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

Nikolai Erokhovets

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arXiv · 2011.11628

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

Nikolai Erokhovets

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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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BibTeX RIS

Nikolai Erokhovets. 2020-11-23. Canonical geometrization of orientable $3$-manifolds defined by vector-colourings of $3$-polytopes. https://arxiv.org/abs/2011.11628

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