Coproducts in brane topology

TL;DR

This paper generalizes the loop product and coproduct to mapping spaces from spheres and manifolds into certain spaces, establishing their algebraic properties such as associativity, commutativity, and Frobenius compatibility.

Contribution

It extends brane topology operations to higher-dimensional spheres and manifolds, demonstrating their algebraic structures and properties in this broader context.

Findings

Extended loop product and coproduct to $k$-manifolds

Proved associativity, commutativity, and Frobenius compatibility

Established the finite codimension embedding property in Gorenstein spaces

Abstract

We extend the loop product and the loop coproduct to the mapping space from the $k$-dimensional sphere, or more generally from any $k$-manifold, to a $k$-connected space with finite dimensional rational homotopy group, $k\geq 1$. The key to extending the loop coproduct is the fact that the embedding $M\rightarrow M^{S^{k-1}}$ is of "finite codimension" in a sense of Gorenstein spaces. Moreover, we prove the associativity, commutativity, and Frobenius compatibility of them.