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On the mapping class groups of 4-manifolds with 1-handles

Jianfeng Lin, Yi Xie, Boyu Zhang

TL;DR

The paper addresses the problem of central elements in mapping class groups of 4-manifolds with a 1-handle by extending Budney-Gabai's W3 framework. It constructs a scanning map from diffeomorphism groups to embedding spaces and builds a cosimplicial/configuration-space framework to study embeddings via a spectral-sequence approach. The main results show that the image of the 1-handle diffeomorphism group into the ambient mapping class group has infinite rank under natural hypotheses, with extensions to homeomorphisms. This work links embedding calculus, configuration-space invariants, and spectral-sequence methods to provide a broad method for detecting infinite-rank centers in mapping class groups of 4-manifolds.

Abstract

We develop a framework that generalizes Budney-Gabai's $W_3$ invariant on $π_0\textrm{Diff}(S^1\times D^3,\partial)$ to 4-manifolds with 1-handles. As applications, we show that if $M=(S^1\times D^3)\natural \hat M$ where $\hat M$ either has the form $I\times Y$ or is a punctured aspherical manifold, then the center of the mapping class group of $M$ is of infinite rank.

On the mapping class groups of 4-manifolds with 1-handles

TL;DR

The paper addresses the problem of central elements in mapping class groups of 4-manifolds with a 1-handle by extending Budney-Gabai's W3 framework. It constructs a scanning map from diffeomorphism groups to embedding spaces and builds a cosimplicial/configuration-space framework to study embeddings via a spectral-sequence approach. The main results show that the image of the 1-handle diffeomorphism group into the ambient mapping class group has infinite rank under natural hypotheses, with extensions to homeomorphisms. This work links embedding calculus, configuration-space invariants, and spectral-sequence methods to provide a broad method for detecting infinite-rank centers in mapping class groups of 4-manifolds.

Abstract

We develop a framework that generalizes Budney-Gabai's invariant on to 4-manifolds with 1-handles. As applications, we show that if where either has the form or is a punctured aspherical manifold, then the center of the mapping class group of is of infinite rank.
Paper Structure (35 sections, 68 theorems, 121 equations)

This paper contains 35 sections, 68 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The scanning map $\mathcal{S}$
  3. The embedding space
  4. The map $\mathcal{S}$
  5. The mapping space
  6. Compactifications of configuration spaces
  7. The mapping spaces
  8. Homotopy groups with base points in a simply connected set
  9. Homotopy groups of the configuration spaces
  10. General setups
  11. Properties of $w_{ij}$
  12. Homotopy groups of configuration spaces
  13. The general setup
  14. Computation of $\pi_3^\mathbb{Q} C_n'\langle M,\partial\rangle$ for $n=2,3$
  15. Computation of $\pi_4^\mathbb{Q} C_n'\langle M,\partial\rangle$
  16. ...and 20 more sections

Key Result

Theorem 1.1

The mapping class group $\pi_0 \mathop{\mathrm{Diff}}\nolimits(S^1\times D^3,\partial)$ is an abelian group of infinite rank.

Theorems & Definitions (162)

  • Theorem 1.1: BG2019Watanabe2020
  • Theorem 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 152 more