In the second chapter, certain cohomology classes have been defined on a foliated manifold. These are obtained by using differential forms related to the characteristic classes of the normal bundle. Moreover, the cohomology classes have formal properties analogous to those of the characteristic classes of a vector bundle. Hence, it is reasonable to try to obtain them by mapping into a classifying space for foliations, that is, some sort of foliated space whose cohomology consists of the characteristic classes of its foliation, and such that all foliations are obtained (up to some reasonable equivalence) by mapping into the classifying space and pulling back its foliation. Even for vector bundles, the classifying space is not a finite dimensional smooth manifold, so it is necessary to generalize the definition to allow foliations of much more general topological spaces, and so that the generalized foliations map contravariantly under a much larger class of mappings. In particular, even for a smooth manifold some class of singularities must be allowed. In the three sections of this chapter, three types of classifying spaces and the accompanying notions of singularity will be discussed.
KeywordsVector Field Vector Bundle Smooth Manifold Cohomology Class Normal Bundle
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