Homotopy types of connected sums of spherical fibre spaces over spheres
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.
scientific article; zbMATH DE number 3942503
scientific article