An introduction to L cohomology
نویسنده
چکیده
After a quick introduction to L cohomology, we discuss recent joint work with Jeff Cheeger where we study, from a mostly topological standpoint, the L-signature of certain spaces with nonisolated conical singularities. The contribution from the singularities is identified with a topological invariant of the link fibration of the singularities, involving the spectral sequence of the link fibration. This paper consists of two parts. In the first, we give an introduction to L cohomology. This is partly based on [8]. We focus on the analytic aspect of L cohomology theory. For the topological story, we refer to [1; 22; 31] and of course the original papers [16; 17]. For the history and comprehensive literature, see [29]. The second part is based on our joint work with Jeff Cheeger [11], which gives the contribution to the L signature from nonisolated conical singularity. It is a pleasure to thank Eugenie Hunsicker for numerous comments and suggestions. 1. L cohomology: what and why What is L cohomology? The de Rham theorem provides one of the most useful connections between the topological and differential structure of a manifold. The differential structure enters the de Rham complex, which is the cochain complex of smooth exterior differential forms on a manifold M , with the exterior derivative as the differential: 0 ! ̋.M / d ! ̋.M / d ! ̋.M / d ! ̋.M /! Partially supported by NSF and CNSF..
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