DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 23–27

THE MODULI SPACE M(M) OF CERTAIN

NON–ORIENTABLE MANIFOLDS OF EVEN DIMENSION

EGIDIO BARRERA-YANEZ

Abstract. Let M be an even dimensional manifold with π_{1}(M) = ℤ_{ℓ} for ℓ = 2^{ν} ≥ 2. We assume that the universal cover M̃ is spin. We shall define N(M) = M̃ × M̃/ℤ_{2ℓ} and establishe a relation between the eta invariant of M and the eta invariant of N(M). We use this to study the moduli space of metrics of positive scalar curvature on the manifold M.

1.0 Introduction. We shall use the eta invariant (which is an analytical invariant) and equivariant spin bordism (which is a topological invariant) to study the space of metrics of positive scalar curvature on a manifold (this is a geometrical question).

Let ℓ := 2^{ν} ≥ 2 henceforth; we are only interested in 2 primary phenomena. Let g_{ℓ} = e^{2π\sqrt{−1}/ℓ} be the canonical generator of the cyclic group ℤ_{ℓ} = \{λ ∈ ℂ : λ^{ℓ} = 1\}. Let ρ_{s}(g_{ℓ}) := e^{2πs\sqrt{−1}/ℓ} define a linear representation of ℤ_{ℓ}.

Let M be a closed manifold of dimension m ≥ 5, let M̃ be the universal cover of M, assume that M̃ has a given spin structure. Let R(M) be the space of metrics on M with positive scalar curvature, let M(M) = R(M)/Diff(M) be the associated moduli space. A homomorphism σ : π_{1}(M) → ℤ_{ℓ} defines a map f_{σ} (which is unique up to homotopy) from M to the classifying space Bℤ_{ℓ} of ℤ_{ℓ}. If π_{1}(M) = ℤ_{ℓ}, then the natural homomorphism σ is the identity map. In 1.1 and 1.2 we define the equivariant spin bordism groups, in 1.3 we present the eta invariant which is the analytical invariant that we use in the study of the moduli space M(M). In 1.4 we define a twisted product N(M) of dimension 2m associated to M which is nonorientable and which has a canonical pin^{c} structure. In 1.8 we shall show that the moduli space of metrics of positive scalar curvature on certain non-orientable manifolds has an infinite number of arc components; see Theorem 1.10 for details.

1.1 Equivariant spin bordism. Let ξ be a real vector bundle over Bℤ_{ℓ}. The equivariant spin bordism groups MSpin_{m}(Bℤ_{ℓ}, ξ) are the equivalence classes of triples (M, f, s) where M is a closed manifold of dimension m which need not be connected, f is a ℤ_{ℓ} structure on M or equivalently a representation of the fundamental group π_{1}(M), and s is a spin structure on T (M) ⊕ f^{∗}ξ; we define the relation (M, f, s) ∼ 0 in MSpin_{m}(Bℤ_{ℓ}, ξ) if there exists a compact manifold X with boundary M so that the structures f and s extend over X. Let M be a manifold with π_{1}(M) = ℤ_{ℓ} whose universal cover M̃ admits a spin structure. Let ρ be a representation of ℤ_{ℓ}. There exists bundles and suitable structures s so that [(M, f, s)] ∈ MSpin_{m}(Bℤ_{ℓ}, ξ). Only two Stiefel Whitney classes ω_{i} for i = 1, 2 play a role. There are 4 cases to consider:

1991 Mathematics Subject Classification. Primary 55N15; Secondary 58G12.

Key words and phrases. Moduli space, eta invariant.

Research partially supported by the Abdus Salam International Centre for Theoretical Physics (Trieste, Italy).

This paper is in final form and no version of it will be submitted for publication elsewhere.

(1) We have ω_{1}(ξ_{0}) = 0 and ω_{2}(ξ_{0}) = 0. We may take ξ_{0} to be the trivial line bundle and identify MSpin_{m}(Bℤ_{ℓ}, ξ_{0}) with the ordinary bordism groups MSpin_{m}(Bℤ_{ℓ}); such a manifold admits a canonical spin structure s.

(2) We have ω_{1}(ξ_{1}) = 0 and ω_{2}(ξ_{1}) ≠ 0. We may take ξ_{1} to be the underlying real 2 plane bundle of the complex line bundle defined by the representation ρ_{1}. Such a manifold M admits a canonical spin^{c} structure s with determinant line bundle given by ρ_{1}.

(3) We have ω_{1}(ξ_{2}) ≠ 0 and ω_{2}(ξ_{2}) = 0. We may take the real line bundle define by ρ_{ℓ/2}. Such a manifold M admits a canonical pin structure s.

(4) We have ω_{1}(ξ_{3}) ≠ 0 and ω_{2}(ξ_{3}) ≠ 0. We may take ξ_{3} = ξ_{1} ⊕ ξ_{2}. Such a manifold M admits a canonical pin^{c} structure s with determinant line bundle given by ρ_{1}.

1.2 Geometric equivariant spin bordism. Let τ := R_{ijji} be the scalar curvature of a Riemannian metric g; we consider quadruples (M, f, s, g) where (M, f, s) are as in §1.1, and g is a metric of positive scalar curvature on M; necessarily m ≥ 2. The geometric equivariant spin bordism groups ^{+}MSpin_{m}(Bℤ_{ℓ}, ξ) are defined similarly to the bordism groups MSpin_{m}(Bℤ_{ℓ}, ξ). For m ≥ 2 we consider quadruples (M, f, s, g) as above and say that (M, f, s, g) ∼ 0 if there exists a compact manifold X with boundary M so that the structures s and f extend over X and so that the metric g extends over X as a metric of positive scalar curvature which is the product near the boundary M.

1.3 The eta invariant. Let M be a compact Riemannian manifold of dimension m. If m is even, we assume that M admits a pin^{c} structure s; if m is odd, we assume that M admits a spin^{c} structure s; see Barrera-Yanez [3] and Gilkey [5] for further details. We assume that M is equipped with a ℤ_{ℓ} structure f. If ρ is a representation of ℤ_{ℓ}, let D_{ρ} be the Dirac operator defined by a metric g and the (s)pin^{c} structures with coefficients in the locally flat bundle of spinors S_{ρ} over M defined by ρ. The operator D_{ρ} has a discrete spectral resolution. Let {λ_{n}} be the eigenvalues of D_{ρ} where each eigenvalue is repeated according to multiplicity; we have |λ_{n}| → ∞. The eta function of Atiyah, Patodi, and Singer [1] is defined by

η(D_{ρ})(z) := _{∑λ_{n}≠0} sign(λ_{n})|λ_{n}|^{−z} + dim ker D_{ρ} .

This sum converges to define a holomorphic function of z for ℜ(z) ≫ 0; it has a meromorphic extension to ℂ which is regular at z = 0 see [5] for details. The eta invariant is a measure of the spectral asymmetry of D_{ρ} which is defined by

η(M)(ρ) := \frac{1}{2}\{η(D_{ρ})(z)\}_{z=0}.

THE MODULI SPACE M(M) OF NON–ORIENTABLE MANIFOLDS 25

The eta invariant is additive with respect to direct sums and extends to the group representation ring RU(ℤ_{ℓ}).

1.4 Twisted Products. Assume that M has fundamental group ℤ_{ℓ}, that the universal cover M̃ of M is spin, and that M is connected. If m is odd, assume that M is orientable; if m is even, assume that M is non-orientable. Thus the deck group action x̃ → g_{ℓ}x̃ preserves the orientation of M̃ if m is odd and reverses the orientation if m is even. Give M̃ ×M̃ the product spin structure. We define a fixed point free action of ℤ_{2ℓ} on M̃ × M̃ by g_{2ℓ} : (x̃, ỹ) → (g_{ℓ}ỹ, x̃). Let

N := N(M) := M̃ × M̃/ℤ_{2ℓ}

be the resulting quotient manifold. If m is odd, the flip reverses the orientation and g_{ℓ} preserves the orientation. If m is even the flip preserves the orientation and g_{ℓ} reverses the orientation. Thus the action of g_{2ℓ} on M̃ × M̃ reverses the orientation so N is not orientable. We refer to [2, 3] for the proof of the following Lemma:

1.5 Lemma. Let m be even, with the notation established above, we have:

(1) If ℓ = 2 and if m is even, then [N(M)] ∈ MSpin_{2m}(Bℤ_{4}, ξ_{3}) admits a canonical pin^{c} structure.

(2) If ℓ ≥ 4 and if m is even, then [N(M)] ∈ MSpin_{2m}(Bℤ_{2ℓ}, ξ_{2}) admits a canonical pin^{−} structure.

(3) If m is odd, then [N(M)] ∈ M_{2m}(Bℤ_{2ℓ}, ξ_{i}) for i = 2, 3 admits a canonical pin^{−} or pin^{c} structure.

We use these structures to define Dirac operators on M and N with coefficients in the line bundle defined by ρ_{s} and let η(M)(ρ_{s}) and η(N)(ρ_{s}) be the resulting eta invariants, see [3,6] for details. Let b = 0 if M admits a (s)pin^{−} structure s so that the associated complex line bundle det(s) is trivial, let b = 1 if M admits a (s)pin^{c} structure s so that the associated complex line bundle det(s) is given by the representation ρ_{1}. Then det(s) = ρ_{b}. If m is odd, Gilkey proved [6] that:

1.6 Theorem. Let m be odd and let M and N be as above. If m ≡ 3 mod 4, let β = 0; if m ≡ 1 mod 4, let β = ℓ/2.

(1) If u = 2s − b + β, then η(N)(ρ_{u}) = η(M)(ρ_{s}) + η(M)(ρ_{s−ℓ/2}) in ℝ/ℤ.

(2) If u = 2s − b + β + 1, then η(N)(ρ_{s}) = 0 in ℝ/ℤ.

(3) If there are no harmonic spinors on M̃, the equalities above hold in ℝ not just ℝ/ℤ.

If m is even we have the following Theorem see [2,3] for details.

1.7 Theorem. Let m be even and let M and N be as above.

(1) If ℓ = 2, then we have:

1-a) If u = 2s − b + m/2 then η(N)(ρ_{u}) = η(M)(ρ_{s}) in ℝ/ℤ.

1-b) If u = 2s − b + 1 + m/2, then η(N)(ρ_{u}) = η(M)(ρ_{s}) in ℝ/ℤ.

(2) If ℓ > 2, then we have:

2-a) If u = 2s − b + m/2 + ℓ/4, then η(N)(ρ_{u}) = η(M)(ρ_{s} + ρ_{s+ℓ/4}) in

ℝ/ℤ.

2-b) If u = 2s − b + m/2 + ℓ/4 + 1, then η(N)(ρ_{u}) = 0 in ℝ/ℤ.

(3) If there are no harmonic spinors on M̃, then these equalities hold in ℝ.

1.8 Moduli space of metrics of positive scalar curvature. Let R(M) be the space of metrics of positive scalar curvature on a manifold M and let M(M) be the moduli space of metrics of positive scalar curvature. We say that two metrics g_{i} for i = 1, 2 are geometrically bordant if [(M, f, s, g_{1})] = [(M, f, s, g_{2})] in +MSpin(Bℤ_{ℓ}, ξ). We shall need the following result due to Botvinnik and Gilkey [4] that uses work of Gromov and Lawson [7], Rosenberg [8], Rosenberg and Stolz [9] and Schoen and Yau [10].

1.9 Theorem. Let M be a closed connected manifold of dimension m ≥ 5 with fundamental group ℤ_{ℓ} so that [(M, f, s)] ∈ MSpin_{m}(Bℤ_{ℓ}, ξ_{i}). If m is odd, assume i = 0, 1; if m is even, assume i = 2, 3. Let ρ ∈ RU(ℤ_{ℓ}).

(1) If [(M, f, s)] = [(M_{1}, f_{1}, s_{1})] in MSpin_{m}(Bℤ_{ℓ}, ξ) where M_{1} admits a metric of positive scalar curvature g_{1}, then M admits a metric of positive scalar curvature g such that [(M, f, s, g)] = [(M_{1}, f_{1}, s_{1}, g_{1})] in^{+}MSpin_{m}(Bℤ_{ℓ}, ξ).

(2) If [(M_{2}, f_{2}, s_{2}, g_{2})] = 0 in ^{+}MSpin_{m}(Bℤ_{ℓ}, ξ), then η(M_{2})(ρ) = 0 in ℝ.

(3) If M admits a metric of positive scalar curvature and if there exists any element [(M_{3}, f_{3}, s_{3}, g_{3})] in ^{+}MSpin_{m}(Bℤ_{ℓ}, ξ) with η(M_{3})(ρ) ≠ 0 in ℝ, then M(M) has an infinite number of components and there exists a countable family of metrics g_{j} of positive scalar curvature on M which are not geometrically bordant.

We can now state the main theorem of this note:

1.10 Theorem. Let X be an even dimensional non-orientable manifold of dimension m ≥ 6 which admits a metric of positive scalar curvature. Assume that the fundamental group of X is ℤ_{2ℓ} and that the universal cover of X is spin. Assume that one of the following cases given below holds.

(1) We have that ℓ = 2, m ≡ 0 mod 4 and ω_{2}(X) ≠ 0.

(2) We have that ℓ = 2, m ≡ 2 mod 4 and ω_{2}(X) = 0.
(3) We have that ℓ = 4, m ≡ 0 mod 4 and ω_{2}(X) ≠ 0.
(4) We have that ℓ ≥ 4, m ≡ 2 mod 4 and ω_{2}(X) = 0.
(5) We have that ℓ ≥ 8, m ≡ 4 mod 8, m ≥ 12 and ω_{2}(X) = 0.

Then the moduli space M(X) of metrics of positive scalar curvature on X has an infinite number of components and there exists a countable family of metrics g_{j} on X which have positive scalar curvature and which are not geometrically bordant.

Proof. We apply Theorem 1.9. We must only construct a manifold N so that N admits a metric of positive scalar curvature, so that N ∈ MSpin_{2m}(Bℤ_{2ℓ}, ξ), and so that η(N)(ρ) ≠ 0 for some ρ. We take N = N(M) where (M, ξ) are as given by:

(1) Let m = 4k, ξ = ξ_{3} and M := ℝℙ^{4k}.
(2) Let m = 4k + 2, ξ = ξ_{2} and M := ℝℙ^{4k+2}.
(3) Let m = 4k, ξ = ξ_{3} and M := (S^{2k} × S^{2k})/ℤ_{4}.

(4) Let m = 4k + 2, ξ = ξ_{2} and M := (S^{2k+1} × S^{2k+1})/ℤ_{ℓ}.

(5) Let m = 8k + 4, ξ = ξ_{2} and M := ((S^{2k+1} × S^{2k+1}) × (S^{2k+1} × S^{2k+1}))/ℤ_{ℓ}. We use the formulas in Theorems 1.6 and 1.7 to show that the eta invariant of N(M) is non-trivial. □

References

[1] Atiyah, M. F., Patodi, V. K., Singer, I., Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambr. Phil. Soc. 77 (1975), 43–69.

[2] Barrera-Yanez, E., The eta invariant, equivariant bordism, connective K-theory and manifolds of positive scalar curvature, Ph. D. Thesis (University of Oregon June 1997).

[3] Barrera-Yanez, E., The eta invariant of twisted products of even dimensional manifolds whose fundamental group is a cyclic 2 group, accepted for publication in DGA.

[4] Botvinnik, B. & Gilkey, P., The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995), 507-517.

[5] Gilkey, P., Invariance Theory, the Heat Equation, and The Atiyah-Singer Index Theorem (2^{nd} edition), ISBN 0-8493-7874-4, CRC Press, Boca Raton, Florida, 1994.

[6] , The eta invariant of pin manifolds with cyclic fundamental group, preprint.

[7] Gromov, M. & Lawson Jr., H. B., Spin and scalar curvature in the presence of the fundamental group I, Annals of Math. 111 (1980), 209–230.

[8] Rosenberg, J., C^{∗}-algebras, positive scalar curvature and the Novikov Conjecture, III, Topology 25 (1986), 319–336.

[9] Rosenberg, J. & Stolz, S., Manifolds of positive scalar curvature, Algebraic Topology and Applications (G. Carlson, R. Cohen, W.-C. Hiang, and J. D. S. Jones, eds), M. R. S. I. Publications, vol. 27, Springer, New York, (1994), 241–267.

[10] Schoen, R. & Yau, S. T., On the structure of manifolds with positive scalar curvature, Manuscr. Math. 28 (1969), 159–183.

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