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In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés globales des variétés differentiables that the cobordism groups could be computed as the homotopy groups of certain Thom spaces $MG$.

Let us denote Thom space of the vector bundle $E$ over the base space $B$ as $T(E)=Thom(B,E)$ following Wikipedia notation.

Since Thom class, the Stiefel–Whitney classes, and the Steenrod operations are all related.

Question: Do we have the following relations $$ H^n(MTG,\mathbb{Z}_2)=H^n(BG,\mathbb{Z}_2)U? $$ or $$ H^n(MG,\mathbb{Z}_2)=H^n(BG,\mathbb{Z}_2)U? $$ where $U$ is the Thom class with $Sq^iU=w_i U$? True or false?

How do we show this?

Attempt: The $MG$ and $MTG$ are two related Thom space, related by $$ MG=Thom(BG,V), \quad MTG=Thom(BG,-V), $$ with $V$ the vector bundle.

The Pontryagin-Thom isomorphism provides a relation between the bordism groups of manifolds with (stable) tangential structure and homotopy groups of the Madsen-Tillman spectrum $MTG$. The $MTG$ is a close cousin of the more usual Thom spectrum $MG$ associated to tangential structure $G$.

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