Homotopy classification of connected sums of sphere bundles over spheres. II
type: Item
instance of: scholarly article
title: Homotopy classification of connected sums of sphere bundles over spheres. II
zbMATH Open document ID: 0531.55007
author: Hiroyasu Ishimoto
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1982-01-01T00:00:00Z
review text: 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 base…
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 3842746
zbMATH Keywords: connected sums of q-sphere bundles over p-spheres up to homotopy equivalence
zbMATH Keywords: J-homomorphism
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195184025
OpenAlex ID: W4248006725
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DOI: 10.2977/PRIMS/1195184025
description: scientific article; zbMATH DE number 3842746