The cohomology of covering spaces of 𝐻-spaces

Abstract

In this note, the cohomology of a covering space of an £T-space is computed, in terms of the cohomology of the i?-space, for coefficients in a field. If the characteristic of the field is different from 2, we also calculate the ring structure. Let X be an arcwise connected space with a continuous multiplication with unit (an u-space). We will suppose that H(X; Zp) is finite dimensional for each q (singular cohomology will be used throughout). Let X be a covering space of X, T: X-+X the covering map, G the group of deck translations of X over X, i.e. the fibre of w. Then X can be given an Ji-space structure so that ir is a multiplicative map. Let us consider the spectral sequence of Leray-Cartan for this covering space. We can obtain it by replacing X by a homotopically equivalent space (again denoted by X) which is a fibre space over K(G, 1) with fibre X, the inclusion of X in X being homotopic to 7r, and the fibre m a p / : X—>K(Gf 1) being multiplicative. The group G acts trivially on H*(X; Zp)f so we have simple coefficients in E2 (with coefficients in Zp). THEOREM. Let p be an odd prime. Then fl**(Z; Zp) =A ® £ as rings, where A =7r*(iI*(X; ZP)=H*(X; Zp)/I, I is the ideal generated by j•*(H*(K(G, 1);ZP)), E is the exterior algebra on n generators x\> • • , x n , where the dimension of Xi = 2p — 1 and 2p are the dimensions of a system of generators of the kernel of ƒ*. If £ = 2, then the same result holds, but only as modules.

Locations

• Bulletin of the American Mathematical Society - View - PDF