On the bialgebra structure of the free loop homology

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.