The rational abelianization of the Chillingworth subgroup of the mapping class group of a surface

Abstract

The Chillingworth subgroup of the mapping class group of a compact oriented surface of genus $g$ with one boundary component is defined as the subgroup whose elements preserve nonvanishing vector fields on the surface up to homotopy. In this work, we determine the rational abelianization of the Chillingworth subgroup as a full mapping class group module. The abelianization is given by the first Johnson homomorphism and the Casson--Morita homomorphism for the Chillingworth subgroup. Additionally, we compute the order of the Euler class of a certain central extension related to the Chillingworth subgroup and determine the kernel of the Casson--Morita homomorphism for the Chillingworth subgroup.