A Thom Spectrum Model for $C_2$-Integral Brown--Gitler Spectra

A Thom spectrum model for a $C_2$-equivariant analogue of integral Brown–Gitler spectra is established and shown to have a multiplicative property. The $C_2$-equivariant spectra constructed enjoy properties analogous to classical nonequivariant integral Brown–Gitler spectra and thus may prove useful for producing $C_2$-equivariant analogues of splittings of $BP⟨1⟩\wedge BP⟨1⟩$ and $bo\wedge bo\mathrm{.}$