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

Abstract

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

Moduli spaces of compact RCD(0,N)-structures

“…Proof of Theorem C. The proof of Theorem C is inspired by the proof of Theorem 1.1 in [33]. First of all, we show that the Albanese map associated to X × T k admits a continuous section; therefore, it induces a surjective map on homotopy groups.…”

“…Theorem C should be put in contrast with that result since it shows that the topology of moduli spaces of RCD(0, N )-structures is not always as trivial. Moreover, Theorem C can also be seen as a non-smooth analogue of Theorem 1.1 in [33]. ), let X be a compact topological space that admits an RCD(0, N )-structure such that π 1 (X) = 0 (see Theorem 1.2 for the definition of π 1 (X)), and let T k be a torus of dimension k ≥ 4 such that k = 8, 9, 10.…”

“…Inspired by the proof of Theorem 5.4 in [33], we introduce the following "shrunk" metrics. Remark 3.1.…”

“…First of all, we show that the Albanese map associated to X × T k admits a continuous section; therefore, it induces a surjective map on homotopy groups. To conclude, we make use of Proposition 5.5 of [33], which gives information on the homotopy groups of the moduli space of flat metrics on T k .…”

“…Yet, there are not as many results on moduli spaces of non-negatively Ricci curved metrics. In 2017, Tuschmann and Wiemeler published the first result on these moduli spaces' homotopy groups (see Theorem 1.1 in [33]). Until recently, most of the results on moduli spaces in metric geometry focused on smooth metrics.…”