Twisted homology operations for $E_{\infty}$-algebras

Abstract

We develop a theory of operations on the twisted homology of $E_{\infty}$-algebras, generalizing a classical theory developed by J.P. May. First we describe a framework suitable for discussing twisted coefficients, which requires working with $E_{\infty}$-algebras in certain categories of functors. In this context, we define twisted versions of the classical Dyer--Lashof operations, as well as a product. Moreover, we prove that these distinguished operations generate all operations on twisted homology by giving a (non-canonical) explicit basis for the homology of free $E_{\infty}$-algebras in terms of these operations. We also make this statement functorial by proving that the homology of a free $E_{\infty}$-algebra is a free object in an appropriate category of objects equipped with an action of the Dyer--Lashof operations. This theory has applications to the study of $E_{\infty}$-spaces with local coefficients, though these are not discussed in detail here.