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

TL;DR

This paper extends a known invariant to 4-manifolds with 1-handles and demonstrates that certain 4-manifolds have mapping class groups with centers of infinite rank, revealing complex symmetries.

Contribution

It generalizes the $W_3$ invariant to a broader class of 4-manifolds with 1-handles and analyzes their mapping class groups.

Findings

Center of the mapping class group is of infinite rank for specified 4-manifolds.

Generalization of the $W_3$ invariant to 4-manifolds with 1-handles.

Provides new insights into the structure of mapping class groups in 4-dimensional topology.

Abstract

We develop a framework that generalizes Budney-Gabai's $W_3$ invariant on $\pi_0\textrm{Diff}(S^1\times D^3,\partial)$ 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.