Stable characteristic classes of smooth manifold bundles

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.Comment: 52 page

The space of non-degenerate closed curves in a Riemannian manifold

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".Comment: 9 pages 1 figur

On the connectivity of finite subset spaces

We show that for an m-connected cell complex X the space exp_k X of non-empty subsets of X of cardinality at most k is (m + k - 2)-connectedComment: 2 page