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Geometric topology of generalized 3-manifolds

Abstract

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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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 71–84, 2005.

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Cavicchioli, A., Repovš, D. & Thickstun, T.L. Geometric topology of generalized 3-manifolds. J Math Sci 144, 4413–4422 (2007). https://doi.org/10.1007/s10958-007-0279-y

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Keywords

  • Manifold
  • Inverse Limit
  • Regular Neighborhood
  • Resolution Theorem
  • Loop Theorem