Volume 11, issue 1 (2007)

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The signature of a fibre bundle is multiplicative mod 4

Ian Hambleton, Andrew Korzeniewski and Andrew Ranicki

Geometry & Topology 11 (2007) 251–314

arXiv: math.AT/0502353

Abstract

We express the signature modulo 4 of a closed, oriented, 4k–dimensional PL manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined by Korzeniewski [Absolute Whitehead torsion, Geom. Topol. 11 (2007) 215–249]. Let F E B be a PL fibre bundle, where F, E and B are closed, connected, and compatibly oriented PL manifolds. We give a formula for the absolute torsion of the total space E in terms of the absolute torsion of the base and fibre, and then combine these two results to prove that the signature of E is congruent modulo 4 to the product of the signatures of F and B.

Keywords
signature, fibre bundle, multiplicative
Mathematical Subject Classification 2000
Primary: 55R25
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Publication
Published: 16 March 2007
Proposed: Tom Goodwillie
Seconded: Colin Rourke and Peter Teichner
Authors
Ian Hambleton
Department of Mathematics & Statistics
McMaster University
Hamilton
Ontario
L8S 4K1
Canada
Andrew Korzeniewski
School of Mathematics
University of Edinburgh
Edinburgh
EH9 3JZ
United Kingdom
Andrew Ranicki
School of Mathematics
University of Edinburgh
Edinburgh
EH9 3JZ
United Kingdom