Contractibility of moduli spaces of RCD(0,2)-structures

Abstract

This paper focuses on RCD(0, 2)-spaces, which can be thought of as possibly non-smooth metric measure spaces with non-negative Ricci curvature and dimension less than 2. First, we establish a list of the compact topological spaces admitting an RCD(0, 2)-structure. Then, we describe the associated moduli space of RCD(0, 2)-structures for each of them. In particular, we show that all these moduli spaces are contractible. CONTENTS1. Introduction 1.1. Basic definitions 1.2. Main results 1.3. Sketch of the proofs Organization of the paper Acknowledgements Preliminaries 2. Equivariance 3. Albanese variety and soul 4. Essential dimension and topological obstructions Computations of Moduli spaces 5. Helpful results 5.1. The case of compact flat manifolds 5.2. The case of surfaces 6. The 1-dimensional case 6.1. The circle S 1 6.2. The closed unit interval I 7. The 2-dimensional case 7.1. The 2-Torus T 2 and the Klein bottle K 2 7.2. The Möbius band M 2 and the cylinder S 1 × I 7.3. The 2-sphere S 2 , the projective plane RP 2 , and the closed disc D References

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

“…As we will observe at the end of Sect. 2.4.1, Theorem A implies the following corollary, which is particularly useful when computing the homeomorphism type of specific examples of moduli spaces (see for example the case of RP 2 in [28]).…”

“…Our next result states the continuity of the Albanese map and the soul map associated to a compact topological space that admits an RCD(0, N )-structure (with N ∈ [1, ∞)). On the first hand, this result is essential when computing the homeomorphism type of specific examples of moduli spaces (see for example the case of the Möbius band M 2 and the finite cylinder S 1 × [0, 1] in [28]). On the other hand, the continuity of the Albanese map will be crucial in the proof of Theorem C.…”

“…Let us recall that if X is a compact topological space that admits an RCD(0, 2)structure, then the moduli space M 0,2 (X ) is contractible (see Theorem 1.1 in [28]). 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.…”

“…However, D eq p does not satisfy the triangle inequality a priori. Nevertheless, it will be sufficient for our purposes to show that D eq p satisfy the following modified triangle inequality: D eq p ( X1 , X3 ) ≤ 4(D eq p ( X1 , X2 ) + D eq p ( X2 , X3 )) (28) for any three equivariant pointed RCD(0, N )-structures Xi = ( X , di , mi , * i ) ∈ R p,eq 0,N ( X ) (i ∈ {1, 2, 3}).…”