Geometriae Dedicata

, Volume 148, Issue 1, pp 15–33

The smooth structure set of Sp × Sq

Original Paper

DOI: 10.1007/s10711-010-9513-8

Cite this article as:
Crowley, D. Geom Dedicata (2010) 148: 15. doi:10.1007/s10711-010-9513-8


We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of Sp × Sq, 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 

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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