Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Theta-invariants of $\mathbb{Z}π$-homology equivalences to spherical 3-manifolds

Hisatoshi Kodani, Tadayuki Watanabe

TL;DR

This work develops a universal, configuration-space–based Θ-invariant, $\widehat{Z}_\Theta^{\mathrm{odd}}$, for $\mathbb{Z}\pi$-homology equivalences to spherical 3-manifolds and shows its universality among Bott–Cattaneo invariants for all finite-dimensional acyclic local systems. By interpreting these invariants in terms of 2-loop $\pi$-decorated graphs, the authors connect topological data to a finite-type framework, enabling explicit dimension bounds via finite group representation theory. They prove that the invariant lands in the $\mathbb{Z}_2$-invariant subspace of $\mathrm{Sym}^3(\mathrm{Ker}\,\varepsilon)$ and establish a canonical surjection to the Bott–Cattaneo invariant space, tying the topological and graph-based pictures together. A complete, case-by-case calculation of the dimensions $\dim\mathscr{A}_\Theta^{\mathrm{odd}}(\mathbb{C}\pi)$ and $\dim\mathscr{A}_\Theta^{\mathrm{odd}}(\mathrm{Ker}\,\varepsilon)$ is carried out for all spherical 3-manifold groups $\pi$, using character theory and a careful treatment of real versus nonreal representations; the leading terms of these dimensions agree across cases, supporting conjectures about lower bounds and the two-loop structure of these invariants. The results yield concrete, group–theoretic constraints on finite-type invariants of $\mathbb{Z}\pi$-homology equivalences and provide tabulated data for principal spherical-group families, including cyclic, binary dihedral, binary polyhedral, and their direct products with $\mathbb{Z}_m$.

Abstract

We study Bott and Cattaneo's $Θ$-invariant of 3-manifolds applied to $\mathbb{Z}π$-homology equivalences from 3-manifolds to a fixed spherical 3-manifold. The $Θ$-invariants are defined by integrals over configuration spaces of two points with local systems and by choosing some invariant tensors. We compute upper bounds of the dimensions of the space spanned by the Bott--Cattaneo $Θ$-invariants and of that spanned by Garoufalidis and Levine's finite type invariants of type 2. The computation is based on representation theory of finite groups.

Theta-invariants of $\mathbb{Z}π$-homology equivalences to spherical 3-manifolds

TL;DR

This work develops a universal, configuration-space–based Θ-invariant, , for -homology equivalences to spherical 3-manifolds and shows its universality among Bott–Cattaneo invariants for all finite-dimensional acyclic local systems. By interpreting these invariants in terms of 2-loop -decorated graphs, the authors connect topological data to a finite-type framework, enabling explicit dimension bounds via finite group representation theory. They prove that the invariant lands in the -invariant subspace of and establish a canonical surjection to the Bott–Cattaneo invariant space, tying the topological and graph-based pictures together. A complete, case-by-case calculation of the dimensions and is carried out for all spherical 3-manifold groups , using character theory and a careful treatment of real versus nonreal representations; the leading terms of these dimensions agree across cases, supporting conjectures about lower bounds and the two-loop structure of these invariants. The results yield concrete, group–theoretic constraints on finite-type invariants of -homology equivalences and provide tabulated data for principal spherical-group families, including cyclic, binary dihedral, binary polyhedral, and their direct products with .

Abstract

We study Bott and Cattaneo's -invariant of 3-manifolds applied to -homology equivalences from 3-manifolds to a fixed spherical 3-manifold. The -invariants are defined by integrals over configuration spaces of two points with local systems and by choosing some invariant tensors. We compute upper bounds of the dimensions of the space spanned by the Bott--Cattaneo -invariants and of that spanned by Garoufalidis and Levine's finite type invariants of type 2. The computation is based on representation theory of finite groups.

Paper Structure

This paper contains 29 sections, 71 theorems, 153 equations, 1 figure, 18 tables.

Table of Contents

  1. Introduction
  2. Background: Finite type invariants of $\mathbb{Z}\pi$-homology equivalences
  3. $\mathbb{Z}\pi$-homology equivalence vs. 3-manifold over $K(\pi,1)$
  4. Preliminaries
  5. Notations and facts about representations of finite groups
  6. Homology of configuration spaces of two points
  7. Twisted de Rham cohomology
  8. The invariant $\widehat{Z}_\Theta^\mathrm{odd}$
  9. The invariant $Z_\Theta^\mathrm{odd}$ of $\mathbb{Z}\pi$-homology equivalences with framings
  10. Framing correction
  11. Bott--Cattaneo's interpretation of the $\Theta$-invariants
  12. Antisymmetric propagators and $\mathbb{Z}_2$-invariance
  13. Computation of $\dim\mathscr{A}_\Theta^\mathrm{odd}(\mathbb{C}\pi)$ and $\dim\mathscr{A}_\Theta^\mathrm{odd}(\mathrm{Ker}\,\varepsilon)$
  14. Formulas for computation of $\dim \mathscr{A}_\Theta^\mathrm{odd}(\mathbb{C}\pi)$ and $\dim\mathscr{A}_\Theta^\mathrm{odd}(\mathrm{Ker}\,\varepsilon)$
  15. Cyclic group $\mathbb{Z}_n$
  16. ...and 14 more sections

Key Result

Theorem 1

Let $N$ be a spherical 3-manifold, and let $\pi=\pi_1N$. Then, configuration space integrals for local systems give a well-defined invariant of $\mathbb{Z}\pi$-homology equivalences to $N$. It is universal among the Bott--Cattaneo $\Theta$-invariant $I_{(\Theta,\rho_1,\rho_2)}(f,A)$ for finite dimensional acyclic local systems $A$ on $N$ and $\pi$-equivariant linear maps $\rho_1\colon \mathbb{C}\

Figures (1)

  • Figure 1: The relations AS, IHX, Orientation reversal, Linearity, and Holonomy. Here $\bar{p}$ is the involution of $\mathbb{Z}\pi$ given by $\bar{g}=g^{-1}$, $p,q,r\in \mathbb{Q}\pi$, $\alpha\in \mathbb{Q}$, and $g\in \pi$.

Theorems & Definitions (136)

  • Definition 1.1: $\mathbb{Z}\pi$-homology equivalence
  • Theorem 1: Theorem \ref{['thm:unframed-inv']}, \ref{['thm:recover-I']}, Corollary \ref{['cor:inv-framed']}
  • Theorem 2: Proposition \ref{['prop:Z2-invariance']}, Theorem \ref{['thm:upper-bound2']}
  • Theorem 1.2: e.g. Sav02
  • Theorem 3: Proof in §\ref{['s:computation_dim']}
  • Remark 1.3
  • Theorem 1.4: Garoufalidis--Levine GL
  • Conjecture 1.5
  • Proposition 1.6
  • proof
  • ...and 126 more