On the signature of fibre bundles and absolute Whitehead torsion

• Main Author: Korzeniewski, Andrew John
• Published: University of Edinburgh 2005
• Subjects: 530.1
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.653547

In 1957 Chern, Hirzebruch and Serre proved that the signature of the total space of a fibration of manifolds is equal to the product of the signatures of the base space and the fibre space if the action of the fundamental group of the base space on the fibre is trivial. In the late 1960s Kodaira, Atiyah and Hirzebruch independently discovered examples of fibrations of manifolds with non-multiplicative signature. These examples are in the lowest possible dimension where the base and fibre spaces are both surfaces. W. Meyer investigated this phenomenon further and in 1973 proved that every multiple of four occurs as the signature of the total space of a fibration of manifolds with base and fibre both surfaces. Then in 1998 H. Endo showed that the simplest example of such a fibration with non-multiplicative signature occurs when the genus of the base space is 111. We will prove two results about the signature of fibrations of Poincaré spaces. Firstly we show that the signature is always multiplicative modulo four, extending joint work with A. Ranicki and I. Hambleton on the modulo four multiplicativity of the signature in a <i>PL</i>-manifold fibre bundle. Secondly we show that if the action of the fundamental group of the base space on the middle-dimensional homology of the fibre with coefficients in Z₂ is trivial, then the signature is multiplicative modulo eight. The main ingredient of the first result is the development of absolute Whitehead torsion; this is a refinement of the usual Whitehead torsion which takes values in the absolute group <i>K</i>₁(<i>R</i>) of a ring <i>R,</i> rather than the reduced group ?₁(<i>R</i>). When applied to the algebraic Poincaré complexes of Ranicki the “sign” term (the part which vanishes in ?₁(<i>R</i>)) will be identified with the signature modulo four. We prove a formula for the absolute Whitehead torsion of the total space of a fibration and a simple calculation yields the first result. The second result is proved by means of an equivalent Pontrjagin square, a refinement of the usual one. We make use of the Theorem of Morita which states that the signature modulo eight is equal to the Arf invariant of the Pontrjagin square. The Pontrjagin square of the total space of the bundles concerned is expressed in terms of the equivalent Pontrjagin square on the base space and this allows us to compute the Arf invariant.