An upper bound on the revised first Betti number and a torus stability result for RCD spaces

Mondello, Ilaria; Mondino, Andrea; Perales, Raquel

doi:10.4171/cmh/540

Cited by 5 publications

(3 citation statements)

References 41 publications

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“…The following proposition is a sort of converse to Corollary 2.1; it introduces the push-forward of an equivariant RCD(0, N )-structure on X (cf. [27,Lemma 2.18] and [26,Lemma 2.24]). Proposition 2.3 Let ( X , d, m) be an RCD(0, N )-structure on X such that π 1 (X ) acts by isomorphisms on ( X , d, m).…”

Section: Corollary 21 Letmentioning

confidence: 99%

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

Mondino

Navarro

2022

Math. Ann.

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The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures. First, we relate the convergence of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures on a space to the associated lifts’ equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures that have non-trivial rational homotopy groups.

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Section: Corollary 21 Letmentioning

confidence: 99%

“…The following corollary of Proposition 2.5 defines the splitting degree of X (cf. [26,Proposition 2.25]).…”

Section: Splittings and Topological Invariantsmentioning

confidence: 99%

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

Mondino

Navarro

2022

Math. Ann.

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“…Another direction is the one of metric measure spaces satisfying a suitable synthetic Ricci curvature lower bound. In this context, a rigidity result à la Bochner and geometric stability results hold, see [24,28,29].…”

Section: Introductionmentioning

confidence: 99%

Torus stability under Kato bounds on the Ricci curvature

Carron

Mondello

Tewodrose

2022

Journal of London Math Soc

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We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.

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