Relative mapping class group of the trivial and the tangent disk bundles over the sphere (Q2470766)

For an oriented connected smooth manifold \(M\) with boundary \(\partial M\), we denote, by \(\widetilde{\pi}_0\text{Diff}(M,\text{rel}\;\partial )\), the group of pseudo-isotopy classes of diffeomorphisms of \(M\) which are the identity on \(\partial M\), and call this group the relative mapping class group of \(M\). The author determines the group structure of the group \(\widetilde{\pi}_0\text{Diff}(S^p\times D^q, \text{rel}\;\partial )\;(1\leq p,q)\), while using the homotopy group \(\pi_k(SO(p+1))\;(k=p,q)\), the \(h\)-cobordism group \(\Theta_m\) of homotopy \(m\)-spheres \((m=q+1, p+q+1)\), and the group \(FC_p^{q+1}\) of pseudo-isotopy classes of orientation preserving embeddings of \(S^p\times D^{q+1}\) into \(S^{p+q+1}\), for example, \(\widetilde{\pi}_0\text{Diff}(S^p\times D^p, \text{rel}\;\partial )\cong \Theta_{p+q+1}\oplus \pi_p(SO(p+1))\) if \(3\leq p=q\). Next, the author tries to determine the relative mapping class group of the tangent unit disc bundle of the standard \(p\)--sphere and succeeds in doing so in the case when \(p\) is odd and \(p\not\equiv 1\mod 8\). In this case, the relative mapping class group coincides with that of \(S^p\times D^p\).