Summary: L2
-cohomology of Spaces with Non-isolated Conical
Singularities and Non-multiplicativity of the Signature
Jeff Cheeger Xianzhe Dai
Abstract
We study from a mostly topological standpoint, the L2
-signature of certain spaces
with non-isolated conical singularities. The contribution from the singularities is identi-
fied with a topological invariant of the link fibration of the singularities. This invariant
measures the failure of the signature to behave multiplicatively for fibrations for which
the boundary of the fibre in nonempty. The result extends easily to cusp singularities
and can be used to compute the L2
cohomology of certain noncompact hyper-k¨ahler
manifolds which admit geometrically fibered end structures.
1 Introduction
In this paper, we study the L2-cohomology and L2-signature for certain spaces with non-
isolated conical singularities. We call these generalized Thom spaces. Appropriately formu-
lated, our results extend easily to cusp singularities as well. Our main theorem identifies
the contribution to the L2-signature from a singular stratum with a topological invariant
of the link fibration of the stratum. As an immediate application, we get a proof of the

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics