$\mathbb{K}$-framings and $\mathbb{K}$-quadratic forms on surfaces

Abstract

We introduce the notions of $\mathbb{K}$-framings, based $\mathbb{K}$-framings and relative $\mathbb{K}$-framings of a compact connected oriented surface $\Sigma$ for any commutative ring $\mathbb{K}$ with unit, and a map which maps a based loop on $\Sigma$ to a homology class of its unit tangent bundle $U\Sigma$, which recovers Johnson's lifting in the case $\mathbb{K} = \mathbb{Z}/2$. This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring $\mathbb{K}$ with unit. If the genus of $\Sigma$ is positive, we have a bijection between the set of $\mathbb{K}$-framings and the set of some twisted cocycles of the mapping class group of the surface $\Sigma$. Through this bijection, in the case where the boundary $\partial\Sigma$ is non-empty and connected, we discuss some relation between $\mathbb{K}$-framings and the extended first Johnson homomorphism.