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4-manifolds with a given boundary

Anthony Conway, Daniel Kasprowski

TL;DR

This work develops a comprehensive obstruction-theoretic framework for classifying 4-manifolds with boundary up to homotopy and, in favorable cases with good fundamental groups, up to homeomorphism. Central to the approach are the Postnikov 2-type and a pair of obstructions: a primary obstruction detected by comparing pulled-back relative intersection forms, and a secondary obstruction b(c0,c1) valued in Hermitian forms modulo a 𝒢-action. The authors establish conditions under which these obstructions vanish, and crucially prove a main technical result that allows arbitrary adjustments to the secondary obstruction via controlled homotopies, effectively enabling the vanishing of the obstruction in broad group-theoretic settings. They then connect these homotopy classifications to homeomorphism classifications through surgery theory, obtaining concrete results for various π (including finite, dihedral,BS groups, and 3-manifold groups) and recovering known simply-connected and Z-case outcomes as special cases. The methodology blends Postnikov theory, relative k-invariants, twisted (co)homology, and pushouts of Poincaré pairs, yielding a robust framework for extending boundary data to the entire 4-manifold and for deciding when boundary homeomorphisms extend to interior homeomorphisms under spin/non-spin and Kirby–Siebenmann constraints.

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 $π$, 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 $π\cong \mathbb{Z}$ and the $\partial X_i$ have torsion Alexander module.

4-manifolds with a given boundary

TL;DR

This work develops a comprehensive obstruction-theoretic framework for classifying 4-manifolds with boundary up to homotopy and, in favorable cases with good fundamental groups, up to homeomorphism. Central to the approach are the Postnikov 2-type and a pair of obstructions: a primary obstruction detected by comparing pulled-back relative intersection forms, and a secondary obstruction b(c0,c1) valued in Hermitian forms modulo a 𝒢-action. The authors establish conditions under which these obstructions vanish, and crucially prove a main technical result that allows arbitrary adjustments to the secondary obstruction via controlled homotopies, effectively enabling the vanishing of the obstruction in broad group-theoretic settings. They then connect these homotopy classifications to homeomorphism classifications through surgery theory, obtaining concrete results for various π (including finite, dihedral,BS groups, and 3-manifold groups) and recovering known simply-connected and Z-case outcomes as special cases. The methodology blends Postnikov theory, relative k-invariants, twisted (co)homology, and pushouts of Poincaré pairs, yielding a robust framework for extending boundary data to the entire 4-manifold and for deciding when boundary homeomorphisms extend to interior homeomorphisms under spin/non-spin and Kirby–Siebenmann constraints.

Abstract

This paper studies the homotopy and homeomorphism classifications of -manifolds with boundary. Given -manifolds and with fundamental group , we consider the problem of extending a homotopy equivalence to a homotopy equivalence . 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 -manifold groups. When the fundamental group is additionally assumed to be good, we use surgery theory to list situations when a homeomorphism extends to a homeomorphism . The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when and the have torsion Alexander module.
Paper Structure (49 sections, 90 theorems, 292 equations)

This paper contains 49 sections, 90 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. The homotopy classification of $4$-manifolds with boundary.
  3. Homeomorphism classifications
  4. The homotopy classification in terms of the Postnikov $2$-type.
  5. Primary and secondary obstructions
  6. If the obstructions vanish, then the homotopy equivalence extends
  7. The primary obstruction and compatible triples
  8. The vanishing of the secondary obstruction I: odd forms.
  9. The vanishing of the secondary obstruction II: (weakly) even forms.
  10. The vanishing of the secondary obstruction III : the main technical result.
  11. Proof of the statements from the introduction
  12. Preparatory results
  13. Poincaré pairs
  14. The second homotopy group
  15. The cap product pairing
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $\pi$ be a group that is either finite abelian with at most $2$ generators, or finite dihedral, or a surface group, or a solvable Baumslag--Solitar group, or a torsion-free $3$-manifold group, let $X_0$ and $X_1$ be $4$-manifolds with $\pi_1(X_j) \cong \pi$ and $\iota_j \colon \pi_1(\partial X_j Additionally, given a $k$-invariant preserving compatible triple $(F,G,h)$ as above, the homotopy e

Theorems & Definitions (219)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • proof
  • Remark 1.7
  • Theorem 1.8
  • ...and 209 more