In topology, there is a dichotomy between two general classes
of spaces and ways of thinking about their roles. On the one hand,
there are the concrete geometric spaces, most importantly the various
types of manifolds. Typical problems one proposes to study about
such spaces are their classification, at least up to cobordism, and
the obstructions to the existence of an equivalence of a given space
with a space with a richer type of structure. Bundles and fibrations
over geometric spaces generally play a central role in the solution
to such problems. On the other hand, there are the classifying spaces
for bundle and fibration theories and other cohomological invariants
of spaces. These are thought of as tools for the analysis of
geometric problems, and it is a familiar fact that theorems on the
classifying space level often translate to yield intrinsic
information on the bundle theory level. Thus, for example, Bott periodicity
originated as a statement about the homotopy types of classifying
spaces, but is most usefully interpreted as a statement about bundles
and fcheir tensor products.