The smooth structure set of $S^p\times S^q$

Abstract

We calculate the smooth structure set of $S^p\times S^q$, $S(p,q)$, for $p,q\ge 2$ and $p+q\ge 5$. As a consequence we show that in general $S(4j-1,4k)$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F:S(4j,4k)-->S^{Top}(4j,4k)$ is not in general a subgroup of the topological structure set.

arXiv comment21 pages. Referee's suggestions included, proof of Lemma 4.1 clarified, typos corrected. To appear in Geometriae Dedicata