An effective proof of the Cartan formula: the even prime (Q2220194)

The author gives an effective proof of the Cartan formula for the action of the Steenrod squares \(Sq^i\) on mod \(2\) cohomology, working at the cochain level. He first provides a suitable universal construction for algebras over \(\mathcal{E}\), the Barratt-Eccles operad at the prime \(2\). This is a chain homotopy \(H : \mathcal{E} (2) \rightarrow \mathcal{E} (4)\) that gives rise to the required Cartan \(i\)-coboundaries. This result is in the spirit of May's general algebraic approach to Steenrod operations [\textit{J. P. May}, Lect. Notes Math. 168, 153--231 (1970; Zbl 0242.55023)]. He then applies this to the classical case, the normalized cochain complex \(N^* (X)\) with \(\mathbb{F}_2\) coefficients of a simplicial set \(X\). This uses the operad morphisms \[ \mathcal{E} \rightarrow \mathrm{Surj} \rightarrow \mathrm{End} (N^* (X)), \] where \(\mathrm{Surj}\) is the operad of surjections, based upon [\textit{C. Berger} and \textit{B. Fresse}, Math. Proc. Camb. Philos. Soc. 137, No. 1, 135--174 (2004; Zbl 1056.55006); \textit{J. E. McClure} and \textit{J. H. Smith}, J. Am. Math. Soc. 16, No. 3, 681--704 (2003; Zbl 1014.18005)].