Two results on end spaces of infinite type surfaces

Mann, Kathryn; Rafi, Kasra

doi:10.48550/arxiv.2201.07690

Cited by 1 publication

(1 citation statement)

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“…Clearly x y implies x y. Also, if we now assume that E is the end space of a surface †, it is not hard to see (using Lemma 3.4 and Proposition 2.2) that x y if and only if there is a homeomorphism of † taking x to y. Theorem 1.2 of [19] says that if x y then there is a homeomorphism of † taking x to y. It follows that and are the same equivalence relation on E if it is the end space of a surface.…”

Section: Topologically Distinguished Pointsmentioning

confidence: 99%

Big mapping class groups with uncountable integral homology

Palmer,

2024

Doc. Math.

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We prove that, for any infinite-type surface S , the integral homology of the closure of the compactly-supported mapping class group PMap c .S/ and of the Torelli group T .S/ is uncountable in every positive degree. By our results in [arXiv:2211.07470] and other known computations, such a statement cannot be true for the full mapping class group Map.S/ for all infinite-type surfaces S. However, we are still able to prove that the integral homology of Map.S/ is uncountable in all positive degrees for a large class of infinite-type surfaces S . The key property of this class of surfaces is, roughly, that the space of ends of the surface S contains a limit point of topologically distinguished points. Our result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor-Bendixson rank ˛, where ˛is a successor ordinal. We also observe an order-10 element in the first homology of the pure mapping class group of any surface of genus 2, answering a recent question of G. Domat.

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Section: Topologically Distinguished Pointsmentioning

confidence: 99%