Steenrod operations in \(shc\)-algebras
Let \(\pi\) be a cyclic group of order a fixed prime \(p\) and \(W\) be the standard free resolution of \(\mathbb{F}_p\) as a \(\mathbb{F}_p[\pi]\)-module. Generalizing May's work, a differential graded algebra \(A\) is called a \(\pi\)-\(shc\)-algebra if \(A\) is an \(shc\) algebra with a morphism of differential graded \(\pi\)-algebras \[ W\otimes\Omega B (A^{\otimes p})\to A, \] whose restriction to \(\Omega\otimes \Omega B (A^{\otimes p})\) consists into the \(shc\)-multiplication \(\Omega B(A^{\otimes p})\to \Omega B A\) followed by the augmentation. An example is given by the singular cochains \(N^*X\) of a 1-connected space with coefficients in \(\mathbb{F}_p\). The authors prove that if \(A\) is a \(\pi\)-\(shc\)-algebra, then the Hochschild homology \(HH_*A\) admits Steenrod operations. They also show that the Jones isomorphism \[ HH_* N^*X\cong H^* (X^{S^1}; \mathbb{F}_p) \] preserves Steenrod operations.
scientific article; zbMATH DE number 2093431