Abstract

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.

Highlights

  • One possible algorithm to solve the homeomorphism problem for 3-manifolds could run as follows

  • Given two triangulated 3-manifolds M1 and M2, perform Pachner moves on M1 to try to establish a homeomorphism with M2

  • The question arises, to what extent will this algorithm work? That is, could the collections of covers of two distinct 3-manifolds have the same structure? This is a manifestation of the wider question of when two groups have the same set of finite quotients

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Summary

Introduction

One possible algorithm to solve the homeomorphism problem for 3-manifolds could run as follows. Hempel [13] gave Seifert fibred families, with geometry H2 × R These examples notwithstanding, the profinite completion of the fundamental group of a low-dimensional manifold is known to contain a large amount of information. We provide the full solution of the profinite rigidity question for closed orientable Seifert-fibred 3-manifolds. The precise statement, when combined with the work in [24], is: Theorem 1.2 Let M be a (closed orientable) Seifert fibre space. – M is profinitely rigid; or – M has the geometry H2 × R, is a surface bundle with periodic monodromy φ and the only 3-manifolds whose fundamental groups have the same finite quotients as π1 M are the surface bundles with monodromy φk, for k coprime to the order of φ. The cyclic group of order p will be consistently denoted Z/ p or Z/ pZ

Goodness
Chain complexes
Profinite completions of 2-orbifold groups
Seifert fibre spaces
Theorems
Preservation of the fibre
Central extensions
Action on cohomology
Full Text
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