4-manifolds with a given boundary

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.