posted on 2015-03-31, 10:29authored byEugenie Hunsicker
Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands τ i of Dai’s τ invariant [J Amer Math Soc 4 (1991) 265–321] that are defined as signatures on the pages of the Leray–Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the τ i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X .
History
School
Science
Department
Mathematical Sciences
Published in
GEOMETRY & TOPOLOGY
Volume
11
Pages
1581 - 1622 (42)
Citation
HUNSICKER, E., 2007. Hodge and signature theorems for a family of manifolds with fibre bundle boundary. Geometry and Topology, 11, pp. 1581 - 1622.
Publisher
Mathematical Sciences Publishers.
Version
VoR (Version of Record)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
Publication date
2007
Notes
This article was published in the journal Geometry and Topology [Mathematical Sciences Publishers]. It is also available at: http://dx.doi.org/10.2140/gt.2007.11.1581