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

MSC:

 57R55 Differentiable structures in differential topology 57R60 Homotopy spheres, PoincarĂ© conjecture 57R80 $$h$$- and $$s$$-cobordism