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Turaev--Viro invariants and profinite completions of surface bundles

Qirong Yang

TL;DR

This work links profinite completions of surface-bundle groups to Turaev--Viro invariants by establishing that, for closed surface bundles $M_f$ and $M_g$, a procongruently conjugacy separable $ abla= abla_1( abla)$ together with a regular isomorphism $ig(ig)$ between profinite completions implies $TV(M_f)=TV(M_g)$ for all spherical fusion categories. The proof threads through untwisted Dijkgraaf--Witten invariants, a profinite-algebraic equivalence, and a TV-based mapping-class representation argument, culminating in a criterion that transfers profinite conjugacy into equality of TV invariants via the TQFT framework. A key technical component shows that if the profinite completions agree and monodromies are congruent in the profinite mapping class group, then the TV invariants, computed from finite-quotient data, must coincide. The paper also discusses regularity results (notably Xu–Liu for cusped manifolds and Dehn-filled closures) which yield broad classes of closed manifolds where the main theorem applies, contributing to the broader understanding of how profinite data governs 3-manifold invariants.

Abstract

We prove that the Turaev--Viro invariants of the two surface bundles over the circle coincide for every spherical fusion category if the surface group is procongruently conjugacy separable and there exists a regular profinite isomorphism between the fundamental groups.

Turaev--Viro invariants and profinite completions of surface bundles

TL;DR

This work links profinite completions of surface-bundle groups to Turaev--Viro invariants by establishing that, for closed surface bundles and , a procongruently conjugacy separable together with a regular isomorphism between profinite completions implies for all spherical fusion categories. The proof threads through untwisted Dijkgraaf--Witten invariants, a profinite-algebraic equivalence, and a TV-based mapping-class representation argument, culminating in a criterion that transfers profinite conjugacy into equality of TV invariants via the TQFT framework. A key technical component shows that if the profinite completions agree and monodromies are congruent in the profinite mapping class group, then the TV invariants, computed from finite-quotient data, must coincide. The paper also discusses regularity results (notably Xu–Liu for cusped manifolds and Dehn-filled closures) which yield broad classes of closed manifolds where the main theorem applies, contributing to the broader understanding of how profinite data governs 3-manifold invariants.

Abstract

We prove that the Turaev--Viro invariants of the two surface bundles over the circle coincide for every spherical fusion category if the surface group is procongruently conjugacy separable and there exists a regular profinite isomorphism between the fundamental groups.
Paper Structure (8 sections, 13 theorems, 21 equations)

This paper contains 8 sections, 13 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spherical fusion category
  4. Turaev--Viro invariants
  5. The Profinite Completion and Dijkgraaf--Witten invariants
  6. Profinite isomorphism
  7. Proof of Theorem \ref{['main TV']}
  8. The regularity

Key Result

Theorem 1.1

Let $M_1,M_2$ be closed connected 3-manifolds with finitely generated fundamental groups. They have the same untwisted Dijkgraaf--Witten invariants for every finite group if and only if their fundamental groups are profinitely isomorphic, i.e. $\widehat{\pi_1(M_1)}\cong \widehat{\pi_1(M_2)}$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 14 more