Real equivariant bordism and stable transversality obstructions for $\mathbb{Z} /2$

In this paper we compute homotopical equivariant bordism for the group ${\mathbb{Z} }/2$, namely $MO_*^{{\mathbb{Z} /2}}$, geometric equivariant bordism ${\mathfrak{N}}^{{\mathbb{Z} /2}}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. We construct these rings from knowledge of their localizations.