Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds

Abstract

We show that in each dimension 4n + 3, n ≥ 1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

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.…”

“…By [31], in each dimension 4k + 3 ≥ 7 there exist infinite sequences of closed manifolds M such that the moduli space M sec≥0 (M) of Riemannian metrics with non-negative sectional curvature on M has infinitely many path components. Taking products of these examples with tori and applying Theorem 1.3, we obtain: 4 In every dimension n ≥ 7 there exist infinite sequences of closed smooth manifolds M of pairwise distinct homotopy type for which the moduli space M sec≥0 (M) of Riemannian metrics with non-negative sectional curvature has infinitely many path components.…”

Aspects of Positive Scalar Curvature and Topology II

“…Indeed, regarding this positive Ricci result, it was later shown by Kapovitch, Petrunin and Tuschmann in [34] that there are closed 7-dimensional manifolds (of the class constructed by Wang and Ziller) for which the moduli space of non-negative sectional curvature metrics has infinitely many path components. A more recent example can be found in [18]. We should also point out that there are a number of interesting results concerning topological non-triviality in the moduli space of non-negative sectional curvature metrics for certain open manifolds; see in particular work by Belegradek, Kwasik and Schultz [5], Belegradek and Hu [4] and very recently Belegradek, Farrell and Kapovitch [3].…”

Homotopy groups of the observer moduli space of Ricci positive metrics

“…Apart from results of Kreck and Stolz in [17] and Wraith in [23] concerning path-connectivity, we know very little about topology of the corresponding moduli spaces of positive Ricci curvature metrics. (In this context we should also mention work of Dessai, Klaus and Tuschmann on moduli spaces of non-negative sectional curvature metrics in [8], and the results of Crowley, Schick and Steimle on the space of Ricci positive metrics on certain manifolds, see [7].) Whether or not there is any non-triviality in the higher homotopy groups of such moduli spaces is still an open question.…”