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arXiv · 2111.00342

Homotopy and Homology at Infinity and at the Boundary

Mohammed Barhoush
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Abstract

In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, $\delta$-hyperbolic space that is acted upon geometrically by a group has trivial homotopy at infinity then the first \v{C}ech homotopy group is trivial. Secondly, we prove that if a hyperbolic group on a finite field has trivial $i^{th}$ homology at infinity then the boundary of the group has trivial $i^{th}$ Steenrod homology. This result turns out to be important in addressing an open problem related to Cannon's conjecture.

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Mohammed Barhoush. 2021-10-30. Homotopy and Homology at Infinity and at the Boundary. https://arxiv.org/abs/2111.00342

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