Mapping class groups of $h$-cobordant manifolds

TL;DR

This paper demonstrates that the mapping class group varies among h-cobordant high-dimensional manifolds, showing it is not an h-cobordism invariant, by analyzing a new moduli space of h-block bundles.

Contribution

It introduces a moduli space of h-block bundles and reveals its differences from ordinary block bundle moduli spaces, establishing the non-invariance of the mapping class group.

Findings

Mapping class groups differ for h-cobordant manifolds

The moduli space of h-block bundles is distinct from that of ordinary bundles

Mapping class group is not an h-cobordism invariant

Abstract

We prove that the mapping class group is not an $h$-cobordism invariant of high-dimensional manifolds by exhibiting $h$-cobordant manifolds whose mapping class groups have different cardinalities. In order to do so, we introduce a moduli space of "$h$-block" bundles and understand its difference with the moduli space of ordinary block bundles.