Self-maps of sphere bundles. II (Q910049): Difference between revisions

Let \(S^ q\to E\to S^{r+1}\) be an oriented orthogonal q-sphere bundle. A fibre preserving map f: \(E\to E\) induces the degree m say on the fibre. In an earlier paper [J. Pure Appl. Algebra 10, 95-99 (1977; Zbl 0367.55017)] the author gave results on the structure, for more general base-space, of the set A(E) of integers m for which there is a fibre preserving map of degree m. In the case where q is odd, homotopical obstructions are given here to the existence of a map of degree m: in the case where \(q=1\), 3 or 7 these conditions become necessary and sufficient. In the case where q is even and the bundle has a cross section, necessary and sufficient homotopical conditions for the existence of a map of degree m are given: in particular there are in this case fibre preserving maps of all odd degrees and all degrees \(m\equiv 0 mod 4\).