Signature cocycles on the mapping class group and symplectic groups

Abstract

Werner Meyer constructed a cocycle in H 2 (Sp(2g, Z); Z) which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 7.2, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N . Using these results, we are able to give a complete answer for N = 2, 4, and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.2010 Mathematics Subject Classification. 20J06 (primary) ; 55R10, 20C33 (secondary).

Mapping class groups of highly connected $$(4k+2)$$-manifolds

“…The extensions (1) and (2) and their variants (3) and (4) have been studied by various authors before, and some special cases of our results were already known:…”

Heisenberg homology on surface configurations

“…Taking the limit 0 ← in the composition of the isomorphisms from equations eqs. (7) to (11), we obtain…”