Classifying Space - Applications

Applications

This still leaves the question of doing effective calculations with BG; for example, the theory of characteristic classes is essentially the same as computing the cohomology groups of BG, at least within the restrictive terms of homotopy theory, for interesting groups G such as Lie groups. As was shown by the Bott periodicity theorem, the homotopy groups of BG are also of fundamental interest. The early work on classifying spaces introduced constructions (for example, the bar construction), that gave concrete descriptions as a simplicial complex.

An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy equivalence in choosing BG. This example shows that classifying spaces may be complicated.

In relation with differential geometry (Chern–Weil theory) and the theory of Grassmannians, a much more hands-on approach to the theory is possible for cases such as the unitary groups that are of greatest interest. The construction of the Thom complex MG showed that the spaces BG were also implicated in cobordism theory, so that they assumed a central place in geometric considerations coming out of algebraic topology. Since group cohomology can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much homological algebra.

Generalizations include those for classifying foliations, and the classifying toposes for logical theories of the predicate calculus in intuitionistic logic that take the place of a 'space of models'.