FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 4, PAGES 71-84

Geometric topology of generalized $3$-manifolds

A. Cavicchioli
D. Repovš
T. L. Thickstun

Abstract

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In this paper, we describe the history and the present status of one of the main classical problems in low-dimensional geometric topology--the recognition of topological $3$-manifolds in the class of all generalized $3$-manifolds (i.e., ANR homology $3$-manifolds). This problem naturally splits into the cell-like resolution problem for $3$-manifolds by means of homology $3$-manifolds and the general-position problem for topological $3$-manifolds. We have also included some open problems.

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Last modified: November 28, 2005