Equivariant maps between cohomology spheres

Abstract

Let G be a compact Lie group. Let X be a Hausdorff compact G-space which is a cohomology sphere over a ring R (cf. Def. 1.1). In Section 1 we define the Euler class of a locally trivial bundle with X as fiber. Using this definition we show that the main result of [7] still holds if the sphere of an orthogonal representation is replaced by a G-space which is a cohomology sphere over R (Ths. 1.8, 1.9). In the second section we consider actions of a finite cyclic group G = Ck. For this group, and X as above, an index of X has been defined in [1] and [2]. First, with the use of the Euler class we define an index of X (cf. [7] for the case where X is the sphere of an orthogonal representation; see Def. 2.2). Next we prove that this index is equal to that introduced by Borisovich and Izrailevich (Th. 2.3), and consequently we obtain a simple geometrical interpretation of the index defined in [1]. Finally, using our definition of the index (via the Euler class) we compute its value for X = S(V ), the sphere of an orthogonal representation V of G = Ck (Th. 3.3). This allows us to find for which V this index is different from 0. From the results of [7] and those quoted above we derive a formula on the degree (modulo k) of a G-equivariant map f : S(W ) → S(V ) between the spheres of representations W and V of G = Ck (Prop. 3.11) (cf. [3], [8]).