Homotopy classification of connected sums of sphere bundles over spheres. II
The author classifies the connected sums of p-sphere bundles over q- spheres \((p,q>1)\) up to homotopy equivalence, which generalizes his previous results [Nagoya Math. J. 83, 15-36 (1981; Zbl 0467.55006)]. Let \(q/2<p<q-1\). Let \(B_ i (i=1,...,r)\) be p-sphere bundles over q-spheres and \(\bar B_ i\) the associated disk bundles. Let \(\alpha_ i\in \pi_{q-1}(SO(p+1))\) be the characteristic element of \(B_ i\), where \(SO(p+1)\) is the rotation group of the Euclidean space \(R^{p+1}\). Let \(\pi_*: \pi_{q-1}(SO(p+1))\to \pi_{q-1}(S^ p)\) be the homomorphism induced by the natural projection \(\pi\) : \(SO(p+1))\to SO(p+1))/SO(p)=S^ p\). The boundary connected sum of the \(\bar B_ i\) can be considered as a handlebody \(W\in {\mathcal H}(m+1,r,q),mp+q\) and the connected sum of the \(B_ i\) as its boundary. Let \(\phi:\quad H_ q(W)\times H_ q(W)\to \pi_ q(S^{p+1})\) be the pairing defined by Wall and \(\alpha\) : \(H_ q(W)\to \pi_{q-1}(SO(p+1))\) the map assigning to each \(x\in H_ q(W)\cong \pi_ q(W)\) the characteristic element of the normal bundle of the embedded q-sphere representing x. A base \(\{w_ 1,...,w_ r\}\) of the free abelian group \(H_ q(W)\) is called admissible if \(\phi(w_ i,w_ j)=0\) for all i,j (\(i\neq j)\). Consider the following diagram for \(\omega \in \pi_{q-1}(S^ p)\): \[ \pi_{p+q-1}(S^{q-1}) \overset{\omega_*}{} \pi_{p+q-1}(S^ p) \overset{J}\leftarrow \pi_{q-1}(SO(p)) \overset{I_*}{} \pi_{p+1}(SO(p+1)), \] where \(\omega_*\) is defined by the composition with \(\omega\), is the \(J\)-homomorphism, and \(i_*\) is induced by the inclusion map. Let \(G(\omega)=i_*(J^{-1}(Im\omega_*))\) and \(\epsilon =\pi_*{\mathbb{O}}\alpha\). Let \(B_ i'(i=1,...,r')\) be another set of p-sphere bundles over q-spheres. If the connected sum of the \(B_ i\) has the same homotopy type as that of the \(B_ i'\), then \(r=r'\). Theorem. The connected sums of \(B_ i\) and of \(B_ i'\) are of the same oriented homotopy type if and only if there exist admissible bases \(\{w_ 1,...,w_ r\}\) and \(\{w_ i,...,w_ r'\}\) such that (1) \(\epsilon(w_ i)=\epsilon '(w_ i')\) and (2) \(\{\alpha(w_ i)\}=\{\alpha '(w_ i')\}\) in \(\pi_{q-1}(SO(p+1))/G(\epsilon(w_ i)) (=\pi_{q-1}(SO(p+1))/G(\epsilon '(w_ i))\); i.e. \(\epsilon_ i=\epsilon_ i'\) and \(\{\alpha_ i\}=\{\alpha_ i'\}\) in \(\pi_{q- 1}(SO(p+1))/G(\epsilon_ i) i=1,...,r\) modulo ''representations''.
connected sums of q-sphere bundles over p-spheres up to homotopy equivalence
scientific article; zbMATH DE number 3842746
scientific article