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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


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.

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