Cohomology Classification of Self Maps of Sphere Bundles over Spheres

Abstract

Let % e TT^CSOCg + l)). We denote the induced g-sphere bundle over the nsphere by £(%) or simply E. The purpose of this note is to study the image of the function [£(%),£(%)] -*Hom(#*(£(%))/Tor, #*(£(%))/Tor) which assigns the induced homomorphism, where H m (X) is the reduced ra-th cohomology group of a space X with values in Z, the group of integers. Let p l : > S Q be the canonical projection. We denote P U C*Q by a <E Tr^CS*).According to [13], where p is the attaching map of the top cell of £(%). Let Y = S Q U a e n . When n = 1 or when q= I and a =0, the function is surjective by [5]. In this note, if we do not specify otherwise, we will always assume q > 2, n > 2, and a -0 provided n = q + l.In this case, note from [5,6] where deg(a: 9 ) = q and deg(t/ n ) =n. Let kj, be integers. When #^w, a self map / of £(%) or 7 is called an M(W) -structure if /*(*<,) = /cr 9 and/*(yj = /y n . Let (fly) be a 2 X 2-matrix whose entries a^ are integers. When q =n, a self map / of E(X) or Y= S n VS n is called an (a^-) -structure with respect to {x n ,y n } if /*(£") = G,uX n +a l2 y n and /*(t/ n ) -fl2A+ G 222/n-When no confusion will occur, we will omit the words "with respect to {x n ,y n }". Notice that when q¥=n, an M(k,0-structure is an M e -structure [4] for any 6: {1,2,...} -> Z with 9 (#) =We will study conditions on the existence of an M (A:,/) -structure and an (a#) -structure on £(%) in § 2 and § 3, respectively. Our results are partial when E(%) does not have a section. To state our results, we need some notations. When E(%) has a section, we denote by f an element of Tr^CSOCg)) such that i'.(?) = X where f : S0(q) -> S0(^ + l) is the inclusion. Let j m denote the identity map of S m and /: ;r r (SO(m)) -» 7r r+m (S m ) the /-homomorphism.