Abstract
We generalize the concept of "foliation" and define k k -flat structures; these are smooth vector bundles with affine connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for k k -flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space.
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