$\tilde{H}$-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory

We study $\tilde{H}$-cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi’s work, none were previously known to distinguish smooth and topological $\tilde{H}$-cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg–Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth $\tilde{H}$-cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to ${\mathbb{Z}}^{\mathrm{\infty }}$.