On the bialgebra structure of the free loop homology

TL;DR

This paper introduces a new algebraic structure on free loop homology, connecting it with intersection and string topology, and defines loop bialgebras for differential graded coalgebras.

Contribution

It develops a novel bialgebra framework for free loop homology and relates it to existing topological products and coproducts.

Findings

Defined a commutative product on homology of special cubical sets.

Lifted this product to free loop homology of geometric realizations.

Calculated the loop bialgebra structure for specific spaces.

Abstract

We introduce a commutative product of degree $-n$ on the homology $H_\ast(X)$ of an $n$-dimensional special cubical set $X$ and lift it on the free loop homology $H_\ast(\Lambda M)$ for $M=|X|$ to be the geometric realization. These products agree with the intersection and string topology products respectively when $M$ is an oriented closed manifold, and we establish the compatibility relation between the string topology product and the standard coproduct on $H_\ast(\Lambda M).$ Motivated by the above relationship we introduce the notion of loop bialgebra for differential graded coalgebras $C$ by means of the coHochschild complex $\Lambda C.$ We calculate the loop bialgebra structure for some spaces.