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I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ spectra. Hu & Kriz described this for $p=2$ about 20 years ago, and for $p$ an odd prime just a couple months ago, in https://arxiv.org/abs/2205.13427 together with Somberg & Zou.

My question is: has it taken this long because of computational limitations, or because people have not needed it so far? In the non-equivariant context, I know of two applications of the Steenrod algebra: showing two spaces are non-equivalent by computing the Steenrod action on their cohomology rings, and the Adams spectral sequence. Are there any other applications I'm not aware of, particularly ones that could be worth developing in the genuine equivariant context? I guess the question applies more generally to figuring out equivariant cohomology operations for genuine $G$-spectra.

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    $\begingroup$ Thom used to the ordinary steenrod algebra to compute the bordism ring. Perhaps an equivariant version? $\endgroup$
    – Thomas Rot
    Commented Sep 6, 2022 at 18:30
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    $\begingroup$ @ThomasRot I went to a talk just today about this given by Jack Carlisle. Neither in his talk about the unoriented case or his paper about the complex case did he reference this, surprisingly. Though it was asked if the equivariant spectrum representing unoriented G-bordism was a wedge of Eilenberg-MacLane spectra, and it seems like it is not known, so there is a potential the equivariant Steenrod algebra comes up in that question. $\endgroup$
    – Connor Malin
    Commented Sep 7, 2022 at 1:59

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