On the Betti numbers of Alexandrov spaces

Liu, Zhong-dong; Zhang, Shen

doi:10.1007/bf02108293

Cited by 7 publications

(4 citation statements)

References 2 publications

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“…The upper bound on the number of boundary components follows from the Betti number theorem for Alexandrov spaces [Liu and Shen 1994] together with an easy homology argument. As any M ∈ ᏹ(n, K − , λ ± , d) is homeomorphic to its extension M, it suffices to give a bound on the number of components of ∂ M. Consider the double 2 M = M ∂ M M. This is again an Alexandrov space of curvature bounded below, since M has a convex boundary.…”

Section: Proof Of Theorem 11(ii) Sincementioning

confidence: 99%

An extension procedure for manifolds with boundary

Wong¹

2008

Pacific J. Math.

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This paper introduces an isometric extension procedure for Riemannian manifolds with boundary, which preserves some lower curvature bound and produces a totally geodesic boundary. As immediate applications of this construction, one obtains in particular upper volume bounds, an upper intrinsic diameter bound for the boundary, precompactness, and a homeomorphism finiteness theorem for certain classes of manifolds with boundary, as well as a characterization up to homotopy of Gromov-Hausdorff limits of such a class.

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Section: Proof Of Theorem 11(ii) Sincementioning

confidence: 99%

An extension procedure for manifolds with boundary

Wong¹

2008

Pacific J. Math.

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“…Proof. As shown by Wong [23], the work of Liu and Shen in [15] bounding the Betti numbers of Alexandrov spaces also bounds the number of boundary components in an Alexandrov space. That is, an Alexandrov space of dimension n, diam ≤ D and curv ≥ k can have C(n, D, k) boundary components.…”

Section: Alexandrov Spaces Of Almost Non-negative Curvaturementioning

confidence: 92%

Almost non-negatively curved 4-manifolds with torus symmetry

Harvey

Searle

2020

Proc. Amer. Math. Soc.

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We prove that if a closed smooth simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. This gives a classification of closed smooth simply-connected 4-manifolds of almost non-negative curvature under the assumption of continuous symmetry.

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“…Working with concentric coverings, Theorem 1.1 yields the following uniform bound on the total Betti number: Corollary 1.2 ([5], [13]). For given n and D, there is a positive integer…”

Section: Introductionmentioning

confidence: 99%

“…In the original work [5], Gromov developed the critical point theory for distance functions to obtain an explicit bound on the total Betti numbers for Riemannian manifolds. The argument in [13] is a natural extension of that in [5] to Alexandrov spaces. Unfortunately our bound is not explicit.…”

Section: Introductionmentioning

confidence: 99%

Collapsing and essential coverings

Yamaguchi

2012

Preprint

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In the present paper, we consider the family of all compact Alexandrov spaces with curvature bound below having a definite upper diameter bound of a fixed dimension. We introduce the notion of essential coverings by contractible metric balls, and provide a uniform bound on the numbers of contractible metric balls forming essential coverings of the spaces in the family. In particular, this gives another view for Gromov's Betti number theorem.

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On the Betti numbers of Alexandrov spaces

Cited by 7 publications

References 2 publications

An extension procedure for manifolds with boundary

An extension procedure for manifolds with boundary

Almost non-negatively curved 4-manifolds with torus symmetry

Collapsing and essential coverings

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