Covering Space - Relations With Classifying Spaces and Group Cohomology

Relations With Classifying Spaces and Group Cohomology

If X is a connected cell complex with homotopy groups πn(X) =0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K(G,1) for G = π1(X).

Moreover, for every n ≥ 0 the group of cellular n-chains Cn(T) (that is, a free abelian group with basis given by n-cells in T) also has a natural ZG-module structure. Here for an n-cell σ in T and for g in G the cell g σ is exactly the translate of σ by a covering transformation of T corresponding to g. Moreover, Cn(T) is a free ZG-module with free ZG-basis given by representatives of G-orbits of n-cells in T. In this case the standard topological chain complex

where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, g m = m for every g ∈ G and every m ∈ Z). This resolution can be used to compute group cohomology of G with arbitrary coefficients.

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