Homotopy types of connected sums of spherical fibre spaces over spheres
type: Item
instance of: scholarly article
title: Homotopy types of connected sums of spherical fibre spaces over spheres
zbMATH Open document ID: 0587.55008
author: Kohhei Yamaguchi
published in: Kodai Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let \(P(\tau)\) denote the Whitehead product \([\tau,\iota_ n]\) in \(\pi_{2n+k}S^ n\) where \(\iota_ n\) generates \(\pi_ nS^ n\) and \(\tau\) is an element in \(\pi_{n+k+1}S^ n\). Let \(X_ h\) denote the total space of an orientable n-sphere fibration with a section over \(S^{n+k+1}\). Let \(\lambda (X_ h)\) denote the coset of P(\(\tau)\) in \(\pi_{2n+k}(S^ n)/P\pi_{n+k+1}(S^ n)\) where \(\tau\) is the adjoint of the classifying map for the fibration \(X_ h\). Since \(X_ h\) is a Poincaré complex, the connected sum \(\#^{r}_{n=1}X_ h\), \(1\leq h\leq r\), is defined [\textit{C. T. C. Wall}, Ann. Math., II. Ser. 86, 213- 245 (1967; Zbl 0153.254)]. The author's main result may be stated as follows: let \(X_ h\) and \(X'_ h\) be total spaces of orientable n- sphere fibrations over \(S^{n+k+1}\) which admit sections. Then the connected sums \(\#^{r}_{h=1}X_ h\) and \(\#^{r}_{h=1}X'_ h\) are of the same homotopy type if and only if there is a matrix A in \(GL_ n({\mathbb{Z}})\) such \({}^ t[\lambda (H'_ 1),...,\lambda (X'_ r)]=A\cdot^ t[\lambda (X_ 1),...,\lambda (X_ r)]\) where the coset \(\pi_{2n+k}(S^ n)/P\pi_{n+k+1}(S^ n)\) is considered as a left \({\mathbb{Z}}\)-module.
Mathematics Subject Classification ID: 55R15
Mathematics Subject Classification ID: 55Q15
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 3942503
zbMATH Keywords: Whitehead product
zbMATH Keywords: n-sphere fibration with a section
zbMATH Keywords: connected sums
zbMATH Keywords: homotopy type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037101
OpenAlex ID: W2074782903
cites work: Note on quasifibrations and fibre bundles
cites work: Homotopy classification of connected sums of sphere bundles over spheres, I
cites work: The Homotopy Theory of Sphere Bundles Over Spheres (I)
cites work: The Homotopy Theory of Sphere Bundles Over Spheres (II)
cites work: Groups of self-equivalences of certain complexes
cites work: On the group of self-equivalences of a mapping cone
cites work: Homotopy types of spherical fibre spaces over spheres
cites work: Manifolds with Few Cells and the Stable Homotopy of Spheres
cites work: A classification theorem for fibre spaces
cites work: Extensions of Mappings Into H -Spaces
cites work: Note on Quasifibrations and Manifolds
cites work: On the classification of sufficiently connected manifolds
cites work: Poincaré complexes. I
cites work: Spherical fibrations and manifolds
cites work: On the self-homotopy equivalences of the wedge of certain complexes
DOI: 10.2996/KMJ/1138037101
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description: scientific article; zbMATH DE number 3942503