Infinite homotopy stable class for 4-manifolds with boundary

TL;DR

This paper constructs an infinite family of topological 4-manifolds with identical stable homeomorphism types and boundary conditions, yet they are distinguished by their homotopy types, revealing complex structures in 4-manifold topology.

Contribution

It demonstrates the existence of infinitely many 4-manifolds with the same stable homeomorphism class and boundary, but distinct homotopy types, for each odd prime q.

Findings

Infinite family of 4-manifolds with same stable homeomorphism class.

Manifolds have isometric equivariant intersection pairings.

Manifolds are not homotopy equivalent via boundary-preserving maps.

Abstract

We show that for every odd prime $q$, there exists an infinite family $\{M_i\}_{i=1}^{\infty}$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $M_i$ have isometric rank one equivariant intersection pairings and boundary $L(2q, 1) # (S^1 \times S^2)$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary.