The String Topology Loop Coproduct and Cohomology Operations

TL;DR

This paper investigates how cohomology operations interact with the string topology loop coproduct, focusing on their failure to commute and identifying cases where they preserve algebraic structures.

Contribution

It provides a description of the failure of cohomology operations to commute with the loop coproduct, especially for operations preserving sums and products.

Findings

Identifies conditions under which cohomology operations commute with the loop coproduct.

Analyzes examples like Steenrod squares and Adams operations.

Provides a framework for understanding the interaction between cohomology operations and string topology.

Abstract

This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory.