The signature of a fibre bundle is multiplicative mod 4

Hambleton, Ian¹ ; Korzeniewski, Andrew² ; Ranicki, Andrew²

¹ Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
² School of Mathematics, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom

Geometry & topology, Tome 11 (2007) no. 1, pp. 251-314.

Résumé

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 $B$.

Détail

DOI : 10.2140/gt.2007.11.251

Keywords: signature, fibre bundle, multiplicative

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