.

The pull-back contains those (m, x, μ) ∈ M × NX̊ with (f(m), ρb(x), 0) = (x, x, μ) in X × NX̊. This is equivalent to x = f(m) = ρb(m) and μ = 0, so that the pull-back is Q. Since all vectors tangent to the fibres of NX̊ are in the image of D(ιπ, id), the intersection is smooth and the excess intersection bundle is the same bundle η as for (f, ρb): M̊ → X̊ × X̊ and δ: X̊ → X̊ × X̊. Hence the right square is η-Cartesian. Theorem 2.17 shows that L(α) is represented by a correspondence of the form X ⟵^{bj} Q → pt, with a suitable class in K_{G}^{∗}(Q) and a suitable K_{G}-orientation on the map Q → pt or, equivalently, the manifold Q. Here we may replace bj by the properly homotopic map ρbj = fj. It remains to describe the K-theory and orientation data.

First, the given K-theory class ξ on M is pulled back to ξ ⊗1 on M ×NX̊ when we take the exterior product with NX̊. In the intersection product, this is pulled back to M along (id, ζρb), giving ξ again, and then to Q along j, giving the restriction of ξ to Q ⊆ M. The unit and counit have 1 as its K-theory datum. Thus the Lefschetz index has ξ|_{Q} ⊗ e(η) ∈ K_{G}^{∗}(Q) as its K-theory datum by Theorem 2.17. The given K_{G}-orientations on E, f and η induce K_{G}-orientations on all maps in Figure 2 that point to the right. This is the K_{G}-orientation on the map Q → pt that we need. We describe it in greater detail after the proof of the theorem. The K_{G}-orientation on the map Q → pt is equivalent to a G-equivariant Spin^{c}- structure on Q. The isomorphism

k̂k _{∗}^{G}(pt, pt) → k̂k _{∗}^{G}(C(pt), C(pt))

described in [15, Theorem 4.2] maps the geometric correspondence just described to the index of the Dirac operator on Q for the chosen Spin^{c}-structure twisted by ξ|_{Q} ⊗ e(η). This gives the last assertion of the theorem.

Since the K_{G}-orientation on Q_{ρb,f} is necessary for computations, we describe it more explicitly now. We still use the notation from the previous proof. We are given K_{G}-orientations on E, f and η. The K_{G}-orientation on f is equivalent to one on the G-vector bundle TM ⊕ f^{∗}(NX) over M because

TX ⊕ NX ≅ X × E

is a K_{G}-oriented G-vector bundle on X. We already discussed during the proof of the theorem that id × (id, ζρ) and (id, ζρ) are normally non-singular embeddings with normal bundle E; this gives the correct K_{G}-orientation for these maps as well. A K_{G}-orientation on the map (f, ζρb): M → X × NX̊ is equivalent to one for TM ⊕ f^{∗}(NX) because the bundle T(X × NX̊) ⊕ pr_{1}^{∗}(NX) over X × NX̊ is isomorphic to the trivial bundle with fibre E ⊕ E and (f, ζρb)^{∗}pr_{1}^{∗}(NX) = f^{∗}(NX). We are already given such a K_{G}-orientation from the K_{G}-orientation of f.

Lemma 2.19. The given K_{G}-orientation on TM ⊕ f^{∗}(NX) is also the one that we get by inducing K_{G}-orientations on (id, ζρb) from (id, ζρ) and on f × id from f and then composing.

Proof. The K_{G}-orientation of f induces one for f × id, which is equivalent to a K_{G}-orientation for

T(M × NX̊) ⊕ (fpr_{1})^{∗}(NX) ≅ (TM ⊕ f^{∗}(NX)) × (NX̊ × E).

This K_{G}-orientation is exactly the direct sum orientation from TM ⊕ f^{∗}(NX) and E; no sign appears in changing the order because E has even dimension.

The map h = (id, ζρ) is a smooth embedding with normal bundle E. Hence we get an extension of vector bundles

TM ⊕ f^{∗}(NX) ↣ h^{∗}(T(M × NX̊) ⊕ (fpr_{1})^{∗}(NX)) ↠ E.

The given K_{G}-orientations on TM ⊕ f^{∗}(NX) and E induce one on the vector bundle in the middle. This is the same one as the pull-back of the one constructed above. This means that the K_{G}-orientation on TM ⊕f^{∗}(NX) induced by h is the given one.

Equation (2.14) provides the following exact sequence of vector bundles over Q:

0 → TQ −−−−−⟶^{Dj,D(ζfj)} j^{∗}(TM) ⊕ (ζfj)^{∗}T(NX̊)

−−−−−−−−−−−−→^{D(f,ζρb),−D(ιπ,id)} (f, ζρb)^{∗}T(X × NX̊) → η → 0.

Since −D(ιπ, id) is injective, we may divide out T(NX̊) and its image to get the simpler short exact sequence

0 → TQ ⟶^{Dj} j^{∗}TM −−−−−⟶^{Df−D(ρb)} f^{∗}TX → η → 0.

Then we add the identity map on j^{∗}f^{∗}(NX) to get

0 → TQ −−⟶^{(Dj,0)} j^{∗}(TM ⊕ f^{∗}NX) −−−−−−−−⟶^{(Df−D(ρb),id)} f^{∗}(TX ⊕ NX) → η → 0.

(2.23)

In the last long exact sequence, the vector bundles j^{∗}(TM ⊕ f^{∗}NX), f^{∗}(TX ⊕ NX) ≅ Q × E and η carry K_{G}-orientations. These together induce one on TQ. This is the K_{G}-orientation that appears in Theorem 2.18. Of course, the resulting geometric cycle should not depend on the auxiliary choice of a K_{G}-orientation on η. Indeed, if we change it, then we change both e(η) and the K_{G}-orientation on TQ, and these changes cancel each other. We now consider some examples of Theorem 2.18.

2.6.1 Self-maps transverse to the identity map

Let X be a compact G-manifold with boundary and let b: X → X be a smooth G-map that is transverse to the identity map. Thus b has only finitely many isolated fixed points and 1−D_{x}b: T_{x}X → T_{x}X is invertible for all fixed points x of b. We turn b into a geometric correspondence α from X to itself by taking M = X, f = id (with standard K_{G}-orientation) and ξ = 1. Since b has only finitely many fixed points, we may choose the collar neighbourhood so small that all fixed points that do not lie on ∂X lie outside the collar neighbourhood, and such that the fixed points of ρb are precisely the fixed points of b not on the boundary of X. Hence ρb = b near all fixed points. Then ρb is also transverse to the diagonal map and Theorem 2.18 applies. The intersection space in Theorem 2.18 is

Q = Q_{ρb,id} = {x ∈ X | ρb(x) = x} = {x ∈ X̊ | b(x) = x},

the set of fixed points of b in X̊. The K-theory class on Q is 1 because ξ = 1 and the intersection is transverse. More precisely, the bundle η is zero-dimensional, and we may give it a trivial K_{G}-orientation for which e(η) = 1. Although Q is discrete, the K_{G}-orientation of the map Q → pt is important extra information: it provides the signs that appear in the familiar Lefschetz fixed-point formula. Equation (2.23) simplifies to

0 → TQ → (TX ⊕ NX)|_{Q} −−−−−⟶^{(id−Db,id)} (TX ⊕ NX)|_{Q} → 0.

We left out η because it is zero-dimensional and carries the trivial K_{G}-orientation to ensure that e(η) = 1. The bundle TQ is also zerodimensional. But a zero-dimensional bundle has non-trivial K_{G}-orientations. The Clifford algebra bundle of a zero-dimensional bundle is the trivial, trivially graded one-dimensional bundle spanned by the unit section. Thus an irreducible Clifford module (spinor bundle) for it is the same as a ℤ/2-graded G-equivariant complex line bundle. Let S be the spinor bundle associated to the given K_{G}-orientation on TX ⊕ NX ≅ E. The exact sequence (2.23) says that the K_{G}-orientation of Q is the ℤ/2-graded G-equivariant complex line bundle ℓ such that (id−Db)^{∗}(S|_{Q})⊗ℓ ≅ S|_{Q} as Clifford modules. This uniquely determines ℓ. Thus ℓ measures whether Db changes orientation or not. This is exactly the sign of the G-equivariant vector bundle automorphism 1 − Db on TX|_{Q}, which is studied in detail in [13]. In particular, it is shown in [13] that ℓ is the complexification of a ℤ/2-graded G-equivariant real line bundle. The ℤ/2-grading gives one sign for each G-orbit in Q, namely, the index of id−Db_{x}. In addition, the sign gives a real character G_{x} → {−1, +1} for each orbit, where G_{x} denotes the stabiliser of a point in the orbit. Twisting the K_{G}-orientation by a line bundle over Q has the same effect as taking the trivial K_{G}-orientation and putting this line bundle on Q. Thus L(α) is represented by the geometric correspondence

X ← (Q, sign(1 − Db|_{Q})) → pt

with the trivial K_{G}-orientation on the map Q → pt. The Lefschetz index of α is the index of the Dirac operator on Q with coefficients in the line bundle sign(1−Db)|_{Q}; this is simply the ℤ/2-graded G-representation on the space of sections of sign(1 − Db)|_{Q}, which is a certain finite-dimensional ℤ/2-graded, real G-representation. If the group G is trivial, then the Lefschetz index is a number and sign(1 − Db) is the family of sign(1 − D_{x}b) ∈ {±1} for x ∈ Q. If X is connected, then all maps X ← pt give the same element in k̂k . Thus L(α) is L-ind(α) times the point evaluation class [X ← pt = pt], and L-ind(α) is the sum of the indices of all fixed points of b in X̊.

2.6.2 Euler characteristics

Now let ξ ∈ K_{G}^{∗}(X) and consider the correspondence with M = X, b = f = id, and the above class ξ. We want to compute the Lefschetz index of the geometric correspondence α associated to ξ. In particular, for ξ = 1 we get the Lefschetz index of the identity element in k̂k _{0}^{G}(X, X), which is the Euler characteristic of X. We only compute the Lefschetz index of ξ ∈ K_{G}^{∗}(X) for X with trivial boundary. Then the map ρ in Theorem 2.18 is the identity map, and id_{X} intersects itself smoothly. The intersection space is Q = X, embedded diagonally into X × X. The excess intersection bundle η is TX. To apply Theorem 2.18, we also assume that X is K_{G}-oriented. Then L(α) is represented by the geometric correspondence

X ⟵−^{id_{X}} (X, ξ ⊗ e(TX)) → pt.

Here e(TX) and the map X → pt both use the same K_{G}-orientation on X. The Lefschetz index of α is represented by

pt ← (X, ξ ⊗ e(TX)) → pt.

By Theorem 2.18, this is the index of the Dirac operator of X with coefficients in ξ ⊗ e(TX). Twisting the Dirac operator by e(TX) gives the de Rham operator: this is the operator d + d^{∗} on differential forms with usual ℤ/2-grading, so that its index is the Euler characteristic of X. Thus (the analytic version of) L(α) is the class in KK_{0}^{G}(C(X), ℂ) of the de Rham operator with coefficients in ξ. This was proved already in [11] by computations in Kasparov’s analytic KK-theory. Now we have a purely geometric proof of this fact, at least if X is K_{G}-oriented. Theorem 2.18 no longer works for X without K_{G}-orientation because there is no K_{G}-orientation on the excess intersection bundle. A way around this restriction would be to use twisted K-theory throughout. We shall not pursue this here, however. We can now clarify the relationship between the Euler class e(TX) ∈ K_{G}^{dim (X)} (X) and the higher Euler characteristic Eul_{X} ∈ KK_{0}^{G}(C(X), ℂ) introduced already in [11]. Since we assume X K_{G}-oriented and without boundary, there is a duality isomorphism K_{G}^{dim (X)} (X) ≅ K_{0}^{G}(X) = KK_{0}^{G}(C(X), ℂ). This duality isomorphism maps e(TX) to Eul_{X}.

2.6.3 Self-maps without transversality

Let X be a compact G-manifold and let b: X → X be a smooth G-map. We want to compute the Lefschetz map on the geometric correspondence

X ⟵^{b} X −−→^{id_{X}} X

with K_{G}-theory class 1 on X.

If b is transverse to the identity map, then this is done already in Section 2.6.1. The case b = id_{X} is done already in Section 2.6.2. Now we assume that b and id_{X} intersect smoothly. We also assume that b has no fixed points on the boundary; then we may choose the collar neighbourhood of ∂X to contain no fixed points of b, so that ρ(x) = x in a neighbourhood of the fixed point subset of b. Furthermore, all fixed points of ρb are already fixed points of b. That b and id_{X} intersect smoothly and away from ∂X means that

Q := {x ∈ X | b(x) = x} = {x ∈ X | ρb(x) = x}

is a smooth submanifold of X̊ and that there is an exact sequence of G-vector bundles over Q:

0 → TQ → TX|_{Q} −−−−−→^{1−D(ρb)} TX|_{Q} → η → 0,

where η is the excess intersection bundle. Remark 2.20. The maps b and id_{X} always intersect smoothly if b: X → X is isometric with respect to a Riemannian metric on X; the reason is that if Db fixes a vector (x, ξ) at a fixed point of b, then b fixes the entire geodesic through x in direction ξ. The vector bundles TQ and η are the kernel and cokernel of the vector bundle endomorphism 1 − D(ρb) on TX|_{Q}. Since both are vector bundles, 1 − D(ρb) has locally constant rank. We may split

TX|_{Q} ≅ ker(id − D(ρb)) ⊕ im(id − D(ρb)) = TQ ⊕ im(id − D(ρb)),
TX|_{Q} ≅ coker(id − D(ρb)) ⊕ coim(id − D(ρb)) = η ⊕ coim(id − D(ρb)).

Since im(φ) ≅ coim(φ) for any vector bundle homomorphism, it follows that η and TQ are stably isomorphic as G-vector bundles. Thus K_{G}-orientations on one of them translate to K_{G}-orientations on the other. Remark 2.21. Given two stably isomorphic vector bundles, there is always a vector bundle endomorphism with these two as kernel and cokernel. Hence we cannot expect η and TQ to be isomorphic.

Corollary 2.22. Let X be a compact G-manifold. Let b: X → X be a smooth G-map without fixed points on ∂X, such that b and id_{X} intersect smoothly. Let the fixed point submanifold Q of b be K_{G}-oriented, and equip the excess intersection bundle with the induced K_{G}-orientation. Then the Lefschetz index of the geometric correspondence

X ⟵^{b} X −−→^{id_{X}} X

with K_{G}-theory class 1 on X is the index of the Dirac operator on Q twisted by e(η).

The Lefschetz map sends the correspondence above to

X ⟵^{bj} Q → pt

with K-theory class e(η) on Q.

2.6.4 Trace computation for standard correspondences

By Corollary 2.8, any element of k̂k _{∗}^{G}(X, X) is represented by a correspondence of the form

X ⟵−−−^{ι∘π∘pr_{1}} NX̊ × X ⟶^{pr_{2}} X

for a unique ξ ∈ K_{G}^{∗}(NX̊ × X). We may view this as a standard form for an element in k̂k _{∗}^{G}(X, X). The map (ρ∘ι∘π∘pr_{1}, pr_{2}) = (ρ∘π)×id: NX̊ ×X → X ×X is a submersion and hence transverse to the diagonal. Thus Theorem 2.18 applies. The space Q_{ρb,f} is the graph of ρπ: NX̊ → X. Thus the Lefschetz map gives the geometric correspondence

X ← NX̊ → pt, ξ|_{NX̊} ∈ K_{G}^{∗}(NX̊),

where we embed NX̊ → NX̊ × X via (id, ρπ) and use the canonical K_{G}-orientation on NX̊. The Lefschetz index in k̂k _{∗}^{G}(pt, pt) ≅ K_{G}^{∗}(pt) is computed analytically as the G-equivariant index of the Dirac operator on NX̊ twisted by ξ|_{NX̊}.

2.6.5 Trace computation for another standard form

Assume now that X has no boundary and is K_{G}-oriented. As we remarked at the end of Section 2.4, any element of k̂k _{∗}^{G}(X, X) is represented by a correspondence

X ⟵^{pr_{1}} X × X ⟶^{pr_{2}} X, ξ ∈ K_{G}^{∗}(X × X).

The same computation as in Section 2.6.4 shows that the Lefschetz map sends this to

X = X → pt, ξ|_{X} ∈ K_{G}^{∗}(X),

where ξ|_{X} is for the diagonal embedding X → X × X. Analytically, this is the K_{G}-homology class of the Dirac operator on X with coefficients ξ|_{X}.

2.6.6 Homogeneous correspondences

We call a self-correspondence X ⟵^{b} M ⟶^{f} X homogeneous if X and M are homogeneous G-spaces. That is, X := G/H and M := G/L for closed subgroups H, L ⊆ G. Then there are elements t_{b}, t_{f} ∈ G with b(gL) := gt_{b}H, f(gL) := gt_{f} H; we need L ⊆ t_{b}Ht_{b}^{−1} ∩ t_{f} Ht_{f}^{−1} for this to be well-defined. Since G/L ≅ G/t_{f}^{−1} Lt_{f} by gL ↦ gLt_{f} , any homogeneous correspondence is isomorphic to one with t_{f} = 1, so that L ⊆ H. We assume this from now on and abbreviate t = t_{b}. Since M and X are compact, the relevant K-theory group RK_{G,X}^{∗} (M) for a homogeneous correspondence is just K_{G}^{∗}(M). The induction isomorphism gives RK_{G,X}^{∗} (M) = K_{G}^{∗}(G/L) ≅ K_{L}^{∗}(pt). A K_{G}-orientation for f: G/L → G/H is equivalent to a K^{H} -orientation for the projection map H/L → pt because f is obtained from this H-map by induction. Thus we must assume a K^{H} -orientation on H/L. Equivalently, the representation of L on T_{1L} (H/L) factors through Spin^{c}. This tangent space is the quotient 𝔥/𝔩, where 𝔥 and 𝔩 denote the Lie algebras of H and L, respectively. Let L′ := H ∩ tHt^{−1} . Then L ⊆ L′ and both maps f, b: G/L → G/H factor through the quotient map p: G/L → G/L′. The geometric correspondence

G/H ⟵^{b} G/L ⟶^{f} G/H, ξ ∈ K_{G}^{∗}(G/L)

is equivalent to the geometric correspondence

G/H ⟵^{b′} G/L′ ⟶^{f′} G/H, ξ′ ∈ K_{G}^{∗}(G/L′)

with ξ′ := p_{!}(ξ) and b′(gL′) = gtH and f′(gL′) = gH. (To construct the equivalence, we first need a normally non-singular map lifting p; then we apply vector bundle modifications on the domain and target of p to replace p by an open embedding; finally, for an open embedding we may construct a bordism as in [15, Example 2.14].) Thus we may further normalise a homogeneous geometric self-correspondence to one with L = H ∩ tHt^{−1} . Now we compute the Lefschetz map for a such a normalised homogeneous selfcorrespondence. First let t ∉ H. Then the image of the map (f, b): G/L → G/H × G/H does not intersect the diagonal. Hence (f, b) is transverse to the diagonal and the coincidence space Q_{b,f} is empty. Thus the Lefschetz map vanishes on a homogeneous correspondence with t ∉ H by Theorem 2.18. Now let t ∈ H. Then b = f: G/L → G/H is the canonical projection map. Our normalisation condition yields L = H and b = f = id in this case; that is, our geometric correspondence is the class in k̂k _{∗}^{G}(G/H, G/H) of some ξ ∈ K_{G}^{∗}(G/H). Thus we have a special case of the Euler characteristic computation in Section 2.6.2. The Lefschetz map gives the class of the geometric correspondence

G/H ⟵_{=}^{id} (G/H, e(TG/H) ⊗ ξ) → pt,

provided G/H is K_{G}-oriented. The Lefschetz index is the index of the de Rham operator with coefficients in ξ. When we identify K_{G}^{∗}(G/H) ≅ K_{H}^{∗} (pt), the Lefschetz index becomes a map

K_{H}^{∗} (pt) → K_{G}^{∗}(pt).

In complex K-theory, this is a map R(H) → R(G). Graeme Segal studied this map in [28, Section 2], where it was denoted by i_{!}. For instance, assume G to be connected and let H = L be its maximal torus. Let t ∈ W := N_{G}H/H, the Weyl group of G. Assume that we are working with complex K-theory, so that K_{G}^{∗}(G/H) ≅ K_{H}^{∗} (pt) ≅ R(H). The Weyl group W acts on G/H by right translations; these are G-equivariant maps.

Taking the correspondences X ⟵−^{w^{−1}} X = X, this gives a representation W → k̂k _{0}^{G}(G/H, G/H). We also map R(H) ≅ K_{G}^{0}(G/H) → k̂k _{0}^{G}(G/H, G/H) using the correspondences X = (X, ξ) = X. These representations of W and R(H) are a covariant pair of representations with respect to the canonical action of W on R(H) induced by the automorphisms h ↦ whw^{−1} of H for w ∈ W. Hence we map

R(H) ⋊ W → k̂k _{0}^{G}(G/H, G/H).

The Lefschetz index R(H) ⋊ W → R(G) maps a · t ↦ 0 for t ∈ W \ {1} and a · 1 ↦ ind_{G} Λ_{a}, where Λ_{a} means the de Rham operator on G/H twisted by a.

2.7 Fixed points submanifolds for torus actions

As another application of our excess intersection formula, we reprove a result that is used in a recent article by Block and Higson [5] to reformulate the Weyl Character Formula in KK-theory. Block and Higson also develop a more geometric framework for equivariant KK-theory for a compact group. For two locally compact G-spaces X and Y, they identify KK_{∗}^{G}(X, Y) with the group of continuous natural transformations Φ_{Z} : K_{G}^{∗}(X × Z) → K_{G}^{∗}(Y × Z) for all compact G-spaces Z; here continuity means that each Φ_{Z} is a K_{G}^{∗}(Z)-module homomorphism. The Kasparov product then becomes the composition of natural transformations. This reduces Kasparov’s equivariant KK-theory to equivariant K-theory. The theory k̂k^{G} does more: it contains the knowledge that all such natural transformations come from geometric correspondences, when geometric correspondences give the same natural transformation, and how to compose geometric correspondences. Thus we get a more concrete KK-theory.

Theorem 2.23. Let T be a compact torus and let X be a smooth, K_{T}-oriented T -manifold with boundary. Let e(TX) ∈ K_{T}^{0}(X) be the Euler class of X for the chosen K_{T}-orientation. Let F ⊆ X be the fixed-point subset of the T -action on X and let j: F → X be the inclusion map. Then F is again a smooth K-oriented manifold with boundary, with trivial T -action, so that the inclusion map j is K_{T}-oriented. Let e(TF) ∈ K^{0}(F) ⊆ K_{T}^{0}(F) be the Euler class of F. The two geometric correspondences

X ⟵−^{id_{X}} (X, e(TX)) −−→^{id_{X}} X,

X ⟵^{j} (F, e(TF)) ⟶^{j} X

represent the same element in k̂k _{0}^{T}(X, X).

This is a generalisation of [5, Lemma 3.1]. We allow Spin^{c}-manifolds instead of complex manifolds. For a Spin^{c}-structure coming from a complex structure, the Euler class is [Λ^{∗}T^{∗}X] ∈ K_{T}^{0}(X), which appears in [5]. The following proof is a translation of the proof in [5] into the category k̂k^{G}.

Proof. The first geometric correspondence above, involving the Euler class of X, is represented by the composition of geometric correspondences

X ⟵−_{=}^{id_{X}} X ⟶^{ζ} TX ⟵^{ζ} X −−→_{=}^{id_{X}} X

by Example 2.13; here ζ denotes the zero section, which is K_{T}-oriented using the given K_{T}-orientation on the T -vector bundle TX. Choose a generic element ξ in the Lie algebra of T, that is, the one-parameter group exp(sξ), s ∈ ℝ, is dense in T. Let α_{t}: X → X denote the action of t ∈ T on X. The action of T maps ξ to a vector field α_{ξ}: X → TX. There is a homotopy of geometric correspondences

X ⟵−_{=}^{id_{X}} X −−→^{α_{sξ}} TX ⟵^{ζ} X −−→_{=}^{id_{X}} X

for s ∈ [0, 1]. For t = 0 we get the composition above, involving e(TX). We claim that for s = 1, the two correspondences intersect smoothly and that the intersection product is the second geometric correspondence in the theorem, involving F and its Euler class. First we show that the fixed-point submanifold F is a closed submanifold. Equip X with a T -invariant Riemannian metric. Let x ∈ F, that is, α_{t}(x) = x for all t ∈ T. Split T_{x}X into

V = {v ∈ T_{x}X | Dα_{t}(x, v) = (x, v) for all t ∈ T }

and its orthogonal complement V^{⊥} . Since the metric is T -invariant,

α_{t}(exp(x, v)) = exp(Dα_{t}(x, v))

for all v ∈ T_{x}X. Since the exponential mapping restricts to a diffeomorphism between a neighbourhood of 0 in T_{x}X and a neighbourhood of x in X, we have exp(x, v) ∈ F if v ∈ V, and the converse holds for v in a suitable neighbourhood of 0. Thus we get a closed submanifold chart for F near x with T_{x}F = V. Hence F is a closed submanifold with

TF = {(x, v) ∈ TX | Dα_{t}(x, v) = (x, v) for all t ∈ T }.

Since ξ is generic, α_{ξ}(x) = 0 in T_{x}X if and only if x ∈ F. Thus F is the coincidence space of the pair of maps ζ, α_{ξ}: X → TX. Let x ∈ F and let v_{1}, v_{2} ∈ T_{x}X satisfy Dζ(x, v_{1}) = Dα_{ξ}(x, v_{2}). Then v_{1} = v_{2} by taking the horizontal components; and the vertical component of Dα_{ξ}(x, v_{2}) vanishes, which means that Dα_{exp (sξ)} (x, v_{2}) = (x, v_{2}) for all s ∈ ℝ. Hence v_{2} ∈ T_{x}F. This proves that ζ and α_{ξ} intersect smoothly. The excess intersection bundle is the cokernel of Dα_{exp (sξ)} −id; since the action of T is by isometries, Dα_{exp (sξ)} −id is normal in each fibre, so that its image and kernel are orthogonal complements. Hence the cokernel is canonically isomorphic to the kernel of Dα_{exp (sξ)} − id. Thus the excess intersection bundle is canonically isomorphic to TF. Hence Theorem 2.17 gives the geometric correspondence X ⟵^{j} (F, e(TF)) ⟶^{j} X as the composition, as desired.

3 The homological Lefschetz index of a Kasparov morphism

The example in Section 2.6.1 shows in what sense the geometric Lefschetz index computations in Section 2 generalise the local fixed-point formula for the Lefschetz index of a self-map. Now we turn to generalisations of the global homological formula for the Lefschetz index. The classical Lefschetz fixed-point formula for a self-map f: X → X contains the (super)trace of the map on the cohomology of X with rational coefficients induced by f. We take rational coefficients in order to get vector spaces over a field, where there is a good notion of trace for endomorphisms. By the Chern character, we may as well take K^{∗}(X) ⊗ ℚ instead of rational cohomology. It is checked in [9] that the Lefschetz index of f ∈ KK_{0}(A, A) for a dualisable C^{∗}-algebra A in the bootstrap class is equal to the supertrace of the map on K_{∗}(A) ⊗ ℚ induced by f. We are going to generalise this result to the equivariant situation for a compact Lie group G. We assume that we are working with complex C^{∗}-algebras, so that k̂k _{∗}^{G}(pt, pt) = KK_{∗}^{G}(ℂ, ℂ) vanishes in odd degrees and is the representation ring R(G) in even degrees. Our methods do not apply to the torsion invariants in KK_{d}^{G}(ℝ, ℝ) for d ≠ 0 in the real case because we (implicitly) tensor everything with ℚ to simplify the Lefschetz index. Furthermore, we work in KK^{G} instead of k̂k^{G} in this section because the category KK^{G} is triangulated, unlike k̂k^{G}. We explain in Remark 3.11 why k̂k^{G} is not triangulated; the triangulated structure on KK^{G} is introduced in [21]. Let S ⊆ R(G) be the set of all elements that are not zero divisors. This is a saturated, multiplicatively closed subset; even more, it is the largest multiplicatively closed subset for which the canonical map R(G) → S^{−1} R(G) to the ring of fractions is injective (see [1, Exercise 9 on p. 44]). The localisation S^{−1} R(G) is also called the total ring of fractions of R(G). Since KK^{G} is symmetric monoidal with unit 𝟙 = ℂ and R(G) = KK_{0}^{G}(ℂ, ℂ), the category KK^{G} is R(G)-linear. Hence we may localise it at S as in [17]. The resulting category T := S^{−1} KK^{G} has the same objects as KK^{G} and arrows

T_{∗}(A, B) := S^{−1} KK_{∗}^{G}(A, B) = S^{−1} R(G) ⊗_{R(G)} KK_{∗}^{G}(A, B).

The category T is S^{−1} R(G)-linear. There is an obvious functor ♮: KK^{G} → T . If A is a separable G-C^{∗}-algebra, then

T_{∗}(ℂ, A) = S^{−1} KK_{∗}^{G}(ℂ, A) ≅ S^{−1} R(G) ⊗_{R(G)} K_{∗}^{G}(A),

where we use the usual R(G)-module structure on K_{∗}^{G}(A) ≅ KK_{∗}^{G}(ℂ, A). There is a unique symmetric monoidal structure on T for which ♮ is a strict symmetric monoidal functor: simply extend the exterior tensor product on KK^{G} S^{−1} R(G)-linearly. Hence if A is dualisable in KK^{G}, then its image in T is dualisable as well, and

♮(trf) = tr(♮f) for all f ∈ KK_{∗}^{G}(A, A).

The crucial point for us is that ♮ tr(f) = tr(♮f) uniquely determines trf because the map

R(G) ≅ KK_{0}^{G}(𝟙, 𝟙) ⟶^{♮} T_{0}(𝟙, 𝟙) ≅ S^{−1} R(G)

is injective. Thus it suffices to compute Lefschetz indices in T . This may be easier because T has more isomorphisms and thus fewer isomorphism classes of objects. Furthermore, the endomorphism ring of the unit T_{∗}(𝟙, 𝟙) = S^{−1} R(G) has a rather simple structure:

Lemma 3.1. The ring S^{−1} R(G) is a product of finitely many fields.

Proof. Let G/ AdG be the space of conjugacy classes in G and let C(G/ AdG) be the algebra of continuous functions on G/ AdG. Taking characters provides a ring homomorphism χ: R(G) → C(G/ AdG), which is well-known to be injective. Hence R(G) is torsion-free as an Abelian group and has no nilpotent elements. Since G is a compact Lie group, R(G) is a finitely generated commutative ring by [28, Corollary 3.3]. Thus R(G) is Noetherian and reduced. This implies that its total ring of fractions is a finite product of fields (see [18, Exercise 6.5]).

The fields in this product decomposition correspond bijectively to minimal prime ideals in R(G). By [28, Proposition 3.7.iii], these correspond bijectively to cyclic subgroups of G/G^{0}, where G^{0} denotes the connected component of the identity element. In particular, S^{−1} R(G) is a field if and only if G is connected.

Example 3.2. Let G be a connected compact Lie group. Let T be a maximal torus in G and let W be the Weyl group, W := N_{G}(T)/T. Highest weight theory provides an isomorphism R(G) ≅ R(T)^{W}. Here R(T) is a ring of integral Laurent polynomials in r variables, where r is the rank of T. Since elements of ℕ_{≥1} are not zero divisors in R(G), the total ring of fractions of R(G) is equal to the total ring of fractions of R(G) ⊗ ℚ. The latter is the ℚ-algebra of W -invariant elements in ℚ[x_{1}, …, x_{r}, (x_{1} ⋯ x_{r})^{−1} ]. This is the algebra of polynomial functions on the algebraic ℚ-variety (ℚ^{×} )^{r}, and the W -invariants give the algebra of polynomials on the quotient variety (ℚ^{×} )^{r}/W. This variety is connected, so that the total ring of fractions S^{−1} R(G) in this case is the field of rational functions on the algebraic ℚ-variety (ℚ^{×} )^{r}/W.

Now we can define an equivariant analogue of the trace of the map on K_{∗}(A)⊗ℚ induced by f ∈ KK_{0}(A, A):

Definition 3.3. Let S^{−1} R(G) = ∏ _{i=1}^{n} F_{i} with fields F_{i}. A module over S^{−1} R(G) is a product ∏ _{i=1}^{n} V_{i}, where each V_{i} is an F_{i}-vector space. In particular, if A is a G-C^{∗}-algebra, then T_{∗}(ℂ, A) = S^{−1} K_{∗}^{G}(A) = ∏ _{i=1}^{n} K_{∗,i}^{G} (A) for certain ℤ/2-graded F_{i}-vector spaces K_{∗,i}^{G} (A). An endomorphism f ∈ T_{0}(A, A) induces grading-preserving endomorphisms K_{∗,i}^{G} (f): K_{∗,i}^{G} (A) → K_{∗,i}^{G} (A).

If the vector spaces K_{∗,i}^{G} (A) are all finite-dimensional, then the (super)trace of K_{∗,i}^{G} (f) is defined to be trK_{0,i}^{G} (f) − trK_{1,i}^{G} (f) ∈ F_{i}, and

trS^{−1} K_{∗}^{G}(f) := (trK_{∗,i}^{G} (f))_{i=1}^{n} ∈ ∏_{i=1}^{n} F_{i} = S^{−1} R(G).

We will see below that dualisability for objects in appropriate bootstrap classes already implies that K_{∗}^{G}(A) is a finitely generated R(G)-module, and then each K_{∗,i}^{G} (A) must be a finite-dimensional F_{i}-vector space.

Theorem 3.4. Let A belong to the thick subcategory of KK^{G} generated by ℂ and let f ∈ KK_{0}^{G}(A, A). Then A is dualisable in KK^{G}, so that trf is defined, and

♮(trf) = trS^{−1} K_{∗}^{G}(f) ∈ S^{−1} R(G).

Thick subcategories are defined in [26, Definition 2.1.6]. The thick subcategory generated by ℂ is, of course, the smallest thick subcategory that contains the object ℂ. We denote the thick subcategory generated by a set A of objects or a single object by ⟨A⟩. As we remarked above, ♮(trf) uniquely determines trf ∈ R(G) because the canonical embedding ♮: R(G) → S^{−1} R(G) is injective. We will prove Theorem 3.4 in Section 3.3. How restrictive is the assumption that X should belong to the thick subcategory of KK^{G} generated by ℂ? The answer depends on the group G. We consider the two extreme cases: Hodgkin Lie groups and finite groups. A Hodgkin Lie group is, by definition, a connected Lie group with simply connected fundamental group; they are the groups to which the Universal Coefficient Theorem and the Künneth Theorem in [27] apply.

Theorem 3.5. Let G be a compact connected Lie group with torsion-free fundamental group. Then a G-C^{∗}-algebra A belongs to the thick subcategory generated by ℂ if and only if

• A, without the G-action, belongs to the bootstrap category in KK, and

• A is dualisable.

We postpone the proof of this theorem until after the proof of Proposition 3.13, which generalises part of this theorem to arbitrary compact Lie groups. The first condition in Theorem 3.5 is automatic for commutative C^{∗}-algebras because the non-equivariant bootstrap category is the class of all separable C^{∗}-algebras that are KK-equivalent to a commutative separable C^{∗}-algebra. Hence Theorem 3.5 verifies the assumptions needed for Theorem 3.4 if A = C_{0}(X) and C_{0}(X) is dualisable in KK^{G}; the latter is necessary for the Lefschetz index to be defined, anyway. In particular, let X be a compact smooth G-manifold with boundary, for a Hodgkin Lie group G. Then X is dualisable in k̂k^{G} by Theorem 2.7, and hence C(X) is dualisable in KK^{G} because the functor k̂k^{G} → KK^{G} is symmetric monoidal. Furthermore, k̂k _{∗}^{G}(X, X) ≅ KK_{∗}^{G}(C(X), C(X)) in this case, so that any endomorphism f ∈ KK_{0}^{G}(C(X), C(X)) comes from some selfcorrespondence in k̂k _{0}^{G}(X, X). We get the following generalisation of the Lefschetz fixed-point formula:

Corollary 3.6. Let G be a Hodgkin Lie group, X a smooth compact G-manifold, possibly with boundary, and f ∈ k̂k _{0}^{G}(X, X). Then tr(f) ∈ R(G) ⊆ S^{−1} R(G) is equal to the supertrace of S^{−1} K_{G}^{∗}(f), acting on the S^{−1} R(G)-vector space S^{−1} K_{G}^{∗}(X).

Notice that S^{−1} R(G) for a Hodgkin Lie group is a field, not just a product of fields. In particular, Corollary 3.6 for the trivial group gives the Lefschetz index formula in [9]. Whereas Theorem 3.4 yields quite satisfactory results for Hodgkin Lie groups, its scope for a finite group G is quite limited:

Example 3.7. For G = ℤ/2 there is a locally compact G-space X with K_{G}^{∗}(X) = 0 but K^{∗}(X) ≠ 0. Equivalently, KK_{∗}^{G}(ℂ, C_{0}(X)) = 0 and KK_{∗}^{G}(C(G), C_{0}(X)) ≠ 0. This shows that C(G) does not belong to ⟨ℂ⟩. Worse, the Lefschetz index formula in Theorem 3.4 is false for endomorphisms of C(G). We have k̂k _{∗}^{G}(G, G) ≅ ℤ[G], spanned by the classes of the translation maps G → G, x ↦ x·g, for g ∈ G, and these are homogeneous correspondences as in Section 2.6.6. Translation by g = 1 is the identity map, and its Lefschetz index is the class of the regular representation of G in R(G). For g ≠ 1, the Lefschetz index is zero because the fixed point subset is empty. However, K_{G}^{∗}(G) = K^{∗}(pt) = ℤ[0] and all translation maps induce the identity map on K_{G}^{∗}(G). Thus the induced map on K_{G}^{∗}(G) is not enough information to compute the Lefschetz index of an endomorphism of G in k̂k^{G}.

3.1 The equivariant bootstrap category

A reasonable Lefschetz index formula should apply at least to KK^{G}-endo-morphisms of C(X) for all smooth compact G-manifolds and thus, in particular, for finite G-sets X. Example 3.7 shows that Theorem 3.4 fails on such a larger category. This leads us to improve the Lefschetz index formula. First we discuss the class of G-C^{∗}-algebras where we expect it to hold. We are going to describe an equivariant analogue of the bootstrap class in KK^{G}. Our class is larger than the class of C^{∗}-algebras that are KK^{G}-equivalent to a commutative C^{∗}-algebra. The latter subcategory is too small because it is not thick. The thick (or localising) subcategory of KK^{G} generated by commutative C^{∗}-algebras is a better choice, but such a definition is not very intrinsic. We will choose an even larger subcategory of KK^{G} because it is not more difficult to treat and has a nicer characterisation.

The category KK^{G} only has countable coproducts because we need C^{∗}-algebras to be separable. Hence the standard notions of compact objects and localising subcategories have to be modified so that they only involve countable coproducts. As in [7, Definition 2.1], we speak of compact_{ℵ_{1}} objects, localising_{ℵ_{1}} subcategories, and compactly_{ℵ_{1}} generated subcategories.

Definition 3.8. Call a G-C^{∗}-algebra A elementary if it is of the form Ind_{H}^{G} 𝕄_{n}ℂ = C(G, 𝕄_{n}ℂ)^{H} for some closed subgroup H ⊆ G and some action of H on 𝕄_{n}ℂ by automorphisms; the superscript H means the fixed points for the diagonal action of H.

Definition 3.9. Let B^{G} ⊆ KK^{G} be the localising_{ℵ_{1}} subcategory generated by all elementary G-C^{∗}-algebras. We call B^{G} the G-equivariant bootstrap category.

An action of H on 𝕄_{n}ℂ comes from a projective representation of H on ℂ^{n}. Such a projective representation is a representation of an extension of H by the circle group. The extension is classified by a cohomology class in H^{2}(H, U(1)). Two actions on 𝕄_{n}ℂ are H-equivariantly Morita equivalent if and only if they belong to the same class in H^{2}(H, U(1)). The G-C^{∗}-algebras Ind_{H}^{G} 𝕄_{n}ℂ for actions of H on 𝕄_{n}ℂ with different cohomology classes need not be KK^{G}- equivalent.

Theorem 3.10. A G-C^{∗}-algebra belongs to the localising_{ℵ_{1}} subcategory generated by the elementary G-C^{∗}-algebras if and only if it is KK^{G}-equivalent to a G-action on a type I C^{∗}-algebra.

Proof. It is already shown in [27, Theorem 2.8] that all G-actions on type I C^{∗}-algebras belong to the localising_{ℵ_{1}} subcategory generated by the elementary G-C^{∗}-algebras. By definition, localising_{ℵ_{1}} subcategories are closed under KK^{G}-equivalence. Elementary G-C^{∗}-algebras are type I C^{∗}-algebras, even continuous trace C^{∗}-algebras. To finish the proof we must show that the G-C^{∗}- algebras that are KK^{G}-equivalent to type I G-C^{∗}-algebras form a localising_{ℵ_{1}} subcategory of KK^{G}. Let T_{1} ⊆ KK^{G} be the full subcategory of type I, separable G-C^{∗}-algebras. If A ∈ T_{1}, then C_{0}(ℝ, A) ∈ T_{1}, so that T_{1} is closed under suspension and desuspension. Let A, B ∈ T_{1} and f ∈ KK_{0}^{G}(A, B). We have KK_{0}^{G}(A, B) ≅ KK_{1}^{G}(A, C_{0}(ℝ, B)), and cycles for the latter group correspond to (equivariantly) semisplit extensions of G-C^{∗}-algebras

C_{0}(ℝ, B) ⊗ K ↣ D ↠ A

with K := K(L^{2}(G×ℕ)). Since B and A are type I, so are C_{0}(ℝ, B)⊗K and D because the property of being type I is inherited by extensions. The semisplit extension above provides an exact triangle isomorphic to

B[−1] → D → A ⟶^{f} B.

Thus there is an exact triangle containing f with all three entries in T_{1}. Furthermore, countable direct sums of type I C^{∗}-algebras are again type I. This implies that the G-C^{∗}-algebras KK^{G}-equivalent to one in T_{1} form a localising_{ℵ_{1}} subcategory of KK^{G}.

Remark 3.11. In the non-equivariant case, any C^{∗}-algebra in the bootstrap class is KK-equivalent to a commutative one. This criterion fails already for G = U(1), as shown by a counterexample in [10]. Since the bootstrap class is the smallest localising subcategory containing ℂ, it follows that the commutative C^{∗}-algebras do not form a localising subcategory. Thus k̂k^{G} is not triangulated: it lacks cones for some maps. In this case, the equivariant bootstrap class is already generated by ℂ and contains all U(1)-actions on C^{∗}-algebras in the non-equivariant bootstrap category. It is shown in [10] that the U(1)-equivariant K-theory of a suitable Cuntz–Krieger algebra with its natural gauge action cannot arise from any U(1)-action on a locally compact space.

Corollary 3.12. The restriction and induction functors KK^{G} → KK^{H} and KK^{H} → KK^{G} for a closed subgroup H in a compact Lie group G restrict to functors between the bootstrap classes in KK^{G} and KK^{H}.

Proof. Restriction does not change the underlying C^{∗}-algebra and thus preserves the property of being type I. Induction maps elementary H-C^{∗}-algebras to elementary G-C^{∗}-algebras, is triangulated, and commutes with direct sums. Hence it maps B^{H} to B^{G}.

Proposition 3.13. An object of B^{G} is compact_{ℵ_{1}} if and only if it is dualisable, if and only if it belongs to the thick subcategory of B^{G} (or of KK^{G}) generated by the elementary G-C^{∗}-algebras.

Proof. The tensor unit ℂ is compact_{ℵ_{1}} because KK_{∗}^{G}(ℂ, A) ≅ K_{∗}^{G}(A) ≅ K_{∗}(G⋉ A) is countable for all G-C^{∗}-algebras A, and the functors A ↦ G ⋉ A and K_{∗} are well-known to commute with coproducts. Furthermore, the tensor product in KK^{G} commutes with coproducts in both variables. Using this, we show that dualisable objects of B^{G} are compact_{ℵ_{1}} . If A is dualisable with dual A^{∗}, then KK^{G}(A, B) ≅ KK^{G}(ℂ, A^{∗} ⊗ B), and since ℂ is compact_{ℵ_{1}} and ⊗ commutes with countable direct sums, it follows that A is compact_{ℵ_{1}} . It follows from [8, Corollary 2.2] that elementary G-C^{∗}-algebras are dualisable and hence compact_{ℵ_{1}} . A compact group has only at most countably many compact subgroups by Lemma 3.14 below; and any of them has at most finitely many projective representations. Hence the set of elementary G-C^{∗}-algebras is at most countable. Therefore, B^{G} is compactly_{ℵ_{1}} generated in the sense of [7, Definition 2.1]. By [7, Corollary 2.4] an object of B^{G} is compact_{ℵ_{1}} if and only if it belongs to the thick subcategory generated by the elementary G-C^{∗}-algebras. The Brown Representability Theorem [7, Corollary 2.2] shows that for every compact_{ℵ_{1}} object A of B^{G} there is a functor Hom(A, ␣) from B^{G} to B^{G} such that

KK^{G}(A ⊗ B, D) ≅ KK^{G}(B, Hom(A, D))

for all B, D in B^{G}. Using exactness properties of the internal Hom functor in the first variable, we then show that the class of dualisable objects in B^{G} is thick (see [7, Section 2.3]). Thus all objects of the thick subcategory generated by the elementary G-C^{∗}-algebras are dualisable.

The following lemma is well-known, see [25].

Lemma 3.14. A compact Lie group has at most countably many conjugacy classes of closed subgroups.

Proof. Let H be a closed subgroup of a compact Lie group G. By the Mostow Embedding Theorem, G/H embeds into a linear representation of G, that is, H is a stabiliser of a point in some linear representation of G. Up to isomorphism, there are only countably many linear representations of G. Each linear representation has finite orbit type, that is, it admits only finitely many different conjugacy classes of stabilisers. Hence there are altogether at most countably many conjugacy classes of closed subgroups in G.

Proof of Theorem 3.5. Let G be a Hodgkin Lie group. The main result of [23] says that A belongs to the localising subcategory of KK^{G} generated by ℂ if and only if A ⋊ G belongs to the non-equivariant bootstrap category (this is special for Hodgkin Lie groups). Since this covers all elementary G-C^{∗}-algebras, we conclude that the localising subcategory generated by ℂ contains B^{G} and is, therefore, equal to B^{G}. The same argument as in the proof of Proposition 3.13 shows that the following are equivalent for an object A of B^{G}:

• A is dualisable;

• A is compact_{ℵ_{1}} ;

• A belongs to the thick subcategory generated by ℂ.

This finishes the proof of Theorem 3.5.

So far we always used the bootstrap class, which is the domain where a Universal Coefficient Theorem holds. The next proposition is a side remark showing that we may also use the domain where a Künneth formula holds.

Definition 3.15. An object A ∈ KK^{G} satisfies the Künneth formula if K_{∗}^{G}(A⊗ B) = 0 for all B that satisfy K_{∗}^{G}(C ⊗B) = 0 for all elementary G-C^{∗}-algebras C.

By results of [20, 24], the assumption in Definition 3.15 is necessary and sufficient for a certain natural spectral sequence that computes K_{∗}^{G}(A ⊗ B) from KK_{∗}^{G}(C, A) and KK_{∗}^{G}(C, B) for elementary C to converge for all B; we have no need to describe this spectral sequence.

Proposition 3.16. Let A ∈ KK^{G} be dualisable with dual A^{∗}. If A or A^{∗} satisfies a Künneth formula, then both A and A^{∗} belong to B^{G}, and vice versa.

Proof. Since B^{G} is generated by the elementary G-C^{∗}-algebras, KK_{∗}^{G}(C, B) = 0 for all elementary G-C^{∗}-algebras C if and only if KK_{∗}^{G}(C, B) = 0 for all C ∈ B^{G}. Any elementary G-C^{∗}-algebra C is dualisable with a dual in B^{G}. Hence K_{∗}^{G}(C ⊗ B) ≅ KK_{∗}^{G}(C^{∗}, B) = 0 for elementary C if KK_{∗}^{G}(C′, B) = 0 for all elementary G-C^{∗}-algebras C′; conversely KK_{∗}^{G}(C, B) ≅ K_{∗}^{G}(C^{∗} ⊗ B) = 0 for elementary C if K_{∗}^{G}(C′⊗ B) = 0 for all elementary G-C^{∗}-algebras C′. Let us denote the class of G-C^{∗}-algebras with these equivalent properties B^{G,⊥} . It follows from [20, Theorem 3.16] that (B^{G}, B^{G,⊥} ) is a complementary pair of localising subcategories. In particular, if KK_{∗}^{G}(A, B) = 0 for all B ∈ B^{G,⊥} , then A ∈ B^{G}. Now assume, say, that A satisfies a Künneth formula. Then KK_{∗}^{G}(A^{∗}, B) ≅ K_{∗}^{G}(A⊗B) = 0 for all B ∈ B^{G,⊥} . Thus A^{∗} ∈ B^{G}. Then K_{∗}^{G}(A^{∗} ⊗B) = 0 for all B ∈ B^{G,⊥} because the class of C with K_{∗}^{G}(C ⊗B) = 0 is localising and contains all elementary C if B ∈ B^{G,⊥} . As above, this implies (A^{∗})^{∗} = A ∈ B^{G}.

The proof of Theorem 3.5 above used that, for a Hodgkin Lie group, B^{G} is already generated by ℂ. For more general groups, we also expect that fewer generators suffice to generate B^{G}. But we only need and only prove a result about topologically cyclic groups here. A locally compact group G is called topologically cyclic if there is an element g ∈ G that generates a dense subgroup of G. A topologically cyclic group is necessarily Abelian. We are interested in topologically cyclic, compact Lie groups here. A compact Lie group is topologically cyclic if and only if it is isomorphic to 𝕋^{r} × F for some r ≥ 0 and some finite cyclic group F (possibly the trivial group), where 𝕋 = ℝ/ℤ ≅ U(1). Here we use that any extension 𝕋^{r} ↣ E ↠ F for a finite cyclic group F splits. This also implies that any projective representation of a finite cyclic groups is a representation.

Theorem 3.17. Let G be a topologically cyclic, compact Lie group. Then the bootstrap class B^{G} ⊆ KK^{G} is already generated by the finitely many G-C^{∗}-algebras C(G/H) for all open subgroups H ⊆ G. Furthermore, an object of B^{G} is compact_{ℵ_{1}} if and only if it is dualisable if and only if it belongs to the thick subcategory generated by C(G/H) for open subgroups H ⊆ G.

Proof. The second statement about compact_{ℵ_{1}} objects in B^{G} follows from the first one and [7, Corollary 2.4], compare the proof of Proposition 3.13. Thus it suffices to prove that the objects C(G/H) for open subgroups already generate B^{G}. For this, we use an isomorphism G ≅ 𝕋^{r} × F for some r ≥ 0 and some finite cyclic subgroup F. Let us first consider the special case r = 0, that is, G is a finite cyclic group. In this case, any subgroup of G is open and again cyclic. We observed above that cyclic groups have no non-trivial projective representations. Thus any elementary G-C^{∗}-algebra is Morita equivalent to C(G/H) for some open subgroup H in G. Hence the assertion of the theorem is just the definition of B^{G} in this case. If F is trivial, then the assertion follows from Theorem 3.5. Now we consider the general case where both F and 𝕋^{r} are non-trivial. The Pontryagin dual Ĝ of G is isomorphic to the discrete group ℤ^{r} × F. If A is a G-C^{∗}-algebra, then G ⋉ A carries a canonical action of Ĝ called the dual action. Similarly, Ĝ ⋉ A for a Ĝ-C^{∗}-algebra A carries a canonical dual action of G. This provides functors KK^{G} → KKĜ and KKĜ → KK^{G}. Baaj–Skandalis duality says that they are inverse to each other up to natural equivalence (see [2, Section 6]). Since both functors are triangulated, this is an equivalence of triangulated categories. If A is type I, then so is G ⋉ A. Hence all objects in BĜ ⊆ KKĜ are KKĜequivalent to a Ĝ-action on a type I C^{∗}-algebra by Theorem 3.10. The group Ĝ is Abelian and hence satisfies a very strong form of the Baum– Connes conjecture: it has a dual Dirac morphism and γ = 1 in the sense of [22, Definition 8.1]. From this it follows that any Ĝ-C^{∗}-algebra A belongs to the localising subcategory of KKĜ that is generated by IndĜ ; Ĥ A for finite subgroups Ĥ ⊆ Ĝ (this is shown as in the proof of [22, Theorem 9.3]). The finite subgroups in ℤ^{r} ×F̂ are exactly the subgroups of F̂, of course. Since we have induction in stages, we may assume Ĥ = F̂. Thus the subcategory of type I Ĝ-C^{∗}-algebras is already generated by IndĜ ; F̂ A for type I F̂-C^{∗}-algebras A. Since F̂ is a finite cyclic group, the discussion above shows that the category of type I F̂-C^{∗}-algebras A is already generated by C_{0}(F̂/Ĥ) for subgroups Ĥ ⊆ F̂. Thus BĜ is generated by the Ĝ-C^{∗}-algebras IndĜ ; F̂ C_{0}(F̂/Ĥ) ≅ C_{0}(Ĝ/Ĥ). The finite subgroups Ĥ ⊆ Ĝ are exactly the orthogonal complements of (finiteindex) open subgroups H ⊆ G. Now G ⋉ C_{0}(G/H) is Morita equivalent to C^{∗}(H) ≅ C_{0}(Ĝ/Ĥ) for any open subgroup H ⊆ G, where Ĥ ⊆ Ĝ denotes the orthogonal complement of H in Ĝ. The dual action on C_{0}(Ĝ/Ĥ) comes from the translation action of Ĝ. Thus the G- and Ĝ-C^{∗}-algebras C_{0}(G/H) and C_{0}(Ĝ/Ĥ) correspond to each other via Baaj–Skandalis duality. We conclude that the G-C^{∗}-algebras C_{0}(G/H) for open subgroups H ⊆ G generate B^{G}.

Let G be topologically cyclic, say, G ≅ 𝕋^{r} × ℤ/k for some r ≥ 0, k ≥ 1. Then open subgroups of G correspond to subgroups of ℤ/k and thus to divisors d of k. The representation ring of G is

R(G) ≅ R(𝕋^{r}) ⊗ R(ℤ/k) ≅ ℤ[x_{1}, …, x_{r}, (x_{1} ⋯ x_{r})^{−1} ] ⊗ ℤ[t]/(t^{k} − 1). (3.1)

Let

t^{k} − 1 = ∏_{d|k} Φ_{d}(t)

be the decomposition into cyclotomic polynomials. Each factor Φ_{d} generates a minimal prime ideal of R(G), and these are all minimal prime ideals of R(G).

The localisation at (Φ_{d}) gives the field ℚ(θ_{d})(x_{1}, …, x_{r}) of rational functions in r variables over the cyclotomic field ℚ(θ_{d}), and the product of these localisations is the total ring of fractions of R(G),

S^{−1} R(G) = ∏_{d|k} ℚ(θ_{d})(x_{1}, …, x_{r}).

(Compare Lemma 3.1.)

Lemma 3.18. Let H ⊊ G be a proper open subgroup. The canonical map

R(G) → KK_{0}^{G}(C(G/H), C(G/H))

from the exterior product in KK^{G} factors through the restriction map R(G) → R(H). The image of C(G/H) in the localisation of KK^{G} at the prime ideal (Φ_{k}) vanishes.

Proof. The exterior product of the identity map on C(G/H) and ξ ∈ R(G) ≅ KK_{0}^{G}(ℂ, ℂ) is given by the geometric correspondence G/H = G/H = G/H with the class p^{∗}(ξ) ∈ K_{G}^{0}(G/H), where p: G/H → pt is the constant map. Now identify K_{G}^{0}(G/H) ≅ K_{H}^{0} (pt) ≅ R(H) and p^{∗} with the restricton map R(G) → R(H) to get the first statement. We have H ≅ 𝕋^{r} × ℤ/d embedded via (x, j) ↦ (x, jk/d) into G ≅ 𝕋^{r} × ℤ/k. If H ≠ G, then d ≠ k. The restriction map R(G) → R(H) annihilates the polynomial (t^{k} − 1)/Φ_{k} = ∏ _{d|k,d≠k} Φ_{d}. This polynomial does not belong to the prime ideal (Φ_{k}) and hence becomes invertible in the localisation of R(G) at (Φ_{k}). Since an invertible endomorphism can only be zero on the zero object, C(G/H) becomes zero in the localisation of KK^{G} at (Φ_{k}).

3.2 Localisation of the bootstrap class

Proposition 3.19. Let G ≅ 𝕋^{r} × ℤ/k be topologically cyclic. Let B_{d}^{G} be the thick subcategory of dualisable objects in the bootstrap class B^{G} ⊆ KK^{G}. Any object in the localisation of B_{d}^{G} at the prime ideal (Φ_{k}) in R(G) is isomorphic to a finite direct sum of suspensions of ℂ.

Proof. By Theorem 3.17 an object of B^{G} is dualisable if and only if it belongs to the thick subcategory generated by C(G/H) for open subgroups H ⊆ G. Lemma 3.18 shows that all of them except ℂ = C(G/G) become zero when we localise at (Φ_{k}). Hence the image of B_{d}^{G} in the localisation is contained in the thick subcategory generated by ℂ. We must show that the objects isomorphic to a direct sum of suspensions of ℂ already form a thick subcategory in the localisation of KK^{G} at (Φ_{k}). The graded endomorphism ring of ℂ in this localisation is

KK_{∗}^{G}(ℂ, ℂ) ⊗_{R(G)} R(G)_{(Φ_{k})} ≅ ℚ(θ_{k})(x_{1}, …, x_{r})[β, β^{−1} ]

with β of degree two generating Bott periodicity. It is crucial that KK_{∗}^{G}(ℂ, ℂ) ≅ F [β, β^{−1} ] for a field F := ℚ(θ_{k})(x_{1}, …, x_{r}). The following argument only uses this fact. We map a finite direct sum A = ⨁ _{i∈I} ℂ[ε_{i}] of suspensions of ℂ to the ℤ/2-graded F -vector spaces V (A) with basis I and generators of degree ε_{i}. For two such direct sums, KK_{0}^{G}(A, B) is isomorphic to the space of gradingpreserving F -linear maps V (A) → V (B) because this clearly holds for a single summand. Now let f ∈ KK_{0}^{G}(A, B) and consider the associated linear map V (f): V (A) → V (B). Choose a basis for the kernel of V (f) of homogeneous elements and extend it to a homogeneous basis for V (A), and extend the resulting basis for the image of V (f) to a homogeneous basis of V (B). This provides isomorphisms V (A) ≅ V_{0} ⊕ V_{1}, V (B) ≅ W_{1} ⊕ W_{2} such that f|_{V_{0}} = 0, f(V_{1}) = W_{1} and f|_{V_{1}} : V_{1} → W_{1} is an isomorphism. The chosen bases describe how to lift the ℤ/2-graded vector spaces V_{i} and W_{i} to direct sums of suspensions of ℂ. Thus the map f is equivalent to a direct sum of three maps f_{0} ⊕ f_{1} ⊕ f_{2} with f_{0} : A_{0} → 0 mapping to the zero object, f_{1} invertible, and f_{2} : 0 → B_{2} with domain the zero object. The mapping cone of f_{0} is the suspension of A_{0}, the cone of f_{2} is B_{2}, and the cone of f_{1} is zero. Hence the cone is again a direct sum of suspensions of ℂ. Furthermore, any idempotent endomorphism has a range object. Thus the direct sums of suspensions of ℂ already form an idempotent complete triangulated category. As a consequence, any object in the thick subcategory generated by ℂ is isomorphic to a direct sum of copies of ℂ.

Proposition 3.20. Let G be a Hodgkin Lie group. Let B_{d}^{G} be the thick subcategory of dualisable objects in the bootstrap class B^{G} ⊆ KK^{G}. Any object in the localisation of B_{d}^{G} at S is isomorphic to a finite direct sum of suspensions of ℂ.

Proof. Theorem 3.5 shows that B_{d}^{G} is the thick subcategory of KK^{G} generated by ℂ. The localisation F := S^{−1} R(G) is a field because G is connected, and the graded endomorphism ring of ℂ in the localisation of KK^{G} at S is F [β, β^{−1} ] with β the generator of Bott periodicity. Now the argument is finished as in the proof of Proposition 3.19.

Remark 3.21. The localisations above use the groups KK^{G}(A, B) ⊗_{R(G)} S^{−1} R(G) for some multiplicatively closed subset S ⊆ End(𝟙) = R(G), following [17]. A drawback of this localisation is that the canonical functor KK^{G} → S^{−1} KK^{G} does not commute with (countable) coproducts. This is why Propositions 3.19 and 3.20 are formulated only for B_{d}^{G} and not for all of B^{G}. Another way to localise B^{G} at S is described in [7, Theorem 2.33]. Both localisations agree on B_{d}^{G} by [7, Theorem 2.33.h]. The construction in [7] has the advantage that the canonical functor from KK^{G} to this localisation commutes with small_{ℵ_{1}} (that is, countable) coproducts. Hence analogues of Propositions 3.19 and 3.20 hold for the whole bootstrap category B^{G}, with small_{ℵ_{1}} coproducts of suspensions of ℂ instead of finite direct sums of suspensions of ℂ.

3.3 The Lefschetz index computation using localisation

Now we have all the tools available to formulate and prove a Lefschetz index formula for general compact Lie groups. We first prove Theorem 3.4, which deals with endomorphisms of objects in the thick subcategory generated by ℂ. Then we formulate and prove the general Lefschetz index formula.

Proof of Theorem 3.4. Since A belongs to the thick subcategory generated by ℂ, it is dualisable in KK^{G} by Proposition 3.13. Hence tr(f) ∈ R(G) is defined for f ∈ KK_{0}^{G}(A, A). The image of tr(f) in S^{−1} R(G) is the Lefschetz index of the image of f in the localisation of KK^{G} at S. The localisation S^{−1} R(G) is a product of fields. It is more convenient to compute each component separately. This means that we localise at larger multiplicatively closed subsets S̄ such that S̄^{−1} R(G) is one of the factors of S^{−1} R(G). In this localisation, the endomorphisms of ℂ form a field again, not a product of fields. If our trace formula holds for all these localisations, it also holds for S^{−1} R(G). Since the endomorphisms of ℂ form a field, the same argument as in the proof of Proposition 3.19 show that, in this localisation, A is isomorphic to a finite sum of copies of suspensions of ℂ. Write A ≅ ⨁ _{i=1}^{n} A_{i} with A_{i} ≅ ℂ[ε_{i}] in S^{−1} KK^{G} for some ε_{i} ∈ ℤ/2. Then f becomes a matrix (f_{ij} ) with f_{ij} ∈ S^{−1} KK_{0}^{G}(A_{j} , A_{i}). The dual of A_{i} ≅ ℂ[ε_{i}] is A_{i}^{∗} ≅ ℂ[ε_{i}] ≅ A_{i}, and the unit and counit of adjunction ℂ ⇆ ℂ[ε_{i}] ⊗ ℂ[ε_{i}] are the canonical isomorphism and its inverse with sign (−1)^{ε_{i}} , respectively; the sign is necessary because the exterior product is graded commutative. Hence the dual of A is isomorphic to A, with unit and counit

ℂ ⇆ A ⊗ A ≅ ⨁_{i,j=1}^{n} A_{i} ⊗ A_{j}

the sum of the canonical isomorphisms ℂ ⇆ A_{i} ⊕ A_{i}, up to signs, and the zero maps ℂ ⇆ A_{i} ⊕ A_{j} for i ≠ j. Thus the Lefschetz index of f is the sum ∑ _{i=1}^{n} (−1)^{ε_{i}} f_{ii} [ε_{i}] as an element in S^{−1} KK_{0}^{G}(ℂ, ℂ). This is exactly the supertrace of f acting on S^{−1} K_{∗}^{G}(A) ≅ ⨁ _{i=1}^{n} S^{−1} R(G)[ε_{i}].

Let G be a general compact Lie group. Let C_{G} denote the set of conjugacy classes of Cartan subgroups of G in the sense of [28, Definition 1.1]. Such subgroups correspond bijectively to conjugacy classes of cyclic subgroups in the finite group G/G^{0}, where G^{0} denotes the connected component of the identity element in G. Thus C_{G} is a non-empty, finite set, and it has a single element if and only if G is connected. The support of a prime ideal 𝔭 in R(G) is defind in [28] as the smallest subgroup H such that 𝔭 comes from a prime ideal in R(H) via the restriction map R(G) → R(H). Given any Cartan subgroup H, there is a unique minimal prime ideal with support H, and this gives a bijection between C_{G} and the set of minimal prime ideals in R(G) (see [28, Proposition 3.7]). More precisely, if H ⊆ G is a Cartan subgroup, then H is topologically cyclic and hence H ≅ 𝕋^{r} × ℤ/k for some r ≥ 0, k ≥ 1. We described a prime ideal (Φ_{k}) in R(H) before Lemma 3.18, and its preimage in R(G) is a minimal prime ideal 𝔭_{H} in R(G). The total ring of fractions S^{−1} R(G) is a product of fields by Lemma 3.1. We can make this more explicit:

S^{−1} R(G) ≅ ∏_{H∈C_{G}} F (R(G)/𝔭_{H}),

where F (␣) denotes the field of fractions for an integral domain.

Definition 3.22. Let A be dualisable in B^{G} ⊆ KK^{G}, let φ ∈ KK_{0}^{G}(A, A), and let H ∈ C_{G}. Let F := F (R(G)/𝔭_{H}) and let K_{H} (A) := K_{∗}^{H} (A) ⊗_{R(H)} F, considered as a ℤ/2-graded F -vector space. Let K_{H} (φ) be the gradingpreserving F -linear endomorphism of K_{H} (A) induced by φ.

Theorem 3.23. Let A be dualisable in B^{G} ⊆ KK^{G}, let φ ∈ KK_{0}^{G}(A, A), and let H ∈ C_{G}. Then the image of tr(φ) in F (R(G)/𝔭_{H}) is the supertrace of K_{H} (φ).

Proof. The map R(G) → F (R(G)/𝔭_{H}) factors through the restriction homomorphism R(G) → R(H) because 𝔭_{H} is supported in H. Restricting the group action to H maps the bootstrap category in KK^{G} into the bootstrap category in KK^{H} by Corollary 3.12, and commutes with taking Lefschetz indices because restriction is a tensor functor. Hence we may replace G by H and take φ ∈ KK_{0}^{H} (A, A) throughout. Since H is topologically cyclic, Proposition 3.19 applies. It shows that in the localisation of KK^{H} at 𝔭_{H}, any dualisable object in B^{G} becomes isomorphic to a finite direct sum of suspensions of ℂ. Now the argument continues as in the proof of Theorem 3.4 above.

4 Hattori–Stallings traces

Before we found the above approach through localisation, we developed a different trace formula where, in the case of a Hodgkin Lie group, the trace is identified with the Hattori–Stallings trace of the R(G)-module map K_{∗}^{G}(f) on K_{∗}^{G}(A). We briefly sketch this alternative formula here, although the localisation approach above seems much more useful for computations. The Hattori–Stallings trace has the advantage that it obviously belongs to R(G). We work in the general setting of a tensor triangulated category (T , ⊗, 𝟙). We assume that T satisfies additivity of traces, that is:

Assumption 4.1. Let A → B → C → A[1] be an exact triangle in T and assume that A and B are dualisable. Assume also that the left square in the following diagram

\begin{matrix}A & B & C & A[1] \\ A & B & C & A[1] \end{matrix} f_{A} f_{B} f_{C} f_{A} [1]

commutes. Then C is dualisable and there is an arrow f_{C} : C → C such that the whole diagram commutes and tr(f_{C}) − tr(f_{B}) + tr(f_{A}) = 0.

Additivity of traces holds in the bootstrap category B^{G} ⊆ KK^{G}. The quickest way to check this is the localisation formula for the trace in Theorem 3.23. It shows that B^{G} satisfies even more: tr(f_{C}) − tr(f_{B}) + tr(f_{A}) = 0 holds for any arrow f_{C} that makes the diagram commute. There are several more direct ways to verify additivity of traces, but all require significant work which we do not want to get into here. The axioms worked out by J. Peter May in [19] are lengthy and therefore rather unpleasant to check by hand. In a previous manuscript we embedded the localising subcategory of KK^{G} generated by ℂ into a category of module spectra. Since additivity is known for categories of module spectra, this implies the required additivity result at least for this smaller subcategory. Another way would be to show that additivity of traces follows from the derivator axioms and to embed KK^{G} into a triangulated derivator. In the following, we will just assume additivity of traces and use it to compute the trace. Let

R := T_{∗}(𝟙, 𝟙) = ⨁_{n∈ℤ} T_{n}(𝟙, 𝟙)

be the graded endomorphism ring of the tensor unit. It is graded-commutative provided T satisfies some very basic compatibility axioms; see [29] for details. If A is any object of T , then M(A) := T_{∗}(𝟙, A) = ⨁ _{n∈ℤ} T_{n}(𝟙, A) is an R-module in a canonical way, and an endomorphism f ∈ T_{n}(A, A) yields a degree-n endomorphism M(f) of M(A). We will prove in Theorem 4.2 below that, under some assumptions, the trace of f equals the Hattori–Stallings trace of M(f) and, in particular, depends only on M(f). Before we can state our theorem, we must define the Hattori–Stallings trace for endomorphisms of graded modules over graded rings. This is well-known for ungraded rings (see [3]). The grading causes some notational overhead. Let R be a (unital) graded-commutative graded ring. A finitely generated free R-module is a direct sum of copies of R[n], where R[n] denotes R with degree shifted by n, that is R[n]_{i} = R_{n+i} . Let F: P → P be a module endomorphism of such a free module, let us assume that F is homogeneous of degree d. We use an isomorphism

P ≅ ⨁_{i=1}^{r} R[n_{i}]  (4.1)

to rewrite F as a matrix (f_{ij} )_{1≤i,j≤r} , where f_{ij} : R[n_{j}] → R[n_{i}] are R-module homomorphisms of degree d. The entry f_{ij} is given by right multiplication by some element of R of degree n_{i} − n_{j} + d. The (super)trace trF is defined as

trF := ∑_{i=1}^{r} (−1)^{n_{i}} trf_{ii};

this is an element of R of degree d. It is straightforward to check that trF is well-defined, that is, independent of the choice of the isomorphism in (4.1). Here we use that the degree-zero part of R is central in R (otherwise, we still get a well-defined element in the commutator quotient R_{d}/[R_{d}, R_{0}]). Furthermore, if we shift the grading on P by n, then the trace is multiplied by the sign (−1)^{n} – it is a supertrace. If P is a finitely generated projective graded R-module, then P ⊕ Q is finitely generated and free for some Q, and for an endomorphism F of P we let

trF := tr(F ⊕ 0: P ⊕ Q → P ⊕ Q).

This does not depend on the choice of Q. A finite projective resolution of a graded R-module M is a resolution

⋯→ P_{ℓ} ⟶^{d_{ℓ}} P_{ℓ−1} −⟶^{d_{ℓ−1}} ⋯−→^{d_{1}} P_{0} −→^{d_{0}} M  (4.2)

of finite length by finitely generated projective graded R-modules P_{j}. We assume that the maps d_{j} have degree one (or at least odd degree). Assume that M has such a resolution and let f: M → M be a module homomorphism. Lift f to a chain map f_{j} : P_{j} → P_{j}, j = 0, …, ℓ. We define the Hattori–Stallings trace of f as

tr(f) = ∑_{j=0}^{ℓ} tr(f_{j}).

It may be shown that this trace does not depend on the choice of resolution. It is important for this that we choose d_{j} of degree one. Since shifting the degree by one alters the sign of the trace of an endomorphism, the sum in the definition of the trace becomes an alternating sum when we change conventions to have even-degree boundary maps d_{j}. Still the trace changes sign when we shift the degree of M.

Theorem 4.2. Let F ∈ T (A, A) be an endomorphism of some object A of T . Assume that A belongs to the localising subcategory of T generated by 𝟙. If the graded R-module M(A) := T_{∗}(𝟙, A) has a finite projective resolution, then A is dualisable in T and the trace of F is equal to the Hattori–Stallings trace of the induced module endomorphism T_{∗}(𝟙, f) of M(A).

Proof. Our main tool is the phantom tower over A, which is constructed in [20]. We recall some details of this construction.

Let M^{⊥} be the functor from finitely generated projective R-modules to T defined by the adjointness property T (M^{⊥} (P), B) ≅ T (P, M(B)) for all B ∈ T . The functor M^{⊥} maps the free rank-one module R to 𝟙, is additive, and commutes with suspensions; this determines M^{⊥} on objects. Since R = T_{∗}(𝟙, 𝟙), T_{∗}(M^{⊥} (P_{1}), M^{⊥} (P_{2})) is isomorphic (as a graded Abelian group) to the space of R-module homomorphisms P_{1} → P_{2}. Furthermore, we have canonical isomorphisms M(M^{⊥} (P)) ≅ P for all finitely generated projective R-modules P. By assumption, M(A) has a finite projective resolution as in (4.2). Using M^{⊥} , we lift it to a chain complex in T , with entries P̂_{j} := M^{⊥} (P_{j}) and boundary maps d̂_{j} := M^{⊥} (d_{j}) for j ≥ 1. The map d̂_{0}: P̂_{0} → A is the pre-image of d_{0} under the adjointness isomorphism T (M^{⊥} (P), B) ≅ T (P, M(B)). We get back the resolution of modules by applying M to the chain complex (P̂_{j}, d̂_{j}). Next, it is shown in [20] that we may embed this chain complex into a diagram

\begin{aligned}A = N_{0} & & N_{1} & & N_{2} & & N_{3} & & ⋯ \\ & P̂_{0} & & P̂_{1} & & P̂_{2} & & P̂_{3} & & ⋯ \end{aligned} ι_{0}^{1} ι_{1}^{2} ι_{2}^{3} ε_{0} ε_{1} ε_{2} d̂_{0} = π_{0} π_{1} π_{2} π_{3} d̂_{1} d̂_{2} d̂_{3}

(4.3)

where the wriggly lines are maps of degree one; the triangles involving d̂_{j} commute; and the other triangles are exact. This diagram is called the phantom tower in [20]. Since P̂_{j} = 0 for j > ℓ, the maps ι_{j}^{j+1} are invertible for j > ℓ. Furthermore, a crucial property of the phantom tower is that these maps ι_{j}^{j+1} are phantom maps, that is, they induce the zero map on T_{∗}(𝟙, ␣). Together, these facts imply that M(N_{j}) = 0 for j > ℓ. Since we assumed 𝟙 to be a generator of T , this further implies N_{j} = 0 for j > ℓ. Therefore, A ∈ ⟨𝟙⟩, so that A is dualisable as claimed. Next we recursively extend the endomorphism F of A = N_{0} to an endomorphism of the phantom tower. We start with F_{0} = F: N_{0} → N_{0}. Assume F_{j} : N_{j} → N_{j} has been constructed. As in [20], we may then lift F_{j} to a map F̂_{j} : P̂_{j} → P̂_{j} such that the square

\begin{matrix}P̂_{j} & N_{j} \\ P̂_{j} & N_{j} \end{matrix} π_{j} F̂_{j} π_{j} F_{j}

commutes. Now we apply additivity of traces (Assumption 4.1) to construct an endomorphism F_{j+1} : N_{j+1} → N_{j+1} such that (F̂_{j}, F_{j} , F_{j+1} ) is a triangle morphism and tr(F_{j}) = tr(F̂_{j}) + tr(F_{j+1} ). Then we repeat the recursion step with F_{j+1} and thus construct a sequence of maps F_{j}. We get

tr(F) = tr(F_{0}) = tr(F̂_{0}) + tr(F_{1}) = ⋯= tr(F̂_{0}) + ⋯+ tr(F̂_{ℓ}) + tr(F_{ℓ+1} ).

Since N_{ℓ+1} = 0, we may leave out the last term. Finally, it remains to observe that the trace of F̂_{j} as an endomorphism of P̂_{j} agrees with the trace of the induced map on the projective module P_{j}. Since both traces are additive with respect to direct sums of maps, the case of general finitely generated projective modules reduces first to free modules and then to free modules of rank one. Both traces change by a sign if we suspend or desuspend once, hence we reduce to the case of endomorphisms of 𝟙, which is trivial. Hence the computation above does indeed yield the Hattori–Stallings trace of M(A) as asserted.

Remark 4.3. Note that if a module has a finite projective resolution, then it must be finitely generated. Conversely, if the graded ring R is coherent and regular, then any finitely generated module has a finite projective resolution. (Regular means that every finitely generated module has a finite length projective resolution; coherent means that every finitely generated homogeneous ideal is finitely presented – for instance, this holds if R is (graded) Noetherian; coherence implies that any finitely generated graded module has a resolution by finitely generated projectives.) Moreover, if R is coherent then the finitely presented R-modules form an abelian category, and this implies (by an easy induction on the triangular length of A) that for every A ∈ ⟨𝟙⟩ = (⟨𝟙⟩_{loc})_{d} the module M(A) is finitely presented and thus a fortiori finitely generated. If R is also regular, each such M(A) has a finite projective resolution. In conclusion: if R is regular and coherent, an object A ∈ ⟨𝟙⟩_{loc} is dualisable if and only if the graded R-module M(A) has a finite projective resolution.

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Ivo Dell’Ambrogio Laboratoire Paul Painlevé Université de Lille 1 Cité Scientifique Bât. M2 F-59655 Villeneuve d’Ascq Cedex, France Heath Emerson Dept. of Math. & Statistics University of Victoria PO BOX 3045 STN CSC Victoria, B. C. Canada V8W 3P4

Ralf Meyer

Mathematisches Institut Universität Göttingen Bunsenstraße 3–5 37073 Göttingen Germany 194

Documenta Mathematica 19 (2014)