Geometry & Topology

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Eugénie Hunsicker

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Abstract

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 τi 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 τi in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X.

Article information

Source
Geom. Topol., Volume 11, Number 3 (2007), 1581-1622.

Dates
Received: 17 February 2006
Revised: 20 November 2006
Accepted: 19 June 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799903

Digital Object Identifier
doi:10.2140/gt.2007.11.1581

Mathematical Reviews number (MathSciNet)
MR2326952

Zentralblatt MATH identifier
1132.58013

Subjects
Primary: 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10] 55N33: Intersection homology and cohomology 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
Secondary: 58J10: Differential complexes [See also 35Nxx]; elliptic complexes 13D22: Homological conjectures (intersection theorems) 32S20: Global theory of singularities; cohomological properties [See also 14E15]

Keywords
$L^2$ Hodge theorem $L^2$ signature theorem tau invariant Novikov additivity Leray–Serre spectral sequence

Citation

Hunsicker, Eugénie. Hodge and signature theorems for a family of manifolds with fibre bundle boundary. Geom. Topol. 11 (2007), no. 3, 1581--1622. doi:10.2140/gt.2007.11.1581. https://projecteuclid.org/euclid.gt/1513799903


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