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Differentiable structures on products of spheres. (English) Zbl 0547.57026
\(S^ n\) denotes the standard n-sphere with the usual differential structure, \(\Sigma^ n\) denotes the homotopy n-sphere, \(\theta^ n\) is the group of h-cobordism classes of homotopy n-spheres under the connected sum operation #, and H(p,k) is the subgroup of \(\theta^ p\) consisting of those homotopy p-spheres \(\Sigma^ p\) such that \(\Sigma^ p\times S^ k\) is diffeomorphic to \(S^ p\times S^ k.\)
The main results of the paper are the following: Theorem. If M is a smooth n-manifold homeomorphic to \(S^ p\times S^ q\times S^ r\), 2\(\leq p\leq q\leq r\), \(n=p+q+r\), then there exist homotopy spheres \(\Sigma^ r\), \(\Sigma^{p+r}\), \(\Sigma^{p+q}\), \(\Sigma^{q+r}\) and \(\Sigma^ n\) such that M is diffeomorphic to \[ [(S^ p\times S^ q\times\Sigma^ r) \#r \Sigma^{p+q}\quad\times \quad\Sigma^ r \#q \Sigma^{p+r}\quad\times \quad S^ q \#p \Sigma^{q+r}\quad\times \quad S^ p] \#\Sigma^ n. \] (Here #m denotes the connected sum of manifolds along an m-cycle.)
Theorem. For 2\(\leq p\leq q\) and r-3\(\leq q\leq r\), the number of differentiable structures on \(S^ p\times S^ q\times S^ r\) is the order of the group \[ [(\theta^{q+r}/H(q+r,p))\quad\times \quad (\theta^{p+r}/H(p+r,q))\quad\times \quad (\theta^{p+q}/H(p+q,r))\quad\times \quad\theta^ n]. \] The paper concludes with several explicit calculations. For example, the number of differentiable structures on \(S^ 4\times S^ 5\times S^ 7\) is shown to be 2 while on \(S^ 4\times S^ 5\times S^ 6\) it is shown to be 16256.
Reviewer: M.V.Mielke
57R55 Differentiable structures in differential topology
57R60 Homotopy spheres, Poincaré conjecture
57R80 \(h\)- and \(s\)-cobordism