Lobachevskii Journal of Mathematics Vol. 8, 2001, 3–18 \begin{matrix}c \\ ◯ \end{matrix}M.A.Bousaidi

M.A.Bousaidi

ON THE ETA INVARIANT, STABLE POSITIVE SCALAR CURVATURE, AND HIGHER Â-GENERA

(submitted by M.A.Malakhaltsev)

Abstract. We show that the eta invariant provides obstructions to the existence, and stable existence, of positive scalar curvature metrics on odd dimensional closed connected spin smooth manifolds. We also prove, using the eta invariant, that the ”stable existence” of a positive scalar curvature metric implies the vanishing of some higher Â-genera.

1. Introduction

This work was motivated by the Gromov-Lawson-Rosenberg conjecture, and is devoted to the study of other types of obstructions to the ”existence” and ”stable existence” of positive scalar curvature metrics. Suppose given an odd dimensional closed connected spin smooth manifold M which carries, ”absolutely” or ”stably”, a positive scalar curvature metric. Our aim is to find some of the constraints induced by this hypothesis on the eta invariant of the Dirac operator on M. The fundamental group enters in the discussion in a vital way.

Let P be a self-adjoint elliptic partial differential operator of order d > 0 on a smooth closed Riemannian manifold M. The eta function of P is defined by :

η(s, P) = ∑_{μ∈Sp(P), μ≠0} sign(μ) |μ|^{−s},

where Sp(P) is the spectrum of P. This function converges for Re(s) > (m/d), and has a meromorphic extension to ℂ with isolated simple poles, on the real axis, in the points : s = (m − n)/d, where m = dimM, and n ≥ 0. It is a deep result that s = 0 is not a pole (the proof of this result requires laborious methods of analysis and algebraic topology); see Atihay-Patodi-Singer [3] for m odd, and Gilkey [10] for m even.

AMS Subject Classification. Primary 53C07, Secondary 58G10.

Key words and phrases. Eta Invariant, Positive Scalar Curvature, K-Theory, Index Theory.

Definition 1.1.

The eta invariant of P is defined to be :

η(P) = \frac{1}{2} \{η(0, P) + dim ker(P)\}.

d The reduced eta invariant is defined to be :

η̃(P) = η(P) mod ℤ.

Remark 1: The eta invariant made its first appearence in the work of Atiyah-Patodi-Singer on index theorems for manifolds with boundary [1], [2] and [3], and has proven to be useful in K-theory, in Cobordism theory and in Differential Geometry. See for instance [5], [6], [7], [8], [11], and [22].

Let M be a smooth closed connected odd dimensional spin manifold, V a Hermitian vector bundle over M, pr : M × [0, 1] → M the projection on the first factor, and let W := pr^{∗}(V) be the pullback of V. Then, we obtain the following results : Theorem 3.1 Suppose given a positive scalar curvature metric g on a closed connected spin smooth odd dimensional manifold M. Let V be a Hermitian line vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose these connections are both flat. Then we have :

η(D_{V} , ´∇, g) = η(D_{V} , ∇, g).

The key role played by the fundamental group π_{1}(M) of M appears clearly in the following theorem : Theorem 3.2 Suppose given a positive scalar curvature metric g on a closed connected spin smooth odd dimensional manifold M and a continuous map h : M → Bπ, where π is a discrete group. Let V be a Hermitian vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose that h(M) is included in a finite subcomplex X of Bπ, and that V is a pullback of a vector bundle over X. Assume the Strong Novikov Conjecture holds for the discrete group π. Then we have :

η̃(D_{V} , ´∇, g) = η̃(D_{V} , ∇, g).

The reduced eta invariant provides also obstructions to the stable existence of positive scalar curvature metrics. A manifold M is said to carry a positive scalar curvature metric stably if for some positive integer r ≥ 1, M × \underbrace{B × ⋯ × B}_{r times} carries a positive scalar curvature metric, where B is a

Bott manifold i.e an 8-dimensional closed simply connected manifold such that index(B) = ∫_{B} Â(B) = 1. This is shown in the following theorems :

Theorem 3.3 Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g. Let V be a Hermitian line vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose these connections are both flat. Then we have :

η̃(D_{V} , ´∇, g_{M}) = η̃(D_{V} , ∇, g_{M}),

where g_{M} is the restriction of g to M. and, Theorem 3.4 Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g, and let h : M → Bπ be a continuous map, where π is a discrete group. Let V be a Hermitian vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose that h(M) is included in a finite subcomplex X of Bπ, and that V is a pullback of a vector bundle over X. Assume the Strong Novikov Conjecture holds for the discrete group π. Then we have :

η̃(D_{V} , ´∇, g_{M}) = η̃(D_{V} , ∇, g_{M}),

where g_{M} is the restriction of g to M.

Next, and as an application of theorems 3.1 and 3.3, we prove that the ”stable existence” of a positive scalar curvature metric on the closed connected spin smooth odd dimensional manifold M implies the vanishing of some higher Â-genera : Theorem 4.1 Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g. Then we have :

⟨Â(M) ∪ y, [M]⟩ = 0 ∀ y ∈ H^{1}(M, ℝ).

We give then a counterexample showing that the result of theorem 4.1 can not be extended using the eta invariant and theorems 3.1, 3.3.

Finally, we demonstrate an analog of theorem 4.1 when the fundamental group is involved : Theorem 4.2 Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g, and let h : M → Bπ be a continuous map, where π is a discrete group. Assume h(M) is included in a finite subcomplex X of Bπ, and the Strong Novikov Conjecture holds for the discrete group π. Then we have :

⟨Â(M) ∪ x_{1} ∪ ⋯ ∪ x_{2j+1}, [M]⟩ = 0 ∀ x_{1}, ⋯, x_{2j+1} ∈ H^{1}(M, ℝ).

We acknowledge with gratitude stimulating conversations with Professor A. Mesnaoui.

2. The ”Vanishing of higher Â-genera” conjecture

This conjecture has a, still unknown, relation with the Novikov conjecture, so some words are in order :

Conjecture 2.1 (Novikov Conjecture). Let M be a closed oriented smooth manifold, π a discrete group, and Bπ its classifying space. Let f : M → Bπ be a continuous map, then the ”higher signatures” ⟨L_{∗}(M) ∪ f^{∗}(a), [M]⟩, where a ∈ H^{∗}(Bπ, ℚ), are oriented homotopy invariants.

The validity of this conjecture has been established for several groups. See, for instance, Novikov [19], Kasparov [14, 15], and Lusztig [18]. The intersection product of Kasparov (see Blackadar [4]) defines a homomorphism :

β : RK_{∗}(Bπ) ⟶ K_{∗}(C^{∗}(π)) ,

where C^{∗}(π) is the group C^{∗}-algebra of π, K_{∗}(C^{∗}(π)) its K-homology groups, and RK_{∗}(Bπ) is the representable K-homology of Bπ. The Strong Novikov Conjecture claims that the homomorphism :

β ⊗ ℚ : RK_{∗}(Bπ) ⊗ ℚ → K_{∗}(C^{∗}(π)) ⊗ ℚ is injective .

The word ”Strong” is justified by the fact that this last conjecture implies the Novikov Conjecture, see Kasparov [15, theorem 6.3].

Conjecture 2.2 (Vanishing of higher Â-genera). Let M be a closed spin smooth manifold carrying a positive scalar curvature metric, π a discrete group, and h : M → Bπ a continuous map. Then, for every x ∈ H^{∗}(Bπ, ℚ),

⟨Â(M) ∪ h^{∗}(x), [M]⟩ = 0.

The next theorem contains some evidence for this conjecture :

Theorem 2.1. Let M be a closed spin smooth manifold which carries a positive scalar curvature metric, π a discrete group, and h : M → Bπ a continuous map.

. Assume the Strong Novikov Conjecture holds for π, then we have :

⟨Â(M) ∪ h^{∗}(x), [M]⟩ = 0 for all x ∈ H^{∗}(Bπ, ℚ).

. Without taking the group π into account, we have :

⟨Â(M) ∪ x_{1} ∪ ⋯ ∪ x_{2i+1}, [M]⟩ = 0 ∀ x_{1}, ⋯, x_{2i+1} ∈ H^{1}(M, ℚ), i ≥ 0.

Proof. The first statement is due to Gromov and Lawson [13] (see Rosenberg [20] for a proof). The second statement follows closely the one given by Rosenberg [20], and is as follows : The result is evidently trivial when M is even-dimensional, so we will treat only the case : M odd-dimensional. Consider some elements x_{1}, ⋯, x_{2i+1} ∈ H^{1}(M, ℚ), then there exist continuous maps g_{j} : M → S^{1} which represent the x_{j}, i.e : There exist elements σ_{j} ∈ H^{1}(S^{1}, ℚ); S^{1} = B ℤ; such that x_{j} = g_{j}^{∗}(σ_{j}) j = 0, ⋯, 2i + 1, see for instance Spanier [21, chapter 8].

Now, let g : M ⟶^{Δ} \underbrace{M × ⋯ × M}_{(2i+1) times} ⟶^{×_{j=1}^{2i+1} g_{j}} \underbrace{S^{1} × ⋯ × S^{1}}_{(2i+1) times} where Δ is the diago-

nal map, and consider the even-dimensional manifold N = M × S^{1}, and the

continuous map f := g × id : N → \underbrace{S^{1} × ⋯ × S^{1}}_{(2i+1) times} ×S^{1} = B(\underbrace{ℤ ⊕ ⋯ ⊕ ℤ}_{(2i+2) times}).

Applying the Fomenko-Miščenko theorem [9], as in Rosenberg [20], we see that :

β(f_{∗}(ch^{−1}(Â(N) ∩ [N]))) = 0.

But the Kasparov map β is an isomorphism for free abelian groups, hence is an isomorphism for the group \underbrace{ℤ ⊕ ⋯ ⊕ ℤ}_{(2i+2) times}.

This gives : f_{∗}(Â(N) ∩ [N]) = 0, (the Chern character ”ch” being an isomorphism), i.e :

(g_{∗} × id_{∗})((Â(M) ´⊗ 1) ∩ ([M] ´⊗ [S^{1}])) = 0,

where 1 ∈ H^{0}(S^{1}), [S^{1}] the fundamental class of S^{1}, and ´⊗ is in order of appearence the cohomological and homological external product (see Spanier [21]). We deduce then that :

g_{∗}(Â(M) ∩ [M]) = 0 in H_{∗}(Bℤ, ℚ).

This is equivalent to :

⟨Â(M) ∪ g^{∗}(a), [M]⟩ = 0 for every a ∈ H^{∗}(B(\underbrace{ℤ ⊕ ⋯ ⊕ ℤ}_{(2i+1) times}), ℚ).

Taking a = σ_{1} ∪ ⋯ ∪ σ_{2i+1}, we get :

⟨Â(M) ∪ x_{1} ∪ ⋯ ∪ x_{2i+1}, [M]⟩ = 0.

This completes the proof of the theorem.

3. The eta invariant

This section contains the obstructions furnished by the eta invariant to the ”existence” ans ”stable existence” of positive scalar curvature metrics. Let M be a smooth closed odd dimensional manifold, V a Hermitian vector bundle over M, pr : M × [0, 1] → M the projection on the first factor, and W := pr^{∗}(V) be the pullback of V. Let ∇ and ´∇ be two unitary connections on V, and endow W with the unitary connection ∇^{×} defined by :

(∇^{×} s)(x, t) := (∇_{t} s(., t))(x) + dt ∧ \frac{∂s(x, t)}{∂t} ,

where s ∈ Ω^{0}(M × [0, 1], W) and ∇_{t} := (1 − t) ∇ + t ´∇. Then, we have the following result :

Proposition 3.1. Suppose given two unitary flat connections ∇ and ´∇ on the Hermitian vector bundle V, and endow W with the unitary connection ∇^{×} described above. Then, we have :

ch_{n}(W) = − n (\frac{i}{2π})^{n} \frac{1}{n!} (t − t^{2})^{n−1} Tr(θ^{2n−1}) ∧ dt ∀ n ≥ 1,

where θ = ώ − ω, ω and ώ being the 1-form connections of the connections ∇ and ´∇ respectively.

Proof. The curvature Ω^{×} of the connection :

∇^{×} := ∇_{t} + dt ∧ \frac{∂}{∂t}

is given by :

\begin{aligned}Ω^{×}(x, t) & = & dω^{×}(x, t) − ω^{×}(x, t) ∧ ω^{×}(x, t) \\ & = & [ω(x) − ώ(x)] ∧ dt + Ω_{t}(x), \end{aligned}

where ω and ώ are respectively the connection 1-forms of ∇ and ´∇, and Ω_{t} is the curvature of the connection ∇_{t}. Moreover, by induction, we obtain :

Tr((Ω^{×})^{n})(x, t) = n Tr([ω(x) − ώ(x)] ∧ Ω_{t}^{n−1}(x)) ∧ dt + Tr(Ω_{t}^{n})(x) ∀n ≥ 1.

We begin by rewriting ω_{t} in the form :

ω_{t} = ω + t (ώ − ω) = ω + t θ.

Next, fix x ∈ M and choose a local frame for T^{∗}M such that ω(x) = 0. Then, we get :

ώ = θ and Ώ = dθ − θ ∧ θ = 0 ⟹ θ ∧ θ = dθ.

It follows that :

Ω_{t} = (t − t^{2}) θ ∧ θ = (t − t^{2}) dθ.

Thus, for n ≥ 1, we get :

Tr((Ω^{×})^{n}) = (t − t^{2})^{n} d(Tr(θ^{2n−1})) − n (t − t^{2})^{n−1} Tr(θ ∧ \underbrace{dθ ∧ ⋯ ∧ dθ}_{(n−1) times}) ∧ dt.

But, in such circumstances, we know that :

ch( ´∇) − ch(∇) = d(Tch( ´∇, ∇)) = 0,

where Tch( ´∇, ∇) is the transgression polynomial of the Chern character ”ch”, and is given in this special case by :

Tch( ´∇, ∇) = ∑_{k≥1} Tch_{k}( ´∇, ∇)
= ∑_{k≥1} (i/2 π)^{k} \frac{(k−1)!}{(2k−1)!} Tr(θ^{2k−1}).

This leads immediately to the result :

d(Tr(θ^{2k−1})) = 0 ∀k ≥ 1,

and shows that :

Tr((Ω^{×})^{n}) = − n (t − t^{2})^{n−1} Tr(θ^{2n−1}) ∧ dt.

This completes the proof of the proposition.

Proposition 3.2. Suppose given two unitary connections ∇ and ´∇ on the Hermitian line vector bundle V, and endow W with the unitary connection ∇^{×} described above. Suppose these two connections are both flat. Then, we have :

∫_{M×[0,1]} Â(M) ∧ ch_{n}(W) = 0 ∀ n ≥ 1.

Proof. According to proposition 3.1, we know that :

\{ \begin{matrix}ch_{n}(W) & = & − \frac{i}{2π} θ ∧ dt if n = 1 \\ & = & 0 ∀ n ≥ 1 . \end{matrix}

Moreover,

∫_{M×[0,1]} Â(M) ∧ ch_{1}(W) = − \frac{i}{2π} ∫_{M×[0,1]} Â(M) ∧ θ ∧ dt
= − \frac{i}{2π} ∫_{M} Â(M) ∧ θ
= 0,

for ∫_{M} Â(M) ∧ θ = 0 thanks to theorem 2.1. This completes the proof.

for the remainder of the paper, we will need the following Bochner-Lichnerowicz formulae :

Lemma 3.1. Let (M,g) be a Riemannian spin smooth manifold with scalar curvature sc(g), and let Δ(M) ⊗ V be the spin bundle of M twisted with any complex Hermitian vector bundle V (endowed with a unitary connection). Then the Dirac operator D_{V} and the connection Laplacian ∇^{∗}∇ of Δ(M) ⊗ V satisfy the identity:

D_{V}^{2} = ∇^{∗}∇ + \frac{1}{4} sc(g) + R^{V} .

See Lawson-Michelsohn [16] for the definition of the quantities involved, and for the classical proof.

Let g be a Riemannian metric on M and let P_{g} be the Dirac operator on M × [0, 1] defined by the product metric. Let P_{g} ⊗ W be the Dirac operator with coefficients in W. Impose the Atiyah-Patodi-Singer global boundary conditions. Then, the first result of this section is stated in the following :

Theorem 3.1. Suppose given a positive scalar curvature metric g on a closed connected spin smooth odd dimensional manifold M. Let V be a Hermitian line vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose these connections are both flat. Then we have :

η(D_{V} , ´∇, g) = η(D_{V} , ∇, g).

Proof. Let ω and ώ be the connection 1-forms of the connections ∇ and ´∇. In the sequel, W will be endowed with the connexion ∇^{×} of proposition 3.1. Then, By the Atiyah-Patodi-Singer index theorem for manifolds with boundary, we have :

index(P_{g} ⊗ W) = ∫_{M×[0,1]} Â(M × [0, 1]) ch(W) − [η(D_{V} , ´∇, g) − η(D_{V} , ∇, g)].

.Claim 1: Â(M × [0, 1]) = Â(M). Since the metric used is a product metric, we have :

Â(M × [0, 1]) = Â(Ω_{M×[0,1]})
= Â(Ω_{M} ⊗ 1 ⊕ 1 ⊗ Ω_{[0,1]})
= Â(Ω_{M}) because Ω_{[0,1]} = 0
= Â(M).

Hence we deduce, thanks to proposition 3.2, that :

index(P_{g} ⊗ W) = η(D_{V} , ∇, g) − η(D_{V} , ´∇, g).

.Claim 2: index(P_{g} ⊗ W) = 0.

It is well known that index(P_{g} ⊗ W) is the spectral flow of the family {D ⊗ V_{t}}_{t∈[0,1]} (see Atiyah-Patodi-Singer [3, p. 93 to 95]). Consider the smooth maps : pr_{t} : M → M × [0, 1], x ↦ (x, t) (t ∈ [0, 1]), and let V_{t} := (pr_{t})^{∗}(W). It is clear that V_{t} is isomorphic to V ∀ t ∈ [0, 1]. V_{t} is endowed with the connection ∇_{t} := (1 − t) ∇ + t ´∇. The Hermitian vector bundles V_{t}, t ∈ [0, 1], are flat line bundles over the closed manifold M since the curvature (∇_{t})^{2} = 0, as one can see in the proof of proposition 3.1. Hence, and thanks to the Lichnerowicz formula above, we see that D_{V_{t}} is invertible ∀ t ∈ [0, 1], and then that index(P_{g} ⊗ W) = 0. This completes the proof of the theorem.

The key role played by the fundamental group π_{1}(M) appears clearly in the following theorem :

Theorem 3.2. Suppose given a positive scalar curvature metric g on a closed connected spin smooth odd dimensional manifold M and a continuous map h : M → Bπ, where π is a discrete group. Let V be a Hermitian vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose that h(M) is included in a finite subcomplex X of Bπ, and that V is a pullback of a vector bundle over X. Assume the Strong Novikov Conjecture holds for the discrete group π. Then we have :

η̃(D_{V} , ´∇, g) = η̃(D_{V} , ∇, g).

Proof. W = pr^{∗}(V), where pr : M × [0, 1] → M is the projection on the first factor, is the pullback of a complex vector bundle over X. Now, thanks to the Kunnëth formula in cohomology, we have :

ch(W) = ∑_{i , j} α_{i} β_{j} where α_{i} ∈ H^{∗}(M; ℝ) and β_{j} ∈ H^{∗}([0, 1]; ℝ)
= k^{∗}(ch(W_{X})) ∈ k^{∗}(H^{∗}(X; ℝ)),

where, k = h ∘ pr : M × [0, 1] → M → X. The fact k = h ∘ pr shows that we must have α_{i} ∈ ((inc) ∘ h)^{∗}(H^{∗}(Bπ; ℝ)) ∀ i, where inc : X ↪ Bπ is the inclusion map. Next, endowing W with some unitary connection ∇^{×} which connects ∇ and ´∇, and applying the Atiyah-Patodi-Singer index theorem as in theorem 3.1, we get :

index(P_{g} ⊗ W) = ∫_{M×[0,1]} Â(M × [0, 1]) ch(W) − [η(D_{V} , ´∇, g) − η(D_{V} , ∇, g)].

But,

∫_{M×[0,1]} Â(M × [0, 1]) ch(W) = ⟨Â(M × [0, 1]) ∪ ch(W) , [M × [0, 1]]⟩
= ∑_{i , j} ⟨Â(M) ∪ α_{i} , [M]⟩ . ⟨β_{j} , [[0, 1]]⟩
= 0 according to theorem 2.1.

This completes the proof of the theorem.

This is the most one can show in this situation, and the problem is that ∥R^{V_{t}}∥ is not small enough to ensure the vanishing of index(P_{g} ⊗ W). In the following example, we present a case in which the vanishing of this index can be guaranteed. Example 1: Consider the trivial line complex vector bundle V = M ×ℂ over M. Let us endow V with the two connections ∇_{s} = d+s i ω and ´∇_{s} = d+s i ώ, where ω , ώ ∈ Λ^{1}(M, ℝ) and s ∈ ℝ. W will be endowed, as above, with the connection ∇_{s}^{×} = (1 − t)∇_{s} + t ´∇_{s} + dt ∧ \frac{∂}{∂t}. The vector bundle V_{t} is endowed with the connection (1 − t)∇_{s} + t ´∇_{s}, and we calculate that R^{V_{t}} = i s c(dω_{t}), where ”c” is Clifford multiplication. Hence, the Bochner-Lichnerowicz formulae for D_{V_{t}} is :

(D_{V_{t}})^{2} = ∇^{∗}∇ + \frac{1}{4} sc(g) + i s c(dω_{t}).

On the other hand, it is well known that ω_{t} defines via the Riemannian metric, a skew-symmetric endomorphism of T M. Hence, c(dω_{t}) is a skew-symmetric action on spinors of T M. Let {iλ_{j}^{t}}_{j} be its eigenvalues. These are pure imaginary functions of M. Hence, there exists a constant k > 0 such that ∀ 0 ≤ s < k, we have :

\begin{aligned}((D_{V_{t}})^{2}u, u) & ≥ & \frac{1}{4} (sc(g)u, u) + s (i c(dω_{t})u, u) \\ & ≥ & k (u, u) ∀ t ∈ [0, 1], \end{aligned}

where (., .) is the inner product on C^{∞}(Δ(M)), Δ(M) being the spinor bundle of M, and u ∈ C^{∞}(Δ(M)). It follows that for 0 ≤ s < k, the operator D_{V_{t}} is invertible ∀t ∈ [0, 1]. Hence, index(P_{g} ⊗ W), being the spectral flow of the family {D ⊗ V_{t}}_{t∈[0,1]}, vanishes. End of Example 1

The reduced eta invariant provides also obstructions to the stable existence of positive scalar curvature metrics. A manifold M is said to carry a positive scalar curvature metric stably if for some positive integer r ≥ 1, M × \underbrace{B × ⋯ × B}_{r times} carries a positive scalar curvature metric, where B is a

Bott manifold i.e an 8-dimensional closed simply connected manifold such that index(B) = ∫_{B} Â(B) = 1.

Theorem 3.3. Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g. Let V be a Hermitian line vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose these connections are both flat. Then we have :

η̃(D_{V} , ´∇, g_{M}) = η̃(D_{V} , ∇, g_{M}),

where g_{M} is the restriction of g to M.

Proof. It is sufficent to suppose that M × B carries a positive scalar curvature metric g. Let V^{B} := pr_{1}^{∗}(V (∇)) where pr_{1} : M × B → M is the projection of the first factor, and V_{×}^{B} := pr^{∗}(V^{B}) where pr : M ×B×[0, 1] → M ×B is the projection (x, b, t) ↦ (x, b). Here, V (∇) is V endowed with the connection ∇. Similarly, we define V́^{B} := pr_{1}^{∗}(V ( ´∇)) and V́_{×}^{B}. We will also use the notation ∇ and ´∇ for their pullbacks by the projection pr_{1} ∘ pr. Consider the product metric g_{M} × g_{B} where g_{M} (resp. g_{B}) is the restriction of g to M (resp. to B), and let G be any metric on M ×B ×[0, 1] which connects g and g_{M} × g_{B}. Let P_{G} be the Dirac operator over M × B × [0, 1] defined by the metric G, then the Atiyah-Patodi-Singer index theorem gives :

\begin{aligned}index(P_{G} ⊗ V_{×}^{B}) & = & ∫_{M×B×[0,1]} Â(M × B × [0, 1]) ch(V_{×}^{B}) \\ & − & [η(D_{V^{B}}^{M×B}, ∇, g_{M} × g_{B}) − η(D_{V^{B}}^{M×B}, ∇, g)], \end{aligned}

where D^{M×B} is the Dirac operator over M × B. In the same manner, for V ( ´∇) we get :

\begin{aligned}index(P_{G} ⊗ V́_{×}^{B}) & = & ∫_{M×B×[0,1]} Â(M × B × [0, 1]) ch(V́_{×}^{B}) \\ & − & [η(DM×B ; V́^{B}, ´∇, g_{M} × g_{B}) − η(DM×B ; V́^{B}, ´∇, g)]. \end{aligned}

Hence substracting the two identities for V and V́ above, and using theorem 3.1, we get :

η(DM×B ; V́^{B}, ´∇, g_{M} × g_{B}) = η(D_{V^{B}}^{M×B}, ∇, g_{M} × g_{B}) mod ℤ. (3.3.1)

Now, writing down the expression of the operator D^{M×B} defined by the metric g_{M} × g_{B}, we get :

η(D_{V^{B}}^{M×B}, ∇, g_{M} × g_{B}) = η(D_{V} , ∇, g_{M}) + [dimker(D_{V} , g_{M}) . dimker(D_{−}^{B}, g_{B})], where D_{+}^{B} is the Dirac operator over B and D_{−}^{B} is its adjoint. The same thing is true for V ( ´∇), so that we get finally :

η(D_{V} , ∇, g_{M}) ≡ η(DV́ , ´∇, g_{M}).

This completes the proof of the theorem.

In the same vein, we prove a similar result involving the fundamental group. In effect, on the basis of theorem 3.2 this time and following faithfully the proof of theorem 3.3 above, we show that :

Theorem 3.4. Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g, and let h : M → Bπ be a continuous map, where π is a discrete group. Let V be a Hermitian vector bundle over M endowed with two unitary connections ∇ and ´∇. Suppose that h(M) is included in a finite subcomplex X of Bπ, and that V is a pullback of a vector bundle over X. Assume the Strong Novikov Conjecture holds for the discrete group π. Then we have :

η̃(D_{V} , ´∇, g_{M}) = η̃(D_{V} , ∇, g_{M}),

where g_{M} is the restriction of g to M.

Remarks 1:

1. The identity (3.3.1) is true for any metric k on M × B instead of g_{M} × g_{B}.

2. The eta invariant can distinguish, in the case of theorems 3.1 and 3.3, between the concepts of ”existence” ans ”stable existence” of positive scalar curvature metrics on M. However it can not do the same in the case of theorems 3.2 and 3.4, except in few cases as the one given in example 1.

4. The vanishing of some higher Â-genera

We show in this section that the ”stable existence” of a positive scalar curvature metric on M implies the vanishing of some higher Â-genera. The results obtained are as follows :

4.1. The case without the Strong Novikov Conjecture.

Theorem 4.1. Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g. Then we have :

⟨Â(M) ∪ y, [M]⟩ = 0 ∀ y ∈ H^{1}(M, ℝ).

Proof. It will be enough to suppose that M × B carries a positive scalar curvature metric g, so let it be so. Let V be a Hermitian line vector bundle over M endowed with two flat unitary connections ∇ and ´∇. The formerly defined Hermitian vector bundle W will be endowed with the connection ∇^{×}, and let W^{B} be the pullback of W, via the projection p̂r : M × [0, 1] × B → M × [0, 1], over M × [0, 1] × B. Consider the projections on the first factor : pr : M × B × [0, 1] → M × B and pr_{1} : M × B → M, and let V^{B} : pr_{1}^{∗}(V). Then, W^{B} = pr^{∗}(V^{B}) and we endow it with the connection (1 − t) pr_{1}^{∗}(∇) + t pr_{1}^{∗}( ´∇) + dt ∧ ∂/∂t. Let P_{g_{M} ×g_{B}} be the Dirac operator over M × B × [0, 1] defined by the product metric. Then, by the Atiyah-Patodi-Singer index theorem, we have :

\begin{aligned}index(P_{g_{M} ×g_{B}} ⊗ W^{B}) & = & ∫_{M×B×[0,1]} Â(M × B × [0, 1]) ∧ ch(W^{B}) \\ & − & \underbrace{[η(D_{V^{B}}^{M×B}, ´∇, g_{M} × g_{B}) − η(D_{V^{B}}^{M×B}, ∇, g_{M} × g_{B})]}_{(∗)}, \end{aligned}

where D^{M×B} is the Dirac operator over M × B. But, according to theorem 3.3 and the mod ℤ - equality (3.3.1), we know that (∗) ∈ ℤ. Hence, we get :

∫_{M×B×[0,1]} Â(M × B × [0, 1]) ∧ ch(W^{B}) ∈ ℤ.

Or equivalently :

∫_{M×[0,1]} Â(M × [0, 1]) ∧ ch(W) ∈ ℤ.

Next, and thanks to the proof of proposition 3.2, this expression yields :

\frac{−i}{2π} ∫_{M} Â(M) ∧ θ ∈ ℤ, where θ = ώ − ω.

Now, consider an element y ∈ H^{1}(M, ℝ), and let θ́ ∈ Λ^{1}(M, ℝ) be one of its differential geometric representatives. θ́ is a closed differential form. Let V = (M × ℂ → M) be the trivial line complex vector bundle over M. We endow V with the two flat unitary connections : . ∇ = d, where ”d” is the differentiation operator on smooth functions f : M → ℂ. . ´∇_{s} = d + s i θ́ = d + s θ , where s ∈ ℝ. Then, the above analysis gives :

\frac{−i}{2π} ∫_{M} Â(M) ∧ s θ ∈ ℤ ∀ s ∈ ℝ.

Or equivalently :

(∫_{M} Â(M) ∧ θ́) . s ∈ 2π . ℤ ∀ s ∈ ℝ.

This last formulae forces the result :

⟨Â(M) ∪ y, [M]⟩ = ∫_{M} Â(M) ∧ θ́ = 0,

and completes the proof of the theorem.

The result of the above theorem is the maximum one can show using the eta invariant and theorem 3.1. In effect, the following counterexample shows that theorem 4.1 can not be extended to include more higher Â-genera. Counterexample 1:

The proof of theorem 4.1 relies heavily on the mod ℤ - equality (*) therein, which in turn is a consequence of theorem 3.1. On the other hand, improving theorem 4.1 by means of this method implies using Hermitian vector bundles V with dimV > 1. However, theorem 3.1 is no longer true when dimV > 1 as will be shown below :

Consider the spherical space form : M := S^{3}/τ(ℤ/3), where

τ : ℤ/3 = ⟨λ = e^{2iπ/3}⟩ ⟶ U(2), τ(λ) = ( \begin{matrix}λ & 0 \\ 0 & λ^{2} \end{matrix} ), and let V_{τ} = ( \begin{matrix}S^{3} ×_{τ} ℂ^{2} \\ ↓ \\ M \end{matrix} )

be the locally flat vector bundle defined by τ. V_{τ} is topologically trivial. Next, we define :

\begin{aligned}g & : & S^{3} & ⟶ & SU(2) \\ & & (z_{1}, z_{2}) & ↦ & ( \begin{matrix}z_{1} & z_{2}^{2} \\ −\overline{z_{2}}^{2} & \overline{z_{1}} \end{matrix} ) . \end{aligned}

It is well known that g defines a global unitary frame s_{g} for the trivial bundle S^{3} × ℂ^{2}, and this allows us to define a connection on it by ∇_{g}(s_{g}) = 0. Moreover, we verify easily that

g(τ(λ)(z_{1}, z_{2})) = τ(λ) . g(z_{1}, z_{2}).

Hence, g descends to give an equivariant frame for V_{τ}, which we denote by g also. It follows that ∇_{g} descends also to a connection on V_{τ} denoted again by ∇_{g}. We endow V_{τ} with the two flat unitary connections ∇_{τ} and ∇_{g}. Their respective connection 1-forms will be denoted by ω_{τ} and ω_{g}. Then, applying the Atiyah-Patodi-Singer index theorem as in theorem 3.1 and using proposition 3.1, we get :

η(D_{V_{τ}} , ∇_{τ}, h) − η(D_{V_{τ}} , ∇_{g}, h) = (1/24π^{2}) ∫_{M} Tr(θ^{3}) mod ℤ,

where h is the metric with constant positive sectional curvature inherited by M, and θ := ω_{τ} − ω_{g}. We see easily that θ = g^{−1}dg. Let us compute Tr(θ^{3}) : First, Tr(θ^{3}) is a left invariant 3-form for the group action, and consequently Tr(θ^{3}) = c dvol, where ”c” is a constant (we are following here the notations of [12, p. 320]). Next, if z_{ν} = x_{ν} + i y_{ν}, then :

g(z_{1}, z_{2}) = x_{1} e_{0} + y_{1} e_{1} + (x_{2}^{2} − y_{2}^{2}) e_{2} + 2 x_{2} y_{2} e_{3} ,

where e_{0} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ), e_{1} = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ), e_{2} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), and e_{3} = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ).

We compute ”c” by evaluating Tr(θ^{3}) at the point P = (0, 0, 1, 0). We compute :

g^{−1} = − e_{2} and dg = e_{0} dx_{1} + e_{1} dy_{1} + 2 e_{3} dy_{2} ,

and this gives :

(g^{−1}dg)^{3} = − 12 e_{0} dx_{1} ∧ dy_{1} ∧ dy_{2} .

Consequently :

Tr(θ^{3}) = − 24 dx_{1} ∧ dy_{1} ∧ dy_{2} = − 24 dvol .

Hence, we get finally :

η(D_{V_{τ}} , ∇_{τ}, h) − η(D_{V_{τ}} , ∇_{g}, h) = (−1/π^{2}) ∫_{M} dvol mod ℤ
= (−1/π^{2}) (1/3) Vol(S^{3}) mod ℤ
= − 2/3  mod ℤ.

This gives the desired counterexample.

4.2. The case with the Strong Novikov Conjecture.

Theorem 4.2. Suppose that the closed connected spin smooth odd dimensional manifold M carries stably a positive scalar curvature metric g, and let h : M → Bπ be a continuous map, where π is a discrete group. Assume h(M) is included in a finite subcomplex X of Bπ, and the Strong Novikov Conjecture holds for the discrete group π. Then we have :

⟨Â(M) ∪ x_{1} ∪ ⋯ ∪ x_{2j+1}, [M]⟩ = 0 ∀ x_{1}, ⋯, x_{2j+1} ∈ H^{1}(M, ℝ).

Proof. It will be enough to suppose that M × B carries a positive scalar curvature metric g, so let it be so. Let V be a Hermitian vector bundle over M endowed with two unitary connections ∇ and ´∇, which is the pullback of a vector bundle over X. The formerly defined vector bundle W is endowed with the connection ∇^{×} as before. Then, as in the proof of theorem 4.1, we get :

∫_{M×[0,1]} Â(M × [0, 1]) ∧ ch(W) ∈ ℤ.

Now, let θ_{l} be one of the differential geometric representatives of the cohomological class x_{l}, 1 ≤ l ≤ 2j + 1. Consider the trivial complex vector bundle V = (M × ℂ^{j+1} → M), and endow it with the two connections : . ∇ = d . . ´∇_{s} = d + s θ, where s ∈ ℝ and :

θ = ( \begin{matrix}i θ_{1} & θ_{2} + i θ_{3} & ⋯ & θ_{2j} + i θ_{2j+1} \\ −θ_{2} + i θ_{3} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ −θ_{2j} + i θ_{2j+1} & 0 & ⋯ & 0 \end{matrix} ) .

Then, dθ = 0, Tr(θ^{2j+1}) = c . θ_{1} ∧ ⋯ ∧ θ_{2j+1} for a constant c ≠ 0. Moreover :

Tr((Ω_{s}^{×})^{n}) = (−1)^{n} n s (st)^{2n−2} Tr(θ^{2n−1}) ∧ dt + (−1)^{n} (st)^{2n} Tr(θ^{2n}).

Hence,

∫_{M×[0,1]} Â(M) ∧ ch(W) = ∑_{n≥1} (\frac{−i}{2π})^{n} \frac{1}{(n − 1)!} \frac{1}{2n − 1} (∫_{M} Â(M) ∧ Tr(θ^{2n−1})) s^{2n−1}.

This is a polynomial in ”s” with values in ℤ and no constant term, ∀ s ∈ ℝ. Hence, its coefficients must vanish. In particular :

⟨Â(M) ∪ x_{1} ∪ ⋯ ∪ x_{2j+1}, [M]⟩ = c^{−1} ∫_{M} Â(M) ∧ Tr(θ^{2j+1}) = 0.

This completes the proof of the theorem.

18 M.A.BOUSAIDI

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Department of mathematics, Mohamed V University, B.P : 1014, RA-BAT, MOROCCO. E-mail address: bousaidi@espacenet.net.ma

Received November, 11, 2000