On a extension of the James-Whitehead theorem about sphere bundles over spheres (Q1116122)

Let \(W_ j\) be a handlebody obtained by gluing r q-handles to a \((p+q+1)\)-disk for \(p=q-2\geq 3\) or \(p=q-3\geq 4,\) and consider Wall's invariant system \((H_ j=H_ q(W_ j)\); \(\phi_ j: H_ j\times H_ j\to Z_ 2=\pi_ q(S^{p+1})\), \(\alpha_ j: H_ j\to \pi_{q- 1}(SO_{p+1}))\). If \(\phi_ j(x,x)=0\), then \(\alpha_ j(x)\in Im[S: \pi_{p-1}(SO_ p)\to \pi_{q-1}(SO_{p+1})]\). Now J: \(\pi_{q- 1}(SO_ p)\to \pi_ k(S^ p)\) \((k=p+q-1)\) induces \(\lambda\) : Im \(S\to Co\ker [P: \pi_ q(S^ p)\to \pi_ k(S^ p)]\); and for the generator \(\theta \in \pi_{q-1}(S^ p)\), we have \(i_*: \pi_ k(S^ p)\to \pi_ k(S^ p\cup_{\theta}D^ q)\) induced by the inclusion i and \({\bar \lambda}=i_*\circ \lambda: Im S\to Co\ker [i_*\circ P].\) Theorem. Assume that \(\phi_ j(x,x)=0\) for any \(x\in H_ j\) \((j=1,2)\). Then the boundaries \(\partial W_ 1\) and \(\partial W_ 2\) are homotopy equivalent if and only if there exists an isomorphism h: \(H_ 1\to H_ 2\) such that (1) \(\lambda \circ \alpha_ 1=\lambda \circ \alpha_ 2\circ h\), when \(\phi_ j=0\); (2) \(\phi_ 1=\phi_ 2\circ (h\times h)\) and \({\bar \lambda}\circ \alpha_ 1={\bar \lambda}\circ \alpha_ 2\circ h\), when rank \(\phi_ j=r\); (3) in addition to (2), \(H_ 1\) is a direct sum \(H'\oplus H''\) orthogonal with respect to \(\phi_ 1\), \(\phi\) \(| H''\times H''\) is non-singular, \(\phi | H'\times H'=0\) and (1) holds on \(H'\) when \(0<rank \phi_ j<r.\) Proofs are outlined in case (2), and the details will be given elsewhere.