Geometriae Dedicata

, Volume 148, Issue 1, pp 15–33 | Cite as

The smooth structure set of S p × S q

  • Diarmuid CrowleyEmail author
Original Paper


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.


Smooth structure set Surgery exact sequence Product of spheres Diffeomorphism classification 

Mathematics Subject Classification (2000)

57R55 57R65 


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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