Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

Abstract

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP 5 has infinitely many path components. We also show that in each dimension 4k + 1 there are at least 2 2k homotopy RP 4k+1 s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k + 3 ≥ 7. Outline of proofsThe goal of this section is to give an outline of the proofs of Theorem A and Theorem B, all the references and details can be found in the following sections.

On the topology of moduli spaces of non-negatively curved Riemannian metrics

“…Especially in recent years there has been intensive activity and substantial further progress on these issues, compare, for example, [2,3,[6][7][8][9][10][11][13][14][15][16][17][18][20][21][22][25][26][27][28][29][30][31][33][34][35][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]58,60,61,[65][66][67][69]…”

“…Indeed, for closed manifolds and the (genuine) moduli spaces of metrics of these types, all results in general dimension that are known so far (compare [30,31]) only show that there are manifolds for which the moduli spaces of metrics with non-negative sectional curvature are not connected and can even have an infinite number of components. Moreover, in [65] an analogous result is shown for spaces of non-negative Ricci curvature metrics.…”

Spaces of positive scalar curvature metrics on totally nonspin manifolds with spin boundary

“…(i) If d ≡ 0, 1 (8) and ∂ M is Spin × Bπ 1 (∂ M)-nullbordant 1 , there exists an h ∈ R + (∂ M) such that R + (M) h is non-empty and the following composition is surjective…”

“…In the case where M is nonspin, but its universal cover is Spin, there are cases where R + (M) is not connected or has even infinitely many path components, see [2,7,8,15,16,23,25]. For totally nonspin manifolds, Kastenholz-Reinhold give an example of a closed, totally nonspin manifold of dimension 6 whose space of psc-metrics has infinitely many components in [21].…”

“…Theorem 2.12 [20, Theorem 4.7], [5, Corollary 1.2], [6, Corollary 1.9] Let N be a compact spin manifold of dimension d ≥ 6, h ∈ R + (∂ N ) and g ∈ R + (N ) h . Let furthermore k ≥ 0 such that d + k + 1 ≡ 1, 2(8). Then the composition…”