Diffeomorphism groups of connected sum of a product of spheres and classification of manifolds (Q578666)

Let M be a closed \(C^{\infty}\) manifold. M is called of type (n,p,r) if it is a simply connected n-dimensional manifold such that \(H_ p(M)\approx H_{n-p}(M)\approx {\mathbb{Z}}^ r\), \(H_ 0(M)\approx H_ n(M)\approx {\mathbb{Z}}\) and \(H_ i(M)=0\) otherwise. \textit{E. C. Turner} [Invent. Math. 8, 69-82 (1969; Zbl 0179.518)] and \textit{H. Sato} [J. Math. Soc. Japan 21, 1-36 (1969; Zbl 0193.527)], gave a classification of manifolds of type (n,p,1), \(p\equiv 3,5,6,7\) (mod 8). In this paper, a complete classification of manifolds of type (n,p,2), \(p\equiv 3,5,6,7\) (mod 8) is given. Theorem: If M is of type (n,p,2) with \(p\equiv 3,5,6,7\) (mod 8), then M is diffeomorphic to \[ S^ p \times D^{q+1}\#_{\partial}S^ p \times D^{q+1}\cup_{h}S^ p \times D^{q+1}\#_{\partial}S^ p \times D^{q+1}\quad, \] where \(n=p+q+1\), \(\#_{\partial}\) means connected sum along the boundary and \[ h: S^ p \times S^ q\#S^ p \times S^ q\quad \to \quad S^ p \times S^ q\#S^ p \times S^ q \] is a diffeomorphism. The number of the differentiable manifolds up to diffeomorphism satisfying the above is equal to twice the order of the group \(\pi_ q(SO(p+1))\oplus \theta^{p+q+1}\). The second half of the above theorem is proved by computing the group of pseudo-diffeotopy classes of diffeomorphisms of \(S^ p\times S^ q\#S^ p\times S^ q\) \((p<q).\) The following generalization is stated. Theorem: Let M be of type (n,p,r) where r is an arbitrary positive number, \(n=p+q+1\) and \(p\equiv 3,5,6,7\) (mod 8). The number of such differentiable manifolds up to diffeomorphism is equal to r times the order of \(\pi_ q(SO(p+1))\oplus \theta^{p+q+1}\).