The signature of a fibre bundle is multiplicative mod 4

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 in Korzeniewski [11]. 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 $B$.

v2: Improved proof of Proposition 9.3

References in corpus (1)

• Absolute Whitehead torsion

Cited by in corpus (5)

• Cohomology of symplectic groups and Meyer's signature theorem
• The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant
• The family signature theorem
• Absolute Whitehead torsion
• Homological congruence formulae for characteristic classes of singular varieties