4-manifolds with a given boundary

2025/10/21 by Anthony Conway, Conway, Anthony, Daniel Kasprowski +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2510.18836

Abstract

This paper studies the homotopy and homeomorphism classifications of 4-manifolds with boundary. Given 4-manifolds X₀ and X₁ with fundamental group π, we consider the problem of extending a homotopy equivalence h \colon ∂ X₀ → ∂ X₁ to a homotopy equivalence X₀ → X₁. 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∂ X₀ →∂ X₁ extends to a homeomorphism X₀ → X₁. The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when π≅ ℤ and the ∂ X_i have torsion Alexander module.

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