Infinite homotopy stable class for 4-manifolds with boundary

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.

v1: 12 pages. v2: 26 pages. The paper underwent a significant rewrite to account for a gap on page 3 of v1, related to whether the union of two spin manifolds is again spin. The main result of the paper is unaffected