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Steenrod originally defined his squares using explicit cochain-level formulas for simplicial mod-2 cochains. To this end, he introduced higher cup products, which control the failure of the usual cup product to be supercommutative on the cochain level. On the other hand, H. Cartan formula tells us how Steenrod squares relate to the usual cup product of mod-2 cohomology classes.

Questions.

(1) How does one show this using Steenrod's definition of Steenrod squares?

(2) Is there some cochain-level relation between cup products of various degrees which implies

Cartan's formula?

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  • $\begingroup$ This may be found in this paper titled "An effective proof of the Cartan formula: The even prime". $\endgroup$
    – Anibal Medina
    Commented May 10, 2021 at 8:08

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I think that what you are searching for is explained in P. May's paper "A general algebraic approach to Steenrod operations". In this paper you will find a very general treatment of cochain level cup-i product operations. In particular P. May introduces the concept of Cartan object and shows that the cohomology of Cartan object comes equipped with the action of Steenrod operations that satisfy the Cartan relations (see proposition 2.6). All the computations are made at the cochain level. The algebra of singular cochains of a topological space is a typical example of a Cartan object.

Moreover, we have an operadic interpretation:

  • take an algebra $A$ over an $E_{\infty}$-algebra, then you have evaluation products $$\mu_r:E_{\infty}(r)\otimes A^{\otimes r}\rightarrow A$$ where each $E_{\infty}(r)$ is a projective resolution of the symmetric group $\Sigma_r$.
  • The evaluation product $\mu_2$ is risponsible for the action of cup-i products.
  • If you take for $E_{\infty}$ the Baratt-Eclles operad $E$ it is a Hopf operad you have a coassiociative diagonal of operads $$\Delta:E\rightarrow E\otimes E.$$
  • Cartan relations come from the Hopf operad structure AND the fact that $E$ is the resolution of the commutative operad $Com$.
  • Cartan relations can be detected in the complex $E(4)$.
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  • $\begingroup$ I have looked at May's paper, but did not find a cochain-level proof of Cartan's formula. I did find there a reference to an interesting paper by G. Hirsch "Quelques proprietes des produits de Steenrod" (1955). Hirsch proves that for any $x$, $a\mapsto a\cup_1 x$ is a derivation of the algebra of mod-2 cochains, but he does not say anything about higher cup products. $\endgroup$
    – Anton Kapustin
    Commented Oct 13, 2016 at 23:19
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    $\begingroup$ There is a recasting in the $H_{\infty}$-ring spectra book in chapter 4 that may be helpful. Also, chapter 4 of Bruner's Adams spectral sequence primer may be helpful. $\endgroup$
    – Sean Tilson
    Commented Oct 14, 2016 at 12:29

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