A linear representation of the mapping class group and the theory of winding numbers

Abstract

This paper describes a linear representation F of the mapping class group , of an orientable surface S with one boundary component. The representation F extends the symplectic representation, and is defined for surfaces of arbitrary genus g> 1. The main tools used to define F are crossed homomorphisms which are defined using nonvanishing vector fields X on S, and the theory of winding numbers of curves on surfaces described by Chillingworth in [1,2]. These crossed homomorphisms were essentially described by Morita in [6]. A geometric interpretation of F is then given. If T1S denotes the unit tangent bundle of S1 then F records the action of on H1(T1S;Z). The kernel of F is then characterized using knowledge of the crossed homomorphisms ex. If matrix entries are taken modulo 2g-2, the representation F factors through the mapping class group of a closed orientable surface of genus g > 1. Thus F induces representations of Dn of for any n[-45 degree rule]2g-2. The Dn were discovered by Sipe in [7, 8], and it is noted that her characterization of the image of Dn carries over to the integer valued case. The structure found in characterizing ker F is then used to study ker Dn. In particular, it is shown that a uotient of ker Dn is a semidirect product for each even n dividing 2g-2.