Homotopy classification of connected sums of sphere bundles over spheres. III
Consider a k-fold connected sum whose factors are p-sphere bundles over q-spheres \((p,q>1)\). In the previous papers [Nagoya Math. J. 83, 15-36 (1981; Zbl 0467.55006); Publ. Res. Inst. Math. Sci. 18, 307-324 (1982)] the author gave a necessary and sufficient condition for such connected sums to be homotopy equivalent. In this paper, in the case where \((p,q)=(8j-1,8j+1)\) and \((p,q)=(n-2,n+1)\), \(n=8j\) or \(8j+1\) and at least one factor admits no cross-section, the author classifies completely such a connected sum up to homotopy equivalence. The precise statement in the case where all factors admit no section is as follows. If \(j>1\) then there is a unique homotopy type represented by k \(B_{(0,1,0)}\) and, if \(j=1\) then homotopy equivalent means diffeomorphic and such a connected sum is represented by either of \[ k B_{(0,1,0)},\quad k B_{(0,1,1)},\quad(k-1) B_{(0,1,0)}\#B_{(0,1,1)}\quad or\quad(k-2) B_{(0,1,0)}\#2 B_{(0,1,1)}. \] Here the subscripts denote appropriate characteristic elements. The theorems in the other case are similar but too complicated to be stated here. The proofs reduce to a homotopy theoretical calculation of the James- Whitehead invariants \(G(\eta)\) or \(G(\eta^ 2)\) which is done using the structure of the J-homomorphism and Whitehead products.
scientific article; zbMATH DE number 3846463
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