Types of perfect matchings in toroidal square grids

Let $T_{m,n}$ be toroidal square grid of size $m\times n$ and let both $m$ and $n$ be even. Let $P$ be a perfect matching of $T_{m,n}$ and let $D(P)$ be the cycle-rooted spanning forest of $P$ obtained by the generalized Temperley's construction. The types of $P$ and $D(P)$ in the first homology group $H_1(\mathbb{T},\mathbb{Z})$ of torus $\mathbb{T}$ with coefficients in $\mathbb{Z}$ has been extensively studied. In this paper we study the types of $P$ and $D(P)$ in the first homology group $H_1(\mathbb{T},\mathbb{F}_2)$ with the coefficients in $\mathbb{F}_2$. Our considerations connect two remarkable results concerning perfect matchings of toroidal square grids, namely Temperley's bijection and the Arf-invariant formula.