Inertial $h$-cobordisms with finite cyclic fundamental group

Abstract:

For $M$ a PL $n$-manifold, $n \geqq 5$, let $I(M)$ be the subset of torsions $\sigma \in \operatorname {Wh} ({\pi _1}M)$ such that the $h$-cobordism $W$ constructed from $M$ with torsion $\sigma$ has its other boundary component PL homeomorphic to $M$. We present three techniques dealing with the determination of $I(M)$ and apply them when ${\pi _1}M = {Z_q}$. We prove: (1) If $n$ is even, ${\pi _1}M \simeq {Z_q},q$ odd, then $I(M) = \operatorname {Wh} ({\pi _1}M)$. (2) If $n$ is odd, then there exists $M$ with ${\pi _1}M \simeq {Z_q}$ such that $I(M) = \operatorname {Wh} ({\pi _1}M)$.