# The mapping class group orbits in the framings of compact surfaces

## TL;DR

This paper classifies the orbits of surface framings under the mapping class group, extending Johnson's work and introducing new invariants for genus 1 surfaces, with implications for the relative case.

## Contribution

It provides a detailed computation of mapping class group orbits in the homotopy set of framings, including new invariants for genus 1 surfaces and analysis of their behavior in relative cases.

## Key findings

- Computed mapping class group orbits for surfaces with boundary.
- Extended Johnson's results with modifications for genus > 1.
- Introduced an additional invariant for genus 1 cases.

## Abstract

We 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 $g > 1$ the computation is some modification of Johnson's results 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 studied for $g > 1$.