ABSTRACT. After a quick introduction to L2 cohomology, we discuss recent joint work with Jeff Cheeger where we study, from a mostly topological standpoint, the L2-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 L2 cohomology. This is partly based on [8]. We focus on the analytic aspect of L2 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 L2 signature from nonisolated conical singularity.

It is a pleasure to thank Eugenie Hunsicker for numerous comments and suggestions.