The smooth structure set of S p × S q

Abstract

We calculate $${\mathcal{S}^{{\it Diff}}(S^p \times S^q)}$$ , the smooth structure set of S p × S q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general $${\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}$$ 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 the forgetful map $${\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}$$ is not in general a subgroup of the topological structure set.