{'fraction_non_alphanumeric': 0.022748815165876776, 'fraction_numerical': 0.018957345971563982, 'mean_word_length': 5.034285714285715, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Collapsing with No Proper Extremal Subsets

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.