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Collapsing with No Proper Extremal Subsets

Published online by Cambridge University Press:  27 June 2025

Karsten Grove
Affiliation:
University of Maryland, College Park
Peter Petersen
Affiliation:
University of California, Los Angeles
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Summary

This is a technical paper devoted to the investigation of collapsing of Alexandrov spaces with lower curvature bound. In a previous paper, the author defined a canonical stratification of an Alexandrov space by the so-called extremal subsets. It is likely that if the limit of a collapsing sequence has no proper extremal subsets, then the collapsing spaces are fiber bundles over the limit space. In this paper a weaker statement is proved, namely, that the homotopy groups of those spaces are related by the Serre exact sequence. A restriction on the ideal boundary of open Riemannian manifolds of nonnegative sectional curvature is obtained as a corollary.

We assume familiarity with the basic notions and results about Alexandrov spaces with curvature bounded below [Burago et al. 1992; Perelman 1994; Perelman and Petrunin 1994], and with earlier results on collapsing with lower curvature bound [Yamaguchi 1991]. For motivation for the collapsing problem, see [Cheeger et al. 1992] and references therein.

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Comparison Geometry , pp. 149 - 156
Publisher: Cambridge University Press
Print publication year: 1997

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