5 - The cohomology of classifying spaces of H-spaces

Summary

The scope of the next paper has been explained in §6. As one of the later papers, it assumes a fair familiarity with the machinery of algebraic topology.

Let G denote an associative H-space with unit (e.g. a topological group). We will show that the relations between G and a classifying space B_G are more readily displayed using a geometric analog of the resolutions of homological algebra. The analogy is quite sharp, the stages of the resolution, whose base is B_G, determine a filtration of B_G. The resulting spectral sequence for cohomology is independent of the choice of the resolution, it converges to H*B_G, and its E₂ term is Ext_H(G)(R, R) (R = ground ring). We thus obtain spectral sequences of the Eilenberg-Moore type in a simpler and more geometric manner.

Geometric resolutions. We shall restrict ourselves to the category of compactly generated spaces. Such a space is Hausdorff and each subset which meets every compact set in a closed set is itself closed (a k-space in the terminology of Kelley). Subspaces are usually required to be closed, and to be deformation retracts of neighborhoods.

Let G be an associative H-space with unit e. A right G-action on a space X will be a continuous map X×G→X with xe = x, x(g₁g₂) = (xg₁)g₂ for all x∈AX, g₁, g₂∈ G. A space X with a right G-action will be called a G-space.