Mapping class groups of $h$-cobordant manifolds

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.