Twisted Homology Operations

Abstract: In the 70s, Fred Cohen and Peter May gave a description of the mod p homology of a free E_n-algebra in terms of certain homology operations, known as Dyer--Lashof operations, and the Browder bracket. These operations capture the failure of the E_n multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for E_{\infty}-algebras. I will also explain computational results that show the existence of new operations for E_2-algebras. Finally, I will discuss examples and applications of this theory.