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On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds

Sam Nariman

TL;DR

This work establishes homological finiteness for the classifying space $ ext{BDiff}(M, ext{rel } abla) $ when $M$ is a reducible 3-manifold built as a connected sum of irreducible pieces with nontrivial boundary, by proving it has finitely many nonzero homology groups, each finitely generated. The authors construct a sphere-complex framework $ oldsymbol{ S}(M)$ and its orbit quotient $[oldsymbol{ S}](M)$ to model the $ ext{Homeo}^{oldsymbol{ullet}}(M, ext{rel } abla) $-action and build a semi-simplicial resolution that enables an inductive argument on the number of prime factors. Central to the method are two-sided bar constructions, parallel-sphere bar models, and Kupers’ bar-resolution framework, which together reduce the analysis to the homological finiteness of fibers and bases in a sequence of fiber sequences. The approach yields strong control over the homology with local coefficients, allowing an inductive passage from the base case of irreducible factors with non-spherical boundary to the full reducible setting. The results provide a concrete homological analogue of Kontsevich’s finite-model conjecture for reducible 3-manifolds and integrate into the broader program linking 3-manifold diffeomorphism groups with finite-type invariants of their prime factors.

Abstract

Kontsevich conjectured that $\text{BDiff}(M, \text{rel }\partial)$ has the homotopy type of a finite CW complex for all compact $3$-manifolds with non-empty boundary. Hatcher-McCullough proved this conjecture when $M$ is irreducible. We prove a homological version of Kontsevich's conjecture. More precisely, we show that $\text{BDiff}(M, \text{rel }\partial)$ has finitely many nonzero homology groups, each finitely generated, when $M$ is a connected sum of irreducible $3$-manifolds that each have a nontrivial and non-spherical boundary.

On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds

TL;DR

This work establishes homological finiteness for the classifying space when is a reducible 3-manifold built as a connected sum of irreducible pieces with nontrivial boundary, by proving it has finitely many nonzero homology groups, each finitely generated. The authors construct a sphere-complex framework and its orbit quotient to model the -action and build a semi-simplicial resolution that enables an inductive argument on the number of prime factors. Central to the method are two-sided bar constructions, parallel-sphere bar models, and Kupers’ bar-resolution framework, which together reduce the analysis to the homological finiteness of fibers and bases in a sequence of fiber sequences. The approach yields strong control over the homology with local coefficients, allowing an inductive passage from the base case of irreducible factors with non-spherical boundary to the full reducible setting. The results provide a concrete homological analogue of Kontsevich’s finite-model conjecture for reducible 3-manifolds and integrate into the broader program linking 3-manifold diffeomorphism groups with finite-type invariants of their prime factors.

Abstract

Kontsevich conjectured that has the homotopy type of a finite CW complex for all compact -manifolds with non-empty boundary. Hatcher-McCullough proved this conjecture when is irreducible. We prove a homological version of Kontsevich's conjecture. More precisely, we show that has finitely many nonzero homology groups, each finitely generated, when is a connected sum of irreducible -manifolds that each have a nontrivial and non-spherical boundary.

Paper Structure

This paper contains 9 sections, 19 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sphere complexes
  3. Reformulation of the main theorem and an inductive strategy
  4. The base case of induction and spherical boundary components
  5. Proof of the technical \ref{['mainlem']}
  6. The homological finiteness of $\eta^{-1}(x)$ for a vertex $x$
  7. Parallel spheres and bar constructions
  8. Kupers' bar resolution for self-embeddings
  9. Higher filtrations and the last step of the proof of \ref{["main'"]}

Key Result

Theorem 1.1

Let $M$ be an orientable $3$-manifold that is a connected sum of compact irreducible $3$-manifolds that are not diffeomorphic to the $3$-ball and each have a nontrivial boundary. Then the classifying space $\mathrm{BDiff}(M, \text{rel }\partial)$ is strongly homologically finite.

Figures (2)

  • Figure 1: $\sigma$ here is a $2$-simplex consisting of 3 separating spheres that are drawn in one dimension lower.
  • Figure 2: Schematic picture in one dimension lower on how $\mathrm{BD}$ acts on $\mathrm{BR}$ and $\mathrm{BL}$

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5: Scharlemann
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 32 more