The mapping class group orbits in the framings of compact surfaces

Abstract

AbstractWe compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.