Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism

The Pontryagin-Thom theorem gives an isomorphism between the cobordism group of framed $n$-dimensional manifolds, $\omega_n$, and the $n^{th}$ stable homotopy group of the sphere spectrum, $\pi_n(\mathbb{S})$. The equivariant analogue of this theorem, gives an isomorphism between the equivariant cobordism group of $V$-framed $G$-manifolds, $\omega_V^G$, and the $V^{th}$ equivariant stable homotopy group of the $G$-sphere spectrum, $\pi_V^G(\mathbb{S})$, for a finite group $G$ and a $G$-representation, $V$. In this paper, we explicitly identify the images of each element of $\omega_1^{C_2}$ and $\omega_\sigma^{C_2}$ in $\pi_1^{C_2}(\mathbb{S})$ and $\pi_\sigma^{C_2}(\mathbb{S})$ under the equivariant Pontryagin-Thom isomorphism.