Homology, Homotopy and Applications, vol.1, No.8, 1999, pp.169–185 ISSN 1512–0139

OPERATIONS AND QUANTUM DOUBLES IN COMPLEX ORIENTED COHOMOLOGY THEORY

VICTOR M BUCHSTABER and NIGEL RAY

(communicated by Hvedri Inassaridze)

Abstract We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the Landweber-Novikov algebra, as represented by a subalgebra of operations in double complex cobordism. We work in the context of Boardman’s eightfold way, which offers an important framework for clarifying the relationship between quantum doubles and the standard machinery of Hopf algebroids of homology cooperations. These considerations give rise to novel structures in double cohomology theory, and we explore the twist operation and extensions of the quantum antipode by way of example.

1. Introduction

In his pioneering work [8], Drinfeld introduced the quantum double construction D(H) for a Hopf algebra H, as an aid to the solution of the Yang-Baxter equations. His work aroused interest in many areas of mathematics, and applications to algebraic topology were begun by Novikov, who proved in [17] that whenever H acts appropriately on a ring R, then the smash product R # H (in the sense of [24]) may be represented as a ring of operators on R. Novikov therefore referred to R#H as the operator double, and observed that his construction applied to the adjoint action of H on its dual H^{∗} whenever H is cocommutative, thereby exhibiting D(H) as the operator double H^{∗} # H. Authors in other fields have recorded similar phenomena, and the subject of doubling constructions is currently very active.

Novikov was motivated by the algebra of cohomology operations in complex cobordism theory, which he constructed as an operator double by choosing H to be the Landweber-Novikov algebra S^{∗}, and R the complex cobordism ring Ω_{∗}^{U}. This viewpoint was suggested by the description of S^{∗} as an algebra of differential operators on a certain algebraic group, due to Buchstaber and Shokurov [4]. Since the Landweber-Novikov algebra is cocommutative, its quantum double is also an operator double, and Buchstaber used this property in [5] to prove the remarkable fact that D(S^{∗}) may be faithfully represented in A_{DU}^{∗}, the ring of operations in double complex cobordism theory. In this sense, the algebraic and geometric doubling procedures coincide. The structures associated with D(S^{∗}) are very different from those traditionally studied by topologists, and one of our aims is to explore their implications for other complex oriented cohomology theories.

By way of historical comment, we recall that complex cobordism originally gained prominence during the late 1960s in the context of stable homotopy theory, but was superseded in the 1970s by Brown-Peterson cohomology because of the computational advantages gained by working with a single prime at a time. Ravenel’s book [20] gives an exhaustive account

Received 7 July 1999; published on 14 July 1999. 1991 Mathematics Subject Classification: 16W30, 55N20, 55N22, 55S25 Key words and phrases: Cohomology operation, complex cobordism, eightfold way, Hopf algebra, Landweber-Novikov algebra, quantum double \begin{matrix}c \\ ◯ \end{matrix} 1999, Victor M Buchstaber and Nigel Ray. Permission to copy for private use granted.

of these events. Work such as [13] has recently led to a resurgence of interest; this has been fuelled by mathematical physics, which was also the driving force behind Drinfeld’s original study of quantum groups.

So far as we are aware, double cobordism theories first appeared in [21], and in the associated work [22] where the double SU-cobordism ring was computed. Since neither source describes the foundations, we begin with a brief but rigorous treatment of the geometric and homotopy theoretic details, establishing double complex cobordism as the natural setting for the dual and the quantum double of the Landweber-Novikov algebra. We couch a major part of our exposition in terms of Boardman’s eightfold way [2], which we believe is still the best available framework for keeping track of all the actions and coactions that we need. We locate the subalgebra D(S^{∗}) of A_{DU}^{∗} in this context, and illustrate the novelty of the resulting structures by studying endomorphisms which extend its antipode. Much of the algebra may be interpreted more geometrically in terms of double complex structures on manifolds of flags, as described in [6].

We use the following notation and conventions without further comment.

We often confuse a complex vector bundle ρ with its classifying map into the appropriate Grassmannian, and write ξ(m) for the universal m-plane bundle over BU (m). We let ℂ^{m} denote the trivial m-plane bundle over any base; if ρ has dimension m and lies over a finite CW complex, we write ρ^{⊥} for the complementary (p − m)-plane bundle in some suitably high dimensional ℂ^{p}.

Our Hopf algebras are intrinsically geometrical and naturally graded by dimension, as are ground rings such as Ω_{∗}^{U}. Sometimes our algebras are not of finite type, and must therefore be topologized when forming duals and tensor products; this has little practical effect, but is explained with admirable care by Boardman [3], for example. Duals are invariably taken in the graded sense and we adapt our notation accordingly. Thus we write A_{U}^{∗} for the algebra of complex cobordism operations, and A_{∗}^{U} for its continuous dual Hom_{Ω_{∗}^{U}} (A_{U}^{∗}, Ω_{∗}^{U}), forcing us to write S^{∗} for the graded Landweber-Novikov algebra, and S_{∗} for its dual Hom_{ℤ}(S^{∗}, ℤ); neither of these notations is entirely standard.

We follow Sweedler’s convention of writing coproducts as δ(x) = ∑ x_{1}⊗x_{2} in any coalgbera. We refer readers to [1], [23], and [25] for comprehensive coverage of basic information

in algebraic topology. So far as Hopf algebras and their actions on rings are concerned, we recommend the books [12], [14], and [16] for background material and a detailed survey of the state of the art.

The authors are indebted to several colleagues for enjoyable and stimulating discussions which have contributed to this article, and in particular to Andrew Baker, Michael Boardman, Francis Clarke, Haynes Miller, Sergei Novikov, and Neil Strickland. Jack Morava deserves special thanks for encouraging us to prepare the material in its current form.

2. Double complex cobordism

In this section we outline the theory of double complex cobordism, considering both the manifold and homotopy theoretic viewpoints. We follow the lead of [5] by writing unreduced bordism functors as Ω_{∗}( ) when emphasizing their geometric origins; if these are of secondary importance, we revert to the notation T_{∗}( ), where T is the appropriate Thom spectrum.

The theory is based on manifolds M whose stable normal bundle is endowed with a fixed splitting ν ≅ ν_{ℓ} ⊕ν_{r} into a left and right component. We may invoke standard procedures [23] to construct the associated bordism and cobordism functors, and to describe them homotopy theoretically in terms of an appropriate Thom spectrum. Nevertheless we explain some of the details.

Given positive integers m and n, we write U(m, n) for the product of unitary groups U(m)× U(n), so that the classifying space BU (m, n) is canonically identified with BU (m) × BU (n).

Thus BU (m, n) carries the complex (m + n)-plane bundle ξ(m, n), defined as ξ(m) × ξ(n) and classified by the Whitney sum map BU (m, n) → BU (m + n). The standard inclusions of U(m) in O(2m) and of U(m, n) in U(m + 1, n + 1) induce a doubly indexed version of a (B, f) structure, but care is required to ensure that the maps are sufficiently compatible over m and n. There are product maps

BU (m, n) × BU (p, q) ⟶ BU (m + p, n + q)  (2.1)

induced by Whitney sum, whose compatibility is more subtle, but serves to confirm that the corresponding (B, f) cobordism theory is multiplicative; it is double complex cobordism theory, referred to in [21] as U × U theory.

We therefore define a double U-structure on M to consist of an equivalence class (M; ν_{ℓ}, ν_{r}) of lifts of ν to BU (m, n), for some values of m and n which are suitably large. The lifts provide the isomorphism ν ≅ ν_{ℓ} ⊕ ν_{r}, where ν_{ℓ} and ν_{r} are classified by the left and right projections onto the respective factors BU (m) and BU (n). If we wish to record a particular choice of m and n, we refer to the resulting U(m, n)-structure. Given such a structure on M, it is convenient to write χ(M) for the U(n, m)-structure (M; ν_{r}, ν_{ℓ}), induced by the switch map BU (m, n) → BU (n, m); we emphasise that M and χ(M) are generally inequivalent. If M has a U(m, n)-structure and N has a U(p, q)-structure, then the product U(m + p, n + q) structure is defined by (M × N; ν_{ℓ}^{M} × ν_{ℓ}^{N}, ν_{r}^{M} × ν_{r}^{N}).

A typical example is given by complex projective space CP^{n}, which admits the double U-structure (−kξ, (k − n − 1)ξ) for each integer k.

The compatibility required of the maps (2.1) is best expressed in the language of May’s coordinate-free functors (as described, for example, in [9]), which relies on an initial choice of infinite dimensional inner product space Z_{∞}, known as a universe. We assume here that Z_{∞} is complex. This language was originally developed to prove that the multiplicative structure of complex cobordism is highly homotopy coherent [15]; its usage establishes that the same is true for double complex cobordism, so long as we consistently embed our double U-manifolds in finite dimensional subspaces V ⊕ W of the universe Z_{∞} ⊕ Z_{∞}. We define the classifying space B(V, W) by appropriately topologizing the set of all subspaces of V ⊕ W which are similarly split. If V and W are spanned respectively by m and n element subsets of some predetermined orthonormal basis for Z_{∞} ⊕ Z_{∞}, we refer to them as coordinatized, and write the classifying space as BU (m, n) to conform with our earlier notation. We then interpret (2.1) as a coordinatized version of Whitney sum, on the understanding that the subspaces of dimension m and p are orthogonal in Z_{∞}, as are those of dimension n and q. The Grassmannian geometry of the universe guarantees the required compatibility.

In our work below we may safely confine such considerations to occasional remarks, although they are especially pertinent when we define the corresponding Thom spectrum and its multiplicative properties.

The double complex cobordism ring Ω_{∗}^{DU}consists of cobordism classes of double U-manifolds, with the product induced as above. The double complex bordism functor Ω_{∗}^{DU}( ) is an unreduced homology theory, defined on spaces X by bordism classes of singular double U-manifolds in X; it admits a canonical involution (also denoted by χ), induced by switching the factors of the normal bundle. Thus Ω_{∗}^{DU}(X) is always a module over Ω_{∗}^{DU}, which is the coefficient ring of the theory, and the product structure ensures that Ω_{∗}^{DU}(X) is both a left and a right Ω_{∗}^{U}-module.

Double complex cobordism Ω_{DU}^{∗}( ) is the dual cohomology functor, which we define geometrically as in [19]. For any double U-manifold X, a cobordism class in Ω_{DU}^{∗}(X) is represented by an equivalence class of compositions

M ⟶^{i} E_{ℓ} ⊕ E_{r} ⟶^{π} X

where π is the projection of a complex vector bundle split into left and right components, and i is an embedding of a double U-manifold M whose normal bundle is split compatibly.

If we ignore the given splitting of each normal bundle and simultaneously identify Z_{∞} ⊕Z_{∞} isometrically with Z_{∞}, we obtain a forgetful homomorphism π: Ω_{∗}^{DU}(X) → Ω_{∗}^{U}(X) for any space X. Conversely, if we interpret a given U(m)-structure as either a U(m, 0)-structure or a U(0, m)-structure, we obtain left and right inclusions ι_{ℓ} and ι_{r}: Ω_{∗}^{U}(X) → Ω_{∗}^{DU}(X), which are interchanged by χ. All three transformations have cobordism counterparts, which are multiplicative. Since π ∘ ι_{ℓ} and π ∘ ι_{r} give the identity, we deduce that π is an epimorphism and that ι_{ℓ} and ι_{r} are monomorphisms, in both bordism and cobordism.

We now turn to the homotopy theoretic viewpoint. In order to allow spectra which consist of a doubly indexed direct systems, as well as to ensure the existence of products which are highly homotopy coherent, it is most elegant to return to the coordinate-free setting. We define the Thom space M(V, W) by the standard construction on B(V, W), and again allow the Grassmannian geometry of the universe to provide the necessary compatibility for both the structure and the product maps. As in (2.1) we give explicit formulae only for coordinatized subspaces.

We write MU(m, n) for the Thom complex of ξ(m, n), which is canonically identified with MU (m) ∧ MU (n). Then the coordinatized structure maps take the form

S^{2(p+q)} ∧ MU (m, n) ⟶ MU (m + p, n + q),  (2.2)

given by Thom complexifying the classifying maps of (ℂ^{p}×ℂ^{q}) ⊕ ξ(m, n). We take this direct system as our definition of the DU spectrum, noting that the Thom complexifications of the maps (2.1) provide a product map μ_{DU}, which is highly coherent, and equipped with a unit by (2.2) in the case m = n = 0. It is a left and right module spectrum over MU by virtue of the systems of maps

MU (p) ∧ MU (m, n) → MU (m + p, n) and
MU (m, n) ∧ MU (q) → MU (m, n + q),

which are also highly coherent by appeal to the coordinate-free setting.

This setting enables us to define smash products of spectra [9], and therefore to write DU as MU ∧ MU. The involution χ is then induced by interchanging factors, and we may represent the bimodule structure by maps MU ∧DU → DU and DU ∧MU → DU, induced by applying the MU product μ_{U} on the left and right copies of MU ∧ MU respectively.

As usual, we define the reduced bordism and cobordism functors by

DU_{k}(X) = \underrightarrow{lim}_{m,n}π_{2(n+m)+k}(MU(m, n) ∧ X)

and DU^{k}(X) = \underrightarrow{lim}_{m,n}\{S^{2(n+m)−k} ∧ X, MU (m, n)\} (2.3)

respectively, for all integers k ⩾ 0, where the brackets { } denote based homotopy classes of maps. The graded group DU^{∗}(X) becomes a commutative graded ring by virtue of the product structure on DU, and the coefficient ring DU_{∗} (or DU^{−∗}) is the homotopy ring π_{∗}(DU). The unreduced bordism and cobordism functors arise as DU_{∗}(X_{+}) and DU^{∗}(X_{+}), by appending a disjoint basepoint. In this context, we write 1 for the element which corresponds to the appropriate generator of DU_{0}. Given a second homology theory E_{∗}( ), we define the homology groups E_{k}(DU) and the cohomology groups E^{k}(DU) by \underrightarrow{lim}_{m,n}E_{2(n+m)+k}(MU (m, n)) and \underleftarrow{lim}_{m,n}E^{2(n+m)+k}(MU (m, n)) respectively, for all integers k ⩾ 0.

The Whitney sum map BU (m, n) → BU (m + n) induces a forgetful map π: DU → MU , which coincides with the product μ_{U}; the inclusions of BU (m) and BU (n) in BU (m + n) induce inclusions ι_{ℓ} and ι_{r}: MU → DU of the left and right factors respectively. All three are maps of ring spectra which extend to the coordinate-free setting, and define multiplicative transformations between the appropriate functors. Clearly ι_{ℓ} and ι_{r} are interchanged by χ, and yield the identity map after composition with π. Given an element θ of MU^{∗}(X) or MU_{∗}(X), we shall often write ι_{ℓ}(θ) and ι_{r}(θ) as θ_{ℓ} and θ_{r} respectively.

Following the descriptions above, we may define the canonical isomorphism i_{U}: MU_{∗}(MU ) → DU_{∗} by identifying both sides with π_{∗}(MU ∧ MU). We shall explain below how MU_{∗}(MU ) is also the Hopf algebroid of cooperations in MU-homology theory; suffice it to say here that its associated homological algebra has been extensively studied in connection with the Adams-Novikov spectral sequence and the stable homotopy groups of spheres. For detailed calculations, however, it has proven more efficient to concentrate on a single prime p at a time, and work with the p -local summand BP_{∗}(BP) given by Brown-Peterson homology [20].

There is a natural isomorphism between the manifold and the homotopy theoretic versions of any bordism functor, stemming from the Pontryagin-Thom construction. Given a map f: M^{k} → X, where M^{k} embedded in S^{k+2(m+n)} with a U(m, n)-structure, the construction is accomplished by collapsing the complement of a normal neighbourhood to ∞ and composing with the Thom complexification of ν × f; the result is a map from S^{k+2(m+n)} to MU (m, n) ∧ X_{+}, and so defines the isomorphism Ω_{k}^{DU}(X_{+}) → π_{k}(DU ∧ X_{+}). Verifying the the necessary algebraic properties requires considerable effort, and depends upon Thom’s transversality theorems. The isomorphism maps the geometric versions of the transformations χ, π, ι_{ℓ}, and ι_{r} to their homotopy theoretic counterparts, and may be naturally extended to the coordinate-free setting.

3. Orientation classes

In this section we characterize DU as the universal example of a spectrum equipped with two complex orientations, and consider the consequences for the double complex bordism and cobordism groups of some well-known spaces in complex geometry. We establish our notation by recalling certain basic definitions and results, which may be found, for example, in [1].

We assume throughout that E is a commutative ring spectrum, with E_{0} isomorphic to ℤ. Then E is complex oriented if the cohomology group E^{2}(CP^{∞}) contains an orientation class x^{E} whose restriction to E^{2}(CP^{1}) is a generator when the latter group is identified with E_{0}. Under these circumstances, we may deduce that E^{∗}(CP^{∞}) consists of formal power series over E_{∗} in the variable x^{E}, whose powers define dual basis elements β_{k}^{E} in E_{2k}(CP^{∞}). If we continue to write β_{k}^{E} for their image under the inclusion of BU (1) in BU (m) (for any value of m, including ∞), then E_{∗}(BU (m)) is the free E_{∗}-module generated by commutative monomials of length at most m in the elements β_{k}^{E}. For 1 ⩽ k ⩽ m, the duals of the powers of β_{1}^{E} define the Chern classes c_{k}^{E} in E^{2k}(BU (m)), which generate E^{∗}(BU (m)) as a power series algebra over E_{∗}; clearly c_{1}^{E} agrees with x^{E} over CP^{∞}. When we pass to the direct limit over m, we obtain

E_{∗}(BU_{+}) ≅ E_{∗}[β_{k}^{E} : k > 0]  and  E^{∗}(BU_{+}) ≅ E_{∗}[[c_{k}^{E} : k > 0]],  (3.1)

where the Pontryagin product in homology is induced by Whitney sum. We write monomial basis elements ∏_{i}(β_{i}^{E})^{ω_{i}} as (β^{E})^{ω} for any sequence ω of nonnegative, eventually zero integers, and their duals as c_{ω}^{E}. Thus c_{ω}^{E} is usually not a monomial, and c_{(k)}^{E} and c_{k}^{E} coincide.

For m ⩾ 1 the Thom complex MU (m) of ξ(m) is the cofiber of the inclusion of BU (m − 1) in BU (m), allowing E_{∗}(MU (m)) and E^{∗}(MU (m)) to be computed from the corresponding short exact sequences of E_{∗}-modules. We may best express the consequences in terms of Thom isomorphisms, for which we first identify the pullback of c_{m}^{E} in E^{2m}(MU (m)) as the Thom class t^{E}(m) of ξ(m), observing that its restriction over the base point is a generator of E^{2m}(S^{2m}) when the latter is identified with E_{0}. Indeed, this property reduces to the defining property for x^{E} when m is 1, thereby identifying x^{E} as the Thom class of ξ(1). The Thom isomorphisms

ϕ_{∗}: E_{k+2m}(MU (m)) → E_{k}(BU (m)_{+})

and ϕ^{∗}: E^{k}(BU (m)_{+}) → E^{k+2m}(MU (m)), (3.2)

for ξ(m) are determined by the relative cap and cup products with t^{E}(m), and define elements b_{k}^{E} in E_{2(k+m)}(MU (m)) as ϕ_{∗}^{−1}(β_{k}), and elements s_{k}^{E} in E^{2(k+m)}(MU (m)) as ϕ^{∗}(c_{k}^{E}); each of these families extends to a set of generators over E_{∗} in the appropriate sense.

When m is 1 the projection CP^{∞} → MU (1) onto the cofiber is a homotopy equivalence, and the induced homomorphism identifies β_{k}^{E} with b_{k−1}^{E} in E-homology. The Thom isomorphisms satisfy ϕ_{∗}(b_{k}^{E}) = β_{k}^{E} and ϕ^{∗}((x^{E})^{k−1}) = (x^{E})^{k} respectively, for all k ⩾ 1.

We stabilize (3.2) by allowing m to become infinite, so that (3.1) yields

E_{∗}(MU ) ≅ E_{∗}[b_{k}^{E} : k > 0]  and  E^{∗}(MU ) ≅ E_{∗}[[s_{k}^{E} : k > 0]],  (3.3)

where b_{k}^{E} lies in E_{2k}(MU ) and s_{k}^{E} in E^{2k}(MU ), for all k ⩾ 0. The multiplicative structure in homology is induced by μ_{U}, but exists in cohomology only as an algebraic consequence of ϕ^{∗}, and not as a cup product. We continue to write monomial basis elements in the b_{k}^{E} as (b^{E})^{ω}, and their duals as s_{ω}^{E}, for any sequence ω. Again, s_{(k)}^{E} and s_{k}^{E} coincide. We write t^{E} in E^{0}(MU ) for the stable Thom class, which corresponds to the element 1 under (3.3), and is represented by a multiplicative map of ring spectra.

We have therefore described a procedure for constructing t^{E} from our initial choice of x^{E}; in fact this provides a bijection between complex orientation classes in E and multiplicative maps MU → E.

Any complex m -plane bundle ρ over a space X has a Thom class t^{E}(ρ) in E^{2m}(M(ρ)), obtained by pulling back the universal example t^{E}(m) along the classifying map. We use this Thom class as in (3.2) to define isomorphisms

ϕ_{∗}: E_{k+2m}(M(ρ)) → E_{k}(X_{+})  and  ϕ^{∗}: E^{k}(X_{+}) → E^{k+2m}(M(ρ)).

If ρ is virtual then M(ρ) is stable, and the Thom isomorphisms assume a similar format to the universal examples (3.1) and (3.3) if we arrange for the bottom cell to have dimension zero.

We remark that MU is complex oriented by letting the homotopy equivalence CP^{∞} → MU (1) represent x^{MU}; the corresponding Thom class t^{U} is represented by the identity map on MU . In fact MU is the universal example, since any Thom class t^{E} induces the ring map MU^{∗}(CP^{∞}) → E^{∗}(CP^{∞}) uniquely specified by x^{MU} ↦ x^{E}, and t_{∗}^{E} on coefficients. In view of these properties, we shall dispense with sub- and superscripts U wherever possible in the universal case. So far as the geometric description of cobordism is concerned, a Thom class t(ρ) in Ω_{U}^{2m}(M(ρ)) is represented by the inclusion of the zero section M ⊂ M(ρ), whenever ρ lies over a U-manifold M.

We combine the Thom isomorphism MU_{∗}(MU ) ≅ MU_{∗}(BU_{+}) with the canonical isomorphism i to obtain an isomorphism h: DU_{∗} ≅ MU_{∗}(BU_{+}) of left MU_{∗}-modules; it has an important geometrical interpretation.

Proposition 3.4. Suppose that an element of Ω_{∗}^{DU} is represented by a manifold M^{k} with double U-structure ν_{ℓ} ⊕ ν_{r}; then its image under h is represented by the singular U-manifold ν_{r}: M^{k} → BU .

Proof. By definition, the image we seek is represented by the composition

S^{k+2(m+n)} → M(ν) → MU (m) ∧ MU (n)
⟶^{1∧δ} MU (m) ∧ MU (n) ∧ BU (n)_{+} ⟶^{μ∧1} MU (m + n) ∧ BU (n)_{+},

where the first map is obtained by applying the Pontryagin-Thom construction to an appropriate embedding M^{k} ⊂ S^{k+2(m+n)}, and the second classifies the double U-structure on M^{k}. We may identify the final three maps as the Thom complexification of the composition

M^{k} ⟶^{ν_{ℓ}⊕ν_{r}} BU (m) × BU (n) ⟶^{1×δ} BU (m)×BU (n) × BU (n)

⟶^{⊕×1} BU (m + n) × BU (n),

which simplifies to ν ⊕ ν_{r}, as sought.

Corollary 3.5. Suppose that an element of Ω_{∗}^{U}(BU_{+}) is represented by a singular U-manifold f: M^{k} → BU (q) for suitably large q; then its inverse image under h is represented by the double U-structure (ν ⊕ f^{⊥}) ⊕ f on M.

Our proof of Proposition 3.4 shows that h is multiplicative, so long as we invest MU_{∗}(BU_{+}) with the Pontryagin product which arises from the Whitney sum map on BU . Moreover, h conjugates the involution χ so as to act on MU_{∗}(BU_{+}), where it interchanges the map f of Corollary 3.5 with ν ⊕ f^{⊥}.

The multiplicative maps ι_{ℓ} and ι_{r}: MU → DU both define complex orientations for the spectrum DU, with corresponding Thom classes t_{ℓ} and t_{r} and orientation classes x_{ℓ} and x_{r} in D^{2}(CP^{∞}). We obtain

DU^{∗}(CP_{+}^{∞}) ≅ DU_{∗}[[x_{ℓ}]] ≅ DU_{∗}[[x_{r}]],

and there are mutually inverse formal power series

x_{r} = ∑_{k⩾0} g_{k}x_{ℓ}^{k+1}  and  x_{ℓ} = ∑_{k⩾0} ḡ_{k}x_{r}^{k+1},  (3.6)

written g(x_{ℓ}) and ḡ(x_{r}) respectively. The elements g_{k} and ḡ_{k} lie in DU_{2k} for all k, and are interchanged by the involution χ; in particular, g_{0} = ḡ_{0} = 1. They generate an important subalgebra G_{∗} of DU_{∗}, whose structure we now adress.

Proposition 3.7. Under the isomorphism h, we have that

h(g_{k}) = β_{k}

in MU_{2k}(BU_{+}), for all k ⩾ 0.

Proof. We may express g_{k} as the Kronecker product ⟨x_{r}, β_{k+1,ℓ}⟩ in DU_{2k}, which is represented by the composition

S^{2(p+k+1)} ⟶^{β_{k+1}} MU (p) ∧ CP^{∞} ⟶^{1∧x} MU (p) ∧ MU (1),

for suitably large p. This stabilizes to b_{k} in MU_{2k}(MU ), and hence to β_{k} in MU_{2k}(BU_{+}), as required.

Corollary 3.8. The subalgebra G_{∗} is polynomial over ℤ.

Proof. This result follows from the multiplicativity of h and the independence of monomials in the β_{k} over ℤ.

We refer to any spectrum D with two complex orientations which restrict to the same element of D_{0} as doubly complex oriented, writing

x_{r}^{D} = g^{D}(x_{ℓ}^{D})  and  x_{ℓ}^{D} = ḡ^{D}(x_{r}^{D})  (3.9)

for the two orientation classes in D^{2}(CP^{∞}). These power series define elements g_{k}^{D} and ḡ_{k}^{D} in D_{∗}, which generate a subalgebra G_{∗}^{D}. Our main examples are doubles of complex oriented spectra E, given by E ∧ E and denoted by D(E); in such cases, we may extend the left and right notation to all groups D(E)_{∗}(X) and D(E)^{∗}(X).

Following the example of E, we construct left and right sets of D_{∗}-generators for D_{∗}(CP^{∞}), D_{∗}(BU (m)), D_{∗}(MU (m)), and for their cohomological counterparts. Thus there are left and right Chern classes c_{k,ℓ}^{D} and c_{k,r}^{D} in D^{2k}(BU (m)) for k ⩽ m, and left and right Thom classes t_{ℓ}^{D}(m) and t_{r}^{D}(m) in D^{2m}(MU (m)); the latter give rise to left and right Thom isomorphisms associated to an arbitrary complex bundle ρ. The elements g_{k}^{D} again provide the link, as the following example shows.

Lemma 3.10. The left and right Thom classes are related by

t_{r}^{D} = t_{ℓ}^{D} + ∑_{ω}(g^{D})^{ω}s_{ω,ℓ}^{D}

in D^{0}(MU ).

Proof. We dualize (3.9) and pass to D_{∗}(BU ), then dualize back again to D^{∗}(BU ) and apply the Thom isomorphism.

When DU is equipped with the orientation classes x_{ℓ} and x_{r} it becomes the universal example of a doubly complex oriented spectrum, since the exterior product t_{ℓ}^{D}t_{r}^{D} is represented by a multiplicative spectrum map t^{D}: DU → D whose induced transformation DU^{2}(CP^{∞}) → D^{2}(CP^{∞}) maps x_{ℓ} and x_{r} to x_{ℓ}^{D} and x_{r}^{D} respectively. It therefore often suffices to consider the case DU (as we might in Lemma 3.10, for example). We shall continue to omit the superscript DU whenever possible in the universal case. We note from the definitions that the homomorphism of coefficient rings DU_{∗} → D_{∗} induced by t^{D} satisfies

g_{k} ↦ g_{k}^{D}  and  ḡ_{k} ↦ ḡ_{k}^{D}  (3.11)

for all k ⩾ 0.

Whenever a complex vector bundle has a prescribed splitting ρ ≅ ρ_{ℓ} ⊕ ρ_{r}, then t^{D}(ρ) acts as a canonical Thom class t_{ℓ}^{D}(ρ_{ℓ})t_{r}^{D}(ρ_{r}), and so defines a Thom isomorphism which respects the splitting. In the universal case, t(ρ) is represented geometrically by the inclusion of the zero section M ⊆ M(ρ_{ℓ} ⊕ ρ_{r}) whenever ρ lies over a double U-manifold M.

As an example, it is instructive to consider the case when D is MU , doubly oriented by setting x_{ℓ}^{D} = x_{r}^{D} = x. The associated Thom class is the forgetful transformation π: DU → MU, since π(x_{ℓ}) = π(x_{r}) = x; we therefore deduce from (3.11) that both π(g_{k}) and π(ḡ_{k}) are zero, for all k > 0.

We also consider the D_{∗}-modules D_{∗}(BU (m, n)) and D_{∗}(MU (m, n)), together with their cohomological counterparts, which may all be described by applying the Künneth formula. For example D^{∗}(BU (m, n)) is a power series algebra, generated by any one of the four possible sets of Chern classes

\begin{aligned}\{c_{j,ℓ}^{D} ⊗ 1, 1 ⊗ c_{k,r}^{D}\}, & \{c_{j,ℓ}^{D} ⊗ 1, 1 ⊗ c_{k,ℓ}^{D}\}, \\ & \{c_{j,r}^{D} ⊗ 1, 1 ⊗ c_{k,ℓ}^{D}\}, or \{c_{j,r}^{D} ⊗ 1, 1 ⊗ c_{k,r}^{D}\}, \end{aligned}  (3.12)

where 1 ⩽ j ⩽ m and 1 ⩽ k ⩽ n. The first of these are the most natural, and we shall choose them whenever possible. The stable versions, in which we take limits over one or both of m and n, are obtained by the obvious relaxation on the range of j and k. We write (β_{ℓ}^{D})^{ψ} ⊗ (β_{r}^{D})^{ω} for the dual basis monomials in D_{∗}(BU (m, n)), and (b_{ℓ}^{D})^{ψ} ⊗ (b_{r}^{D})^{ω} for their images in D_{∗}(MU (m, n)) under the Thom isomorphism induced by t^{D}(m, n); we then let s_{ψ,ℓ}^{D} ⊗ s_{ω,r}^{D} denote the corresponding basis elements in D^{∗}(MU (m, n)). As before, it often suffices to consider the universal example DU.

We may translate much of the above into the language of formal group laws [10], observing that DU_{∗} is universal amongst rings equipped with two formal group laws linked by a strict isomorphism whose coefficients are the elements g_{k}.

4. The eightfold way

In this section we consider the operations and cooperations associated with our spectrum E, specializing to MU as required. We study algebraic and homotopy theoretic aspects in the framework of Boardman’s eightfold way [2], relating the actions and coactions involving E_{∗}(E) to the theory of the double spectrum D(E), with a view to placing the quantum double in the traditional framework. All comments concerning the singly oriented E apply equally well to D unless otherwise stated.

For any integer n, the cohomology group E^{n}(E) consists of homotopy classes of spectrum maps s: E → S^{n} ∧ E, and therefore encodes E-theory cohomology operations of degree n. Thus E^{∗}(E) is a noncommutative, graded E_{∗}-algebra with respect to composition of maps, and realizes the algebra A_{E}^{∗} of stable E-cohomology operations. It is important to observe that E^{∗}(E) is actually a bimodule over the coefficients E_{∗}, which act naturally on the left (as used implicitly above), but also on the right. The same remarks apply to E_{∗}(E), on which the product map μ_{E} induces a commutative E_{∗}-algebra structure; the two module structures are then defined respectively by the left and right inclusions η_{ℓ} and η_{r} of the coefficients in E_{∗}(E) ≅ π_{∗}(E ∧ E). We refer to E_{∗}(E) as the algebra A_{∗}^{E} of stable E-homology cooperations, for reasons described below.

In fact E_{∗}(X) is free and of finite type for all spaces and spectra X that we consider. This reduces the topologizing of E^{∗}(X) to the accommodation of formal power series in certain computations, and ensures that μ_{E} induces a cocommutative coproduct δ^{E}: E^{∗}(E) → E^{∗}(E)ˆ⊗_{E_{∗}}E^{∗}(E); the tensor product must be completed whenever E^{∗}(E) fails to be of finite type [3].

We consider the E_{∗}-algebra map t_{∗}^{E}: E_{∗}(MU) → E_{∗}(E) induced by the Thom class t^{E}, and define monomials (p^{E})^{ω} as t_{∗}^{E}(b^{E})^{ω}. When E is singly oriented we restrict attention to cases where E_{∗}(E) is a free E_{∗}-module and t_{∗}^{E} is an epimorphism, so that E_{∗}(E) is a quotient of the polynomial algebra E_{∗}[p_{k}^{E} : k ⩾ 0]. It follows that E^{∗}(E) is given as a subalgebra of the dual by Hom_{E_{∗}}(E_{∗}(E), E_{∗}), and that we may interpret a generic operation s as such a homomorphism. The composition product dualizes to a noncocommutative coproduct δ_{E}: E_{∗}(E) → E_{∗}(E)⊗_{E_{∗}} E_{∗}(E) (where ⊗_{E_{∗}} is taken over the right action on the left factor), with counit given by projection onto the coefficients. Together with the left and right units η_{ℓ} and η_{r}, and the antipode χ_{E} induced by interchanging the factors in E_{∗}(E), this coproduct turns E_{∗}(E) into a cogroupoid object in the category of E_{∗}-algebras. Such an object generalizes the notion of Hopf algebra, and is known as a Hopf algebroid; for a detailed discussion, see [20].

Given our assumptions on E, the monomials (p^{E})^{ω} generate E_{∗}(E) over E_{∗} although they may be dependent; nevertheless, they appear naturally in several formulae below. To describe a basis over E_{∗}, we consider each example on its merits and choose an appropriate set of homogeneous polynomials e^{α}, noting that the new set of indices α may well be smaller. Each (p^{E})^{ω} may then be expressed as a linear combination of the e^{α}, whose coefficients generally lie in E_{∗}, rather than ℤ. Maintaining our earlier conventions, we let e_{α} denote the topological basis for E^{∗}(E) dual to the e^{α}.

For any space or spectrum X, we first consider the standard action

E^{∗}(E) ⊗_{E_{∗}} E^{∗}(X) ⟶ E^{∗}(X),  (4.1)

(where ⊗_{E_{∗}} is taken over the right action on E^{∗}(E)). It is the map of left E_{∗}-modules which arises from considering the elements of E^{∗}(E) as selfmaps of E, and we write it functionally; when X is E it reduces to the composition product in E^{∗}(E), and when X is a point (or the sphere spectrum), to the action of E^{∗}(E) on the coefficient ring E_{∗}. The Cartan formula asserts that the product map in E^{∗}(X) is a homomorphism of left E^{∗}(E)-modules with respect to the standard action, and was restated by Milnor in the form

s(yz) = ∑ s_{1}(y)s_{2}(z)  where  δ^{E}(s) = ∑ s_{1} ⊗ s_{2}  (4.2)

for all y and z in E^{∗}(X). Following Novikov [17], we refer to any such module with property (4.2) as a Milnor module.

Given our freeness assumptions we may dualize and conjugate (4.1) to obtain seven alternative structures, whose unification is the aim of Boardman’s eightfold way. We consider four of these (together with a fifth and sixth which are different), selecting from [2] without comment and ignoring issues of sign because our spaces and spectra have no cells in odd dimensions.

The second structure is the E_{∗}-dual of (4.1), the Adams coaction

ψ: E_{∗}(X) ⟶ E_{∗}(E) ⊗_{E_{∗}} E_{∗}(X)  (4.3)

(where ⊗_{E_{∗}} is taken over the right action on E_{∗}(E)), which reduces to the coproduct δ_{E} when X is E. For each operation s and each y in E^{∗}(X), the duality is specified by

⟨s(y), a⟩ = ∑_{α}⟨s, e^{α}⟨y, a^{α}⟩⟩,  (4.4)

where a lies in E_{∗}(X) and a^{α} is defined by ψ(a) = ∑_{α} e^{α} ⊗ a^{α}.

If we assume that X is a spectrum (or stable complex), we may interpret π_{∗}(X ∧ E) as X_{∗}(E), and consider the isomorphism c: E_{∗}(X) ≅ X_{∗}(E) of conjugation. Our third structure is the right coaction

ψ^{c}: X_{∗}(E) ⟶ X_{∗}(E) ⊗_{E_{∗}} E_{∗}(E)  (4.5)

of right E_{∗}-modules (where ⊗_{E_{∗}} is taken over the right action of the scalars on X_{∗}(E)); it is evaluated by ψ(ca) = ∑_{α} ca^{α} ⊗ χ_{E}(e^{α}), conjugating (4.3). When X is E, then c reduces to χ_{E} and ψ becomes δ_{E}, as before.

Fourthly, (4.1) dualizes partially over E_{∗} to give the Milnor coaction

ρ: E^{∗}(X) ⟶ E^{∗}(X)ˆ⊗_{E_{∗}}E_{∗}(E)  (4.6)

of E_{∗}-modules. As Milnor famously observed, the Cartan formulae ensure that ρ is an algebra map, making E^{∗}(X) a Hopf comodule over E_{∗}(E). For each operation s and each x in E^{∗}(X), the partial duality satisfies

s(y) = ∑_{α}⟨s, e^{α}⟩y_{α},  (4.7)

where y_{α} is defined by ρ(y) = ∑_{α} y_{α} ⊗e^{α}; thus y_{α} = e_{α}(y). In view of the completion required of the tensor product in (4.6), we describe ρ more accurately as a formal coaction.

A fifth possibility is provided by the left action

E^{∗}(E) ⊗_{E_{∗}} E_{∗}(X) ⟶ E_{∗}(X)

(where ⊗_{E_{∗}} is taken over the right action of the scalars on X_{∗}(E)), which is defined by analogy with (4.1) in terms of spectrum maps. It is evaluated by partially dualizing the Adams coaction, giving

s_{ℓ}(a) = ∑_{α}⟨s, χ_{E}(e^{α})⟩a^{α},  (4.8)

with notation as above. Given y and a as before, the left action satisfies

⟨y, s_{ℓ}(a)⟩ = ⟨s, c(y_{∗}a)⟩,  (4.9)

where y_{∗}: E_{∗}(X) → E_{∗}(E) is the induced homomorphism.

For our sixth and seventh structures we again assume that X is stable, so the selfmaps of E induce a left action

E^{∗}(E) ⊗_{E_{∗}} X_{∗}(E) ⟶ X_{∗}(E)  (4.10)

(where ⊗_{E_{∗}} is taken over the right action of the scalars on both factors), and a right action

X^{∗}(E) ⊗_{E_{∗}} E^{∗}(E) ⟶ X^{∗}(E)  (4.11)

(where ⊗_{E_{∗}} is taken over the right action on X^{∗}(E)). Neither of these is discussed explicitly by Boardman, although (4.10) appears regularly in the literature, and coincides with (4.1) when X is the sphere spectrum; it is evaluated by partially dualizing (4.5), to obtain

s_{r}(d) = ∑_{α} d^{α}⟨s, e^{α}⟩,  (4.12)

where d lies in X_{∗}(E) with ψ(d) = ∑_{α} d^{α} ⊗ e^{α}. The actions (4.10) and (4.11) are related by

⟨w, s_{r}(d)⟩ = ⟨(w)s, d⟩,  (4.13)

where w lies in X^{∗}(E); this should be compared with (4.9), and justifies the interpretation of (4.10) as a right action (on the left!).

When X is E, (4.9) may be rewritten as

⟨y, s_{ℓ}(a)⟩ = ⟨s, χ_{E}(y_{r}(a))⟩,  (4.14)

whilst (4.13) reduces to the right action of E^{∗}(E) on its dual E_{∗}(E). Given any d in E_{∗}(E), we may evaluate the coproduct δ_{E}(d) as

∑_{α} e_{α,r}(d) ⊗ e^{α} = ∑_{α} χ_{E}(e^{α}) ⊗ e_{α,ℓ}(d).  (4.15)

We now translate aspects of the eightfold way into the language of doubles, using the Etheory canonical isomorphism i_{E}: E_{∗}(E) → D(E)_{∗}. The definition of the (p^{E})^{ω} ensures that the homomorphism x_{∗}^{E}: E_{∗+2}(CP^{∞}) → E_{∗}(E) maps β_{k+1}^{E} to p_{k}^{E}. Dualization yields

s(x^{E}) = ∑_{k⩾0}⟨s, p_{k}^{E}⟩(x^{E})^{k+1},

which combines with (4.7) to confirm that ρ acts by

ρ(x^{E}) = ∑_{k⩾0}(x^{E})^{k+1} ⊗ p_{k}^{E}  (4.16)

in E^{∗}(CP^{∞})ˆ⊗_{E_{∗}}E_{∗}(E).

Lemma 4.17. Given any of our spaces or spectra X, the Milnor coaction (4.6) extends to D(E)^{∗}(X) by means of the commutative diagram

[figure: E^{∗}(X); E^{∗}(X)ˆ⊗_{E_{∗}}E_{∗}(E); D(E)^{∗}(X); D(E)^{∗}(X)ˆ⊗_{D(E)_{∗}}D(E)_{∗}; ρ; ι_{r}; ι_{ℓ}⊗i_{E}; ≅] .

Proof. Commutativity follows from the fact that the Milnor coaction factors through the map E^{∗}(X) → Hom_{E_{∗}}(E^{∗}(X), E^{∗}(E)), defined by taking induced homorphisms.

Corollary 4.18. The canonical isomorphism identifies p_{n}^{E} with g_{n}^{D(E)} and χ_{E}(p_{n}^{E}) with ḡ_{n}^{D(E)} in D(E)_{∗}, for all n ⩾ 0.

Proof. Comparing Lemma 4.17 with (4.16), these equations follow immediately from (3.9).

The formulation of ρ in Lemma 4.17 is almost explicit in [11], and has obvious mutants for each of the actions and coactions which involves E_{∗}(E).

Formula (4.16) is used by Boardman to define the elements p_{n}^{E}, and extends to a representation of the entire algebra A_{E}^{∗} on the bottom cell of the infinite smash product spectrum ∧_{∞}CP^{∞}. It also leads to a description of the structure maps for A_{∗}^{E} in terms of the p_{n}^{E}.

Proposition 4.19. The coproduct and antipode of the Hopf algebroid E_{∗}(E) are given by

δ_{E}(p_{n}^{E}) = ∑_{k⩾0}(p^{E})_{n−k}^{k+1} ⊗ p_{k}^{E}  and  χ_{E}(p_{n}^{E}) = (p^{E})_{n}^{−(n+1)}

respectively.

Proof. Since ρ is a coaction, we have that ρ⊗1(ρ(x^{E})) = 1⊗δ^{E}(ρ(x^{E})) as maps E^{∗}(CP^{∞}) → E^{∗}(CP^{∞})ˆ⊗_{E_{∗}}E_{∗}(E) ⊗_{E_{∗}} E_{∗}(E), and the formula for δ^{E} ensues. Since χ_{E} applies to (3.9) by interchanging the factors of D(E), we may use Corollary 4.18 to give i_{E}(χ_{E}(p_{n}^{E})) = ḡ_{n}^{E}, and the result follows by Lagrange inversion.

These formulae are of limited use in cases where the (p^{E})^{ω} are not independent, since they contain a great deal of inbuilt redundancy. Readers will recognize the spectrum BP as a pertinent example.

There is no such problem with the universal example MU , for which the e^{ω} are simply the original basis monomials b^{ω} in A_{∗}^{U}. The dual operations are written s_{ω}, as described in §3, and are known as the Landweber-Novikov operations. We consider the integral spans S^{∗} and S_{∗} of the s_{ω} and b^{ω} respectively, so that A_{U}^{∗} ≅ Ω_{∗}^{U} ⊗ S^{∗} and A_{∗}^{U} ≅ Ω_{∗}^{U} ⊗ S_{∗} as Ω_{∗}^{U}-modules, where S_{∗} is the polynomial algebra ℤ[b_{k} : k ⩾ 0]. The Ω_{∗}^{U}-duality between A_{U}^{∗} and A_{∗}^{U} therefore restricts to an integral duality between S^{∗} and S_{∗}, for which no topological considerations are necessary because S^{∗} has finite type.

The coproduct and antipode of A_{∗}^{U} are given as δ(b_{n}) = ∑_{k⩾0}(b)_{n−k}^{k+1} ⊗ b_{k} and χ(b_{n}) =

(b)_{n}^{−(n+1)} by Corollary 4.19, and therefore restrict to S_{∗}. Since the left unit and the counit also make sense over ℤ, we deduce that S_{∗} is a Hopf subalgebra of the Hopf algebroid. Duality ensures that S^{∗} is also a Hopf algebra, with respect to composition of operations and the Cartan formula

δ(s_{ω}) = ∑_{ω_{1}+ω_{2}=ω} s_{ω_{1}} ⊗ s_{ω_{2}},  (4.20)

which is dual to the product of monomials. Of course S^{∗} is the Landweber-Novikov algebra. Alternatively, and following the original constructions, we may use the action of S^{∗} on Ω_{U}^{∗}(∧_{∞}CP^{∞}) to prove directly that S^{∗} is a Hopf algebra. Many of our actions and coactions restrict to S^{∗} and S_{∗} and will be important below. We emphasize that A_{U}^{∗} has no Ω_{∗}^{U}-linear antipode, and that the antipode in S^{∗} is induced from the antipode in S_{∗} by ℤ-duality.

Having identified A_{U}^{∗} additively as Ω_{∗}^{U} ⊗ S^{∗} and noted that both factors are subalgebras, the remaining multiplicative structure is determined by the commutation rule for products sx, where s and x lie in S^{∗} and Ω_{∗}^{U} respectively. Recalling (4.1), (4.2), and (4.12) we obtain

sx = ∑ s_{1,r}(x)s_{2},  (4.21)

and write the resulting algebra as Ω_{∗}^{U} # S^{∗}, where the right action of S^{∗} on Ω_{∗}^{U} is understood. This is an important case of the smash product [24], and an analogous algebra may be constructed from any Milnor module over a Hopf algebra; it is Novikov’s operator double [17].

Choosing E and X to be MU in (4.13) and (4.14) provides the left and right action of A_{U}^{∗} on its dual. If s and y lie in A_{U}^{∗} and u in A_{∗}^{U}, we have

⟨y, s_{ℓ}(u)⟩ = ⟨s, χ(y_{r}(u))⟩  and  ⟨y, s_{r}(u)⟩ = ⟨ys, u⟩.  (4.22)

Alternatively, by appealing to (4.8) and (4.12) we may write

s_{ℓ}u = ∑⟨s, χ(u_{1})⟩u_{2}  and  s_{r}u = ∑⟨s, u_{2}⟩u_{1}.  (4.23)

By restriction we obtain identical formulae for the left and right actions of S^{∗} on A_{∗}^{U} and on S_{∗}. In the latter case, ℤ-duality allows us to rewrite the left action as

⟨y, s_{ℓ}(u)⟩ = ⟨χ(s)y, u⟩,

thereby (at last) according it equivalent status to the right action.

The adjoint actions of S^{∗} on A_{∗}^{U} and S_{∗} are similarly defined by

\begin{matrix}⟨y, ad(s)(u)⟩ = ∑ & ⟨χ(s_{1})ys_{2}, u⟩  and \\ ad(s)(u) = ∑⟨χ(s_{1}), u_{1}⟩⟨s_{2}, u_{3}⟩u_{2}, \end{matrix}  (4.24)

which give rise to the adjoint Milnor module structure on A_{∗}^{U} and S_{∗}. By way of example, we combine Corollary 4.19 with (4.15) to produce

\begin{matrix}s_{ϵ(k),ℓ}(b_{n}) = & (k − n − 1)b_{n−k},  s_{ϵ(k),r}(b_{n}) = (b)_{n−k}^{k+1} \\ and  ad(s_{ϵ(k)})(b_{n}) = (k − n − 1)b_{n−k} + (b)_{n−k}^{k+1} \end{matrix}  (4.25)

for all 0 ⩽ k ⩽ n.

An alternative interpretation of S_{∗} lies at the heart of our next section, and follows directly from Proposition 4.18 in the case E = MU .

Proposition 4.26. The subalgebra G_{∗} of Ω_{∗}^{DU} is identified with the dual of the Landweber-Novikov algebra S_{∗} in A_{∗}^{U} under the canonical isomorphism.

In more general cases we insist that the integral span of the e^{α} forms a subring S_{∗}^{E} of A_{∗}^{E}, which is invariant under χ_{E} and maps to ℤ under the counit. Therefore A_{∗}^{E} is isomorphic to E_{∗} ⊗ S_{∗}^{E} as left E_{∗}-algebras. The integral span of the e_{α} is the dual subcolagebra S_{E}^{∗} of A_{E}^{∗}, which admits the involution dual to χ_{E}, and features in the dual isomorphism between A_{E}^{∗} and E_{∗} ⊗ S_{E}^{∗} as left E_{∗}-coalgebras. We may interpret this decomposition as a weak form of smash product E_{∗} # S_{E}^{∗}, since S_{E}^{∗} acts on the coefficient ring E_{∗} according to (4.1) and satisfies the Milnor condition (4.2); the only products not so defined are those internal to S_{∗}^{E}. A typical example to bear in mind is BP, where S_{∗}^{BP} is the polynomial algebra over ℤ_{(p)} on the 2(p^{j} − 1)-dimensional Adams generators t_{j}, and S_{BP}^{∗} is the dual coalgebra spanned over ℤ_{(p)} by the Quillen operations r_{α} [18]. The subalgebra G_{∗}^{BP} of D(BP)_{∗} is certainly not identified with S_{∗}^{BP} under the canonical isomorphism i_{BP}.

5. Double cohomology operations

In this section we consider the algebra of cohomology operations A_{D(E)}^{∗}, studying various subalgebras isomorphic to A_{E}^{∗} and locating an additive subgroup which becomes the quantum double D(S^{∗}) in the universal example DU. We conclude by investigating extensions of the partial endomorphism on A_{DU}^{∗} provided by the antipode in D(S^{∗}). Throughout the section we understand that D denotes an arbitrary double spectrum D(E), in which E satisfies the conditions imposed in Section 4.

Since the Künneth formula identifies E_{∗}(D) with E_{∗}(E) ⊗_{E_{∗}} E_{∗}(E), the elements (p^{E})^{α} in E_{∗}(E) give rise to elements (p_{ℓ}^{E})^{α} and (p_{r}^{E})^{α} in D_{∗}(E), and therefore to four possible choices of generators for D_{∗}(D). These arise from the four sets of generators for D_{∗}(MU) given in (3.12), by applying the map induced in D-homology by the Thom class t_{ℓ}^{D}t_{r}^{D}. We deduce that A_{∗}^{D} is a quotient of the polynomial algebra D_{∗}[p_{j,ℓ}^{D} ⊗ 1, 1 ⊗ p_{k,r}^{D} : j, k ⩾ 0], and that the algebra A_{D}^{∗} of D-theory operations is the appropriate dual D^{∗}(D). These act and coact according to the eightfold way.

Since D is E ∧E, an element s of A_{E}^{∗} yields operations s∧1 and 1∧s in A_{D}^{∗}, which commute by construction; in terms of (3.12) these correspond to s_{ℓ} ⊗ 1 and 1 ⊗ s_{r} respectively, and are therefore consistent with our preferred choice of D_{∗}-basis elements e_{α,ℓ} ⊗ 1 and 1 ⊗ e_{α,r}. We denote the subalgebras consisting of all s_{ℓ} ⊗ 1 and 1 ⊗ s_{r} by A_{E,ℓ}^{∗} ⊗ 1 and 1 ⊗ A_{E,r}^{∗} respectively, and refer to them as the left and right copies of A_{E}^{∗}. They act on the coefficient groups D_{∗} in accordance with (4.1), and feature in the D_{∗}-coalgebra decomposition of A_{D}^{∗} as D_{∗} ⊗ S_{E,ℓ}^{∗} ⊗ S_{E,r}^{∗}; since both actions are Milnor, we may interpret this decomposiiton as a weak form of smash product D_{∗} # (S_{E,ℓ}^{∗} ⊗ S_{E,r}^{∗}), by analogy with our previous description of A_{E}^{∗}. These actions have an important alternative description in terms of (4.13) and (4.14), which follows immediately from the definitions.

Proposition 5.1. The canonical isomorphism identifies the left and right actions of A_{E}^{∗} on A_{∗}^{E} with the actions of A_{E,ℓ}^{∗} and A_{E,r}^{∗} on D_{∗}, respectively.

By restriction we obtain Milnor actions of the subcoalgebras S_{E,ℓ}^{∗} and S_{E,r}^{∗} on D_{∗}, but the image of S_{∗}^{E} under i_{E} is not generally preserved. The coproduct δ_{E}: S_{E}^{∗} → S_{E,ℓ}^{∗}⊗S_{E,r}^{∗} describes a diagonal copy S_{E,d}^{∗} of S_{E}^{∗} in A_{D}^{∗}, which is a subcoalgebra by virtue of cocommutativity.

The D_{∗}-duality between A_{D}^{∗} and A_{∗}^{D} confirms that the latter is isomorphic to D_{∗} ⊗ S_{∗}^{E,ℓ} ⊗ S_{∗}^{E,r} as D_{∗}-algebras, and restricts to the integral duality between S_{E,ℓ}^{∗} ⊗ S_{E,r}^{∗} and S_{∗}^{E,ℓ} ⊗ S_{∗}^{E,r}.

In the universal case, S_{ℓ}^{∗} ⊗ S_{r}^{∗} is actually a Hopf subalgebra of A_{∗}^{DU}, whilst the dual Hopf algebra S_{∗,ℓ} ⊗ S_{∗,r} is a sub-Hopf algebra of the Hopf algebroid A_{∗}^{DU}, and is isomorphic to ℤ[b_{j,ℓ} ⊗ b_{k,r} : j, k ⩾ 0]. In consequence, A_{DU}^{∗} is the genuine operator double Ω_{∗}^{DU} # (S_{ℓ}^{∗} ⊗ S_{r}^{∗}), and the actions of S_{ℓ}^{∗} and S_{r}^{∗} on Ω_{∗}^{DU} are already familiar, as shown by the following extension of Proposition 5.1.

Proposition 5.2. The subalgebra G_{∗} of Ω_{∗}^{DU} is closed with respect to the actions of S_{ℓ}^{∗} and S_{r}^{∗}; in particular, the action of the diagonal subalgebra S_{d}^{∗} is identified with the adjoint action of S^{∗} on S_{∗} under the canonical isomorphism.

Proof. Combining Proposition 4.26 with the fact that S^{∗} is multiplicatively closed, we identify the left and right actions of S^{∗} on S_{∗} (as in (4.22)) with the restrictions to G_{∗} of the respective actions of S_{ℓ}^{∗} and S_{r}^{∗} on Ω_{∗}^{DU}. The result then follows from the dual fact that S_{∗} is comultiplicatively closed.

Corollary 5.3. The quantum double D(S^{∗}) is isomorphic to a subalgebra of A_{DU}^{∗} [5]: as such, the universal R matrix is given by ∑_{ω}(s_{ω_{1},ℓ} ⊗ s_{ω_{2},r}) ⊗ g^{ω} in A_{DU}^{∗} ⊗ A_{DU}^{∗}; the square of the antipode acts as conjugation by the element ∑_{ω} b̄^{ω} # (s_{ω_{1},ℓ} ⊗ s_{ω_{2},r}); the ring Ω_{DU}^{∗}(X) is a crossed S^{∗}-bimodule for any space X; and solutions V_{X} to the Yang-Baxter equations are given by endomorphisms of the form V_{X}(x ⊗ y) = ∑_{ω} b^{ω}y ⊗ (s_{ω_{1},ℓ} ⊗ s_{ω_{2},r})(x) of Ω_{DU}^{∗}(X) ⊗ Ω_{DU}^{∗}(X) .

Proof. Since S^{∗} is cocommutative we may construct D(S^{∗}) as the operator double S_{∗} # S^{∗}, with respect to the adjoint action [16], [17]. By Proposition 5.2, the latter is isomorphic to the subalgebra G_{∗} # S_{d}^{∗} of Ω_{∗}^{DU} # (S_{ℓ}^{∗} ⊗ S_{r}^{∗}), which we have already identified as A_{DU}^{∗}. The remaining facts then follow directly from the definitions; for terminology, we refer to [12].

Both Hopf algebras S_{ℓ}^{∗} ⊗ S_{r}^{∗} and D(S^{∗}) are subalgebras of A_{DU}^{∗}, and also of the operator double G^{∗} #(S_{ℓ}^{∗} ⊗S_{r}^{∗}); this is not itself a Hopf algebra, however, because the action of S_{ℓ}^{∗} ⊗S_{r}^{∗} on G_{∗} is not a coalgebra map [12]. Nevertheless, it has a considerable amount of algebraic structure which extends that of D(S^{∗}), and gives rise to left Ω_{∗}^{U}-linear structure maps on A_{DU}^{∗}. We briefly consider this situation with respect to antipodes.

Suppose we are given Hopf algebras H and T with bijective antipodes χ_{H} and χ_{T} respectively, and a Milnor action of T on H. Then the algebra H #T has an additive basis of elements h # t, and we may define a unique linear automorphism θ on H # T by

θ(h # t) = χ_{T} (t) · χ_{H}(h);  (5.4)

if T is cocommutative and the Milnor coaction is also a coalgebra map, then H # T is a Hopf algebra, and θ is its antipode [12]. This analysis applies to D(S^{∗}) by taking H = S_{∗} and T = S^{∗} with respect to the adjoint action, and shows that the antipode is given by

χ_{D(S^{∗})}(b # s) = ∑⟨χ(s_{3}), b_{3}⟩⟨s_{2}, b_{1}⟩χ(b_{2}) # χ(s_{1}).  (5.5)

Theorem 5.6. We may extend χ_{D(S^{∗})} to a left Ω_{∗}^{U}-automorphism of A_{DU}^{∗}; in particular, there is an extension θ satisfying

θ(g # s) = θ(s) · θ(g) and θ(s · g) = ∑ θ(s_{2})(θ(s_{1})(g)) # θ(s_{3})

and an extension θ′ satisfying

θ′(g # s) = ∑ θ′(θ′(s_{1})(g)) # θ′(s_{2}) and θ′(s · g) = θ′(g) # θ′(s),

where g # s is a generator of G^{∗} # (S_{ℓ}^{∗} ⊗ S_{r}^{∗}).

Proof. We first translate (5.5) into A_{DU}^{∗} by Corollary 5.3, and let θ be given by (5.4) in the case H = G_{∗} and T = S_{ℓ}^{∗} ⊗ S_{r}^{∗}; the formula for θ(s · g) then follows by direct calculation. To obtain θ′, we assume that it is antimultiplicative on products s · g and deduce the formula for θ′(g # s) by direct calculation. Since the terms g # s generate A_{DU}^{∗} as a left Ω_{∗}^{U}-module, these observations suffice.

The point about θ and θ′ is that neither is antimultiplicative on all products.

We conclude with two interrelated remarks concerning the detailed structure of A_{DU}^{∗}; first we express the alternative choices (3.12) for Ω_{∗}^{DU}-bases in terms of the preferred basis, and then we consider the description of the involutory operation χ defined by switching the factors of MU ∧ MU.

Proposition 5.7. In A_{DU}^{∗}, we may express s_{ω,r} ⊗ 1 and 1 ⊗ s_{ω,ℓ} as

∑_{ψ} g^{ψ}s_{ψ,ℓ} ∘ s_{ω,ℓ} ⊗ 1  and  ∑_{ψ} ḡ^{ψ} ⊗ s_{ψ,r} ∘ s_{ω,r}

respectively, in terms of the preferred Ω_{∗}^{DU}-basis.

Proof. Applying Lemma 3.10 in DU^{0}(MU ), we obtain

s_{ω}^{∗}(t_{r}) = s_{ω}^{∗}( ∑_{ψ} g^{ψ}s_{ψ,ℓ})t_{r}

in D^{0}(MU ), and the first equation follows from the Künneth formula. For the second equation, we apply χ.

Proposition 5.8. The involution χ is given by

∑_{ψ,ω} g^{ψ}ḡ^{ω}s_{ψ,ℓ} ⊗ s_{ω,r}

as an element of A_{DU}^{0}.

Proof. The Thom class t in A_{DU}^{0} corresponds to t_{ℓ} ⊗ t_{r} in DU^{0}(MU ) ⊗ DU^{0}(MU ) under the Künneth formula. Since χ switches t_{ℓ} and t_{r} and is determined by its action on t, it suffices to write t_{r} ⊗ t_{ℓ} in terms of t_{ℓ} ⊗ t_{r}. By Lemma 3.10 we obtain ( ∑_{ψ} g^{ψ}s_{ψ,ℓ})t_{ℓ} ⊗ ( ∑_{ω} ḡ^{ω}s_{ω,r})t_{r}, as required.

In the context of (3.6), χ gives rise to the elements g_{n} which extend Ω_{∗}^{U} to Ω_{∗}^{DU}; it also represents the permutation group 𝔖_{2} in A_{DU}^{∗}.

Propositions 5.2, 5.7, and 5.8 and Theorem 5.6 have direct analogues for those spectra E and D(E) for which i_{E}(S_{∗}^{E}) coincides with the subalgebra G_{∗}^{E} of D(E)_{∗}. More generally, the best we can do is to identify the sub-left E_{∗}-module S_{∗}^{E} ⊗ S_{E,d}^{∗} as some sort of weak double for S_{E}^{∗}, and consider the associated antipode as an antiautomorphism. In these cases the relationship between t_{ℓ}^{D} and t_{r}^{D} is governed by that between the bases e_{α,ℓ} and e_{α,r}.

6. Further developments

The quantum algebraic viewpoint highlights novel aspects of A_{E}^{∗} and A_{∗}^{E}, and we conclude with a few examples which we shall develop elsewhere [7]. We restrict attention to universal cases, and refer to [16] for terminology.

We may interpret the dual of the Landweber-Novikov algebra in many ways. For example, the projection t_{∗}^{H}: Ω_{∗}^{U}(MU ) → H_{∗}(MU ) restricts to an isomorphism on S_{∗}, with t_{∗}^{H}(b^{ω}) = (b^{H})^{ω}. This isomorphism is often implicit in the literature, and invests H_{∗}(MU ) with a coproduct and antipode which are purely algebraic; its dual identifies H^{∗}(MU) with the Landweber-Novikov algebra itself. The Hurewicz homomorphism Ω_{∗}^{U} → H_{∗}(MU ) therefore realizes A_{U}^{∗} as a subalgebra of the Heisenberg double S_{∗} # S^{∗} (often written H(S^{∗})) where S^{∗} acts on its dual by the right action (4.23).

The appearance of D(S^{∗}) as an algebra of double cobordism operations is actually part of a decomposition theorem, which describes A_{DU}^{∗} as an appropriate smash product of A_{U}^{∗} and D(S^{∗}). More generally, for any N ⩾ 2 we may introduce N-fold cobordism theory Ω_{∗}^{NU}( ), represented by the N-fold smash product spectrum MU^{∧N}; its algebra of cohomology operations A_{NU}^{∗} has many different subalgebras isomorphic to D(S^{∗}), which contribute to a decomposition theorem expressing A_{NU}^{∗} as an iterated smash product of A_{U}^{∗} and N − 1 copies of D(S^{∗}). When N is sufficiently large, A_{NU}^{∗} contains a remarkable array of subalgebras and Hopf subalgebras, such as the dual of D(S^{∗}) and the quantum and Heisenberg doubles of tensor powers of S^{∗}.

The Hopf algebroids A_{∗}^{NU} decompose dually, as smash coproducts.

References

1. J. Frank Adams. Stable Homotopy and Generalized Homology. Chicago Lectures in Mathematics. University of Chicago Press, 1974.

2. J. Michael Boardman. The eightfold way to BP-operations. In Current Trends in Algebraic Topology, volume 2, Part 1 of CMS Conference Proceedings, pages 187–226. American Mathematical Society, 1982.

3. J. Michael Boardman. Stable operations in generalized cohomology. In I. M. James, editor, Handbook of Algebraic Topology, pages 585–686. North-Holland, Amsterdam, 1995.

4. V. M. Buchstaber and A. B. Shokurov. The Landweber-Novikov algebra and formal vector fields on the line. Funktsional Analiz i Prilozhen, 12:1–11, 1978.

5. Victor M. Buchstaber. Semigroups of maps into groups, operator doubles, and complex cobordisms. In S. P. Novikov, editor, Topics in Topology and Mathematical Physics, volume 170 of American Mathematical Society Translations (2), pages 9–31, 1995.

6. Victor M. Buchstaber and Nigel Ray. Flag manifolds and the Landweber-Novikov algebra. Geometry & Topology, www.maths.warwick.ac.uk/gt/ 2:79–101, 1998.

7. Victor M. Buchstaber and Nigel Ray. Multiple cobordism theory and iterated doubling constructions. In preparation, Manchester University, 1999.

8. V. G. Drinfeld. Quantum groups. In Proceedings of the International Congress of Mathematicians, Berkeley, volume 1, pages 789–820. American Mathematical Society, 1987.

9. A. D. Elmendorf. The Grassmannian geometry of spectra. Journal of Pure and Applied Algebra, 54:37–94, 1988.

10. Michiel Hazewinkel. Formal Groups and Applications. Academic Press, 1978.

11. John R. Hunton. The Boardman homomorphism. In Mila Cenkl and Haynes Miller, editors, Proceedings of the Čech Centennial Homotopy Conference, June 1993, volume 181 of Contemporary Mathematics, pages 251–262. American Mathematical Society, 1995.

12. Christian Kassel. Quantum Groups, volume 155 of Graduate Texts in Mathematics. Springer Verlag, 1995.

13. Toshiyuki Katsura, Yuji Shimizu, and Kenji Ueno. Complex cobordism ring and conformal field theory over Z. Mathematische Annalen, 291:551–571, 1991.

14. Shahn Majid. Foundations of Quantum Group Theory. Cambridge University Press, 1995.

15. J. Peter May, Frank Quinn, and Nigel Ray. E_{∞} ring spectra. Springer Lecture Notes in Mathematics, 577:63–86, 1977.

16. Susan Montgomery. Hopf Algebras and Their Actions on Rings, volume 82 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1993.

17. Sergei P. Novikov. Various doubles of Hopf algebras: Operator algebras on quantum groups, and complex cobordism. Uspekhi Akademii Nauk SSSR, 47:189–190, 1992.

18. Daniel G. Quillen. On the formal group laws of unoriented and complex cobordism. Bulletin of the American Mathematical Society, 75:1293–1298, 1969.

19. Daniel G. Quillen. Elementary proofs of some results of cobordism theory using Steenrod operations. Advances in Mathematics, 7:29–56, 1971.

20. Douglas C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. Academic Press, 1986.

21. Nigel Ray. SU and Sp Bordism. PhD thesis, Manchester University, 1969.

22. Nigel Ray and Robert Switzer. On SU × SU bordism. Oxford Quarterly Journal of Mathematics, 21:137–150, 1970.

23. Robert E. Stong. Notes on Cobordism Theory. Princeton University Press, 1968.

24. Moss E. Sweedler. Hopf Algebras. Mathematical Lecture Note Series. W. A. Benjamin Inc, 1969.

25. Robert M. Switzer. Algebraic Topology, Homotopy and Homology, volume 212 of Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1976.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at ftp://ftp.rmi.acnet.ge/pub/hha/volumes/1999/n8/n8.(dvi,ps,dvi.gz,ps.gz)

Victor M Buchstaber buchstab@mech.math.msu.su

Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russia

Nigel Ray nige@ma.man.ac.uk

Department of Mathematics, University of Manchester, Manchester M13 9PL, England