Torus cannot collapse to a segment

Mikhail G. Katz

doi:10.1007/s00022-020-0525-8

Torus cannot collapse to a segment

Mikhail G. Katz

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.

Original language	English
Article number	13
Journal	Journal of Geometry
Volume	111
Issue number	1
DOIs	https://doi.org/10.1007/s00022-020-0525-8
State	Published - 1 Apr 2020

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1007/s00022-020-0525-8

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AB - In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.

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JF - Journal of Geometry

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Torus cannot collapse to a segment

Abstract

All Science Journal Classification (ASJC) codes

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