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Some Topological Results of Ricci Limit Spaces

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Some Topological Results of Ricci Limit Spaces SOME TOPOLOGICAL RESULTS OF RICCI LIMIT SPACES JIAYIN PAN, JIKANG WANG Abstract. We study the topology of a Ricci limit space (X,p), which is the Gromov-Hausdorff limit of a sequence of complete n-manifolds (Mi,pi) with Ric ≥−(n − 1). Our first result shows that, if Mi has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then X is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of X if Mi has a uniform diameter bound; this improves a result in [8]. 1. Introduction We study the topology of Ricci limit spaces. Let (Mi,pi) be a sequence of pointed complete Riemannian n-manifolds with a uniform Ricci curvature lower bound Ric ≥ −(n − 1). Passing to a subsequence if necessary, we can assume (Mi,pi) converges to (X,p) in the pointed Gromov-Hausdorff sense. We call (X,p) a Ricci limit space. (X,p) is non-collapsing if Vol(B1(pi)) has a uniform lower bound. The regularity theory of Ricci limit spaces has been studied extensively by Cheeger, Colding, and Naber [1, 2, 3, 4, 5, 7]. On the topology of (X,p), Sormani and Wei proved that any Ricci limit space has a universal cover [17, 18], but it is unknown whether the universal cover is simply connected by their work. Recently, Wei and the first author proved that any non-collapsing Ricci limit space is semi-locally simply connected [16]; in particular, such a limit space has a simply connected universal cover. More detailed discussions about the local topology of a Ricci limit space can be found in [16]. In this paper, we prove two topological properties of a Ricci limit space. The first result is that any Ricci limit space with non-collapsing local rewinding volume is semi-locally simply connected. More precisely, in the collapsing case, if the local universal cover of every r-ball in manifolds is non-collapsing, then the limit space arXiv:2103.11344v1 [math.DG] 21 Mar 2021 is semi-locally simply connected. We start with notations in Ricci bounded covering geometry; see [6, 11, 12] for further reference. Let Br(pi) be the open ball with center pi and radius r. We ^ denote B¯r(pi) as the closure of Br(pi). Let Br(pi) be the (incomplete) universal ^ cover of Br(pi) and we choose a pointp ˜i ∈ Br(pi) such that π(˜pi)= pi. Given the ^ pull-back Riemannian metric on Br(pi), we call the volume of Br/4(˜pi) the local rewinding volume of Br(pi), denoted by V ol(Br(pi)). For any metric space X and any x ∈ Xg, we follow the notation in [16] to define the local 1-contractibility radius: ρ(t, x) = inf{∞,ρ ≥ t| any loop in Bt(x) is contractible in Bρ(x)}. J. Pan is partially supported by AMS Simons travel grant. 1 2 X is semi-locally simply connected, if for any point x ∈ M, there is an open neighborhood containing x such that any loop in this neighborhood is contractible in X. In particular, if for all x ∈ M there exists t (depending on x) such that ρ(t, x) is finite, then M must be semi-locally simply connected. Now we state our first result: Theorem 1.1. Let (Mi,pi) be a sequence of complete Riemannian manifolds con- verging to (X,p) such that for all i, (1) B4(pi) ∩ ∂Mi = ∅ and the closure of B4(pi) is compact, (2) Ric ≥−(n − 1) on B4(pi), V ol(B4(pi)) ≥ v > 0. g Then lim ρ(t, x)=0 holds for all x ∈ B1/800(p). t→0 Theorem 1.1 implies the following. Corollary 1.2. Let (Mi,pi) be a sequence of complete Riemannian n-manifolds with Ric ≥−(n − 1) and converging to (X,p). If there exist r and v such that for all i and any x ∈ Mi, V ol(Br(x)) ≥ v > 0. Then (X,p) is semi-locally simply connected. g As mentioned, the universal cover of a Ricci limit space always exists [17, 18]. As a corollary, the universal cover of X in Corollary 1.2 is always simply connected. To prove Theorem 1.1, we establish the existence of a slice for pseudo-group ac- tions on incomplete Ricci limits. Then Theorem 1.1 follows from this slice theorem and Pan-Wei’s result [16]. Theorem 1.3. Let (Mi,pi) be a sequence of connected Riemannian manifolds con- verging to (X,p) such that for all i, B4(pi) ∩ ∂Mi = ∅ and the closure of B4(pi) is ^ compact. Let (B4(pi), p˜i) be the universal cover of B4(pi) and let Γi be the funda- mental group of B4(pi). The pseudo-group Gi = {γ ∈ Γi|d(γp˜i, p˜1) ≤ 1/100} acts on B1(˜pi). Passing to a subsequence, (B¯1(˜pi), p˜i, Gi) converges to (B¯1(˜p), p,˜ G) in the equivariant Gromov-Hausdorff sense. Then the following holds. For any x˜ ∈ B1/800(˜p), there is a slice S at x˜ for pseudo-group G-action, that is, S satisfies: (1) x˜ ∈ S and S is Gx˜-invariant, where Gx˜ is the isotropy subgroup at x˜; (2) S/Gx˜ is homeomorphic to a neighborhood of x. Note that we don’t assume B1(˜pi) has a volume lower bound in Theorem 1.3. The idea to find the slice is that we extend G to a Lie group Gˆ, which acts homeomorphically on an extended space. Then we apply Palais’s slice theorem [14] for this Gˆ-action. In this construction, we will show that the extended group and space are locally homeomorphic old ones, thus we can get the slice atx ˜. See Sections 4 and 5 for details. The next result of this paper is about the description of universal covers of Ricci limit spaces. Now we assume that diam(Mi) ≤ D for some D > 0. Let Mi be the f universal cover of Mi with fundamental group Γi. Passing to a subsequence, we obtain the following convergence 3 GH (Mi, p˜i, Γi) −−−−→ (Y, p,˜ G) f π π y GH y (Mi,pi) −−−−→ (X,p), Ennis and Wei showed that there is ǫ = ǫ(X) > 0 such that ǫ GH ǫ (Mi, p˜i, Γi ) −→ (Y, p,˜ H ) f ǫ ǫ and the universal cover of X is isometric Y/H [8], where Γi is generated by elements {g ∈ Γi | d(q,gq) ≤ ǫ for some q ∈ Mi}. f We give a description of Hǫ from the G-action on Y . Theorem 1.4. Let Mi be a sequence of closed Riemannian n-manifolds with Ric(Mi) ≥−(n − 1), diam(Mi) ≤ D. Suppose that GH (Mi, p˜i, Γi) −−−−→ (Y, p,˜ G) f πi π y GH y (Mi,pi) −−−−→ (X,p). Let H be the group generated by G0 and all isotropy subgroups of G, where G0 is the identity component subgroup of G. Then Y/H is the universal cover of X. Consequently, π1(X,p) is isomorphic to G/H. Note that H is a subgroup of Hǫ. Theorem 1.4 shows that they are indeed the same. This improvement relies on the fact that Isom(Y ) is a Lie group [2, 7]. We organize the paper as follows. In Section 2, we recall some preliminaries, which include Ricci limit spaces and Palais’s slice theorem. In Section 3, we study the limit pseudo-group actions on a closed ball of the Ricci limit space. To prove Theorem 1.3 next, we will use the limit pseudo-group G to construct a Lie group Gˆ in Section 4, then we construct a Gˆ-space and apply Palais’s theorem to find a slice on the new space in Section 5. We prove Theorem 1.4 in Section 6. The authors would like to thank Xiaochun Rong for many helpful discussions. Contents 1. Introduction 1 2. Preliminaries 4 3. Pseudo-group 6 4. Groupfication of G 9 5. Construction of a Gˆ-space 12 6. Proof of Theorem 1.4 15 References 17 4 2. Preliminaries 2.1. Ricci limit spaces. One important result about Ricci limit spaces that we need in this paper is their isometry groups are always Lie groups. [2] proves the non-collapsing case, then [7] proves the general case. Theorem 2.1. [2, 7] The isometry group of any Ricci limit space is a Lie group. We recall some of elements in the proof of Theorem 2.1 since we need them later in Section 3. Let (X,p) be a Ricci limit space. For any integer k ≤ n, Rk is the set of k points where every tangent cone is isometric to R . We say y ∈ (Rk)ǫ,δ, if for all k k Rk 0 <r<δ we have dGH (Br(y),Br (0)) < ǫr, where Br (0) is the r-ball in . Let (Rk)ǫ = ∪δ(Rk)ǫ,δ. Note that Rk = ∩ǫ>0(Rk)ǫ. A topological group is said to have small subgroups if every neighborhood of the identity contains a non-trivial subgroup. By [10, 19], a topological group with no small groups is a Lie group. For the isometry group of a metric space X, we can measure the smallness of a subgroup by its displacement. For any subgroup H of Isom(X) and x ∈ X, we define ρH (x) = sup d(x, hx), DH,r(x) = sup ρH (w) h∈H w∈Br(x) be the displacement function of H on Br(x).
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