#### 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 $F\to E\to B$ 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
Primary: 55R25
##### 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