4-manifolds with a given boundary

TL;DR

This paper explores the classification of 4-manifolds with boundary by analyzing when boundary homotopy equivalences extend to the entire manifold, using surgery theory and group properties.

Contribution

It provides broad conditions under which boundary homotopy equivalences extend to the whole 4-manifold, including new results for various fundamental groups.

Findings

Extended homotopy equivalences for broad classes of groups.

Identified conditions for boundary homeomorphisms to extend to the interior.

Recovered and generalized previous classification results.

Abstract

This paper studies the homotopy and homeomorphism classifications of $4$-manifolds with boundary. Given $4$-manifolds $X_0$ and $X_1$ with fundamental group $\pi$, we consider the problem of extending a homotopy equivalence $h \colon \partial X_0 \to \partial X_1$ to a homotopy equivalence $X_0 \to X_1$. We solve this problem in broad settings for a class of groups that includes free groups, finite cyclic groups, finite dihedral groups, solvable Baumslag-Solitar groups, and many $3$-manifold groups. When the fundamental group is additionally assumed to be good, we use surgery theory to list situations when a homeomorphism $h\colon\partial X_0 \to\partial X_1$ extends to a homeomorphism $X_0 \to X_1$. The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when $\pi \cong \mathbb{Z}$ and the $\partial X_i$ have torsion Alexander module.