Classifying Space - Motivation

Motivation

An example for G infinite cyclic is the circle as X. When G is a discrete group, another way to specify the condition on X is that the universal cover Y of X is contractible. In that case the projection map

becomes a fiber bundle with structure group G, in fact a principal bundle for G. The interest in the classifying space concept really arises from the fact that in this case Y has a universal property with respect to principal G-bundles, in the homotopy category. This is actually more basic than the condition that the higher homotopy groups vanish: the fundamental idea is, given G, to find such a contractible space Y on which G acts freely. (The weak equivalence idea of homotopy theory relates the two versions.) In the case of the circle example, what is being said is that we remark that an infinite cyclic group C acts freely on the real line R, which is contractible. Taking X as the quotient space circle, we can regard the projection π from R = Y to X as a helix in geometrical terms, undergoing projection from three dimensions to the plane. What is being claimed is that π has a universal property amongst principal C-bundles; that any principal C-bundle in a definite way 'comes from' π.