Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes

2020/11/23 by Erokhovets, Nikolai
#52B10 #52B70 #53C15 #57R91 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2011.11628

Abstract

In short geometrization conjecture of W. Thurston (finally proved by G.~Perelman) says that any oriented 3-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal paper (1991) M. W. Davis and T. Januszkiewicz introduced a wide class of n-dimensional manifolds -- small covers over simple n-polytopes. We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable 3-manifold defined by a vector-colouring of a simple 3-polytope, in particular for a small cover. The proof is based on analysis of results in this direction obtained before by different authors.

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