# Mod 2 power operations revisited

@article{Wilson2019Mod2P, title={Mod 2 power operations revisited}, author={Dylan Wilson}, journal={arXiv: Algebraic Topology}, year={2019} }

In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $\mathbb{E}_{\infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.

## One Citation

### Integral models for spaces via the higher Frobenius

- Mathematics
- 2019

We give a fully faithful integral model for spaces in terms of $\mathbb{E}_{\infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent…

## References

SHOWING 1-10 OF 19 REFERENCES

### Homology operations and power series

- MathematicsGlasgow Mathematical Journal
- 1983

Bullett and Macdonald [1] have used power series to simplify the statement and proof of the Adem relations for Steenrod cohomology operations. In this paper I give a similar treatment of May's…

### Power operations and coactions in highly commutative homology theories

- Mathematics
- 2013

Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p,…

### Reduced Powers of Cohomology Classes

- Mathematics
- 1952

We shall present a set of new operations which interrelate the elements of the various dimensional cohomology groups of a space. They are topologically invariant, and provide sharper methods for…

### Representing Tate cohomology of G-spaces

- MathematicsProceedings of the Edinburgh Mathematical Society
- 1987

Tate cohomology of finite groups [5] is very good at emphasising periodic cohomological behaviour and hence at the study of free actions on spheres [8]. Tate cohomology of spaces was introduced by…

### Q-rings and the homology of the symmetric groups

- Mathematics

ly this defines a symmetric tensor product on the category of left Kmodules. It follows that K is a cocommutative bialgebra. More precisely, the comultiplication δ : K → K ⊗K can be defined as the…

### Stable power operations

- Mathematics
- 2020

For any $E_\infty$ ring spectrum $E$, we show that there is an algebra $\mathrm{Pow}(E)$ of stable power operations that acts naturally on the underlying spectrum of any $E$-algebra. Further, we show…

### Rational and p-adic homotopy theory

- Mathematics
- 2016

These are notes for a course taught at the Max-Planck Institute for Mathematics in Bonn in Summer 2016. They are in extremely preliminary form. Comments, corrections and remarks are welcome.

### Topological cyclic homology

- MathematicsHandbook of Homotopy Theory
- 2020

Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace…

### Stable $\infty$-Operads and the multiplicative Yoneda lemma

- Mathematics
- 2016

We construct for every $\infty$-operad $\mathcal{O}^\otimes$ with certain finite limits new $\infty$-operads of spectrum objects and of commutative group objects in $\mathcal{O}$. We show that these…