On the mapping class groups of \(\#_r(S^p \times S^p)\) for \(p = 3, 7\) (Q658331)

For \(p=3\) or 7 let \(M_r=\#_r(\mathbb S^p\times\mathbb S^p)\), \(\text{Aut}(M_r)\) be the isotopy classes of orientation-preserving diffeomorphisms modulo the isotopy classes of diffeomorphisms of \(\mathbb D^{2p}\) fixed on the boundary, \(\mathcal E(M_r)\) the homotopy classes of orientation-preserving homotopy equivalences and \(\phi_r:H_p(M_r)\times H_p(M_r)\to\mathbb Z\) the intersection form. Then the extensions \[ \begin{tikzcd} 0\rar & H_p(M_r) \dar\rar & \Aut(M_r) \dar\rar & \Aut(\phi_r) \dar\rar & 1\\ 0 \rar & H_p(M_r)\otimes\pi_{2p}(\mathbb S^p) \rar & \mathcal E(M_r) \rar & \Aut(\phi_r) \rar & 1 \end{tikzcd} \] split if and only if \(r=1\).