## The loop homology algebra of discrete torsion

Let $M$ be a closed oriented manifold with a finite group action by $G$.

We denote its Borel construction by $M_{G}$. As an extension of string

topology due to Chas-Sullivan, Lupercio-Uribe-Xicot$\’{e}$ncatl

constructed a graded commutative associative product (loop product) on $

H_{*}(LM_{G})$, which plays a significant role in the “orbifold string

topology” . They also showed that the constructed loop product is an

orbifold invariant. In this talk, we describe the orbifold loop product

by determining its "twisting" out of the ordinary loop product in term

of the group cohomology of $G$, when the action is homotopically trivial.

Through this description, the orbifold loop homology algebra can be

seen as R. Kauffmann's “algebra of discrete torsion”, which is a group

quotient object of Frobenius algebra. As a cororally, we see that the

orbifold loop product is a non-trivial orbifold invariant.

By: Yasuhiko Asao

By: Prof. Demetrios CHRISTODOULOU - ETH-Zentrum, Zürich - CH

By: Prof. Demetrios CHRISTODOULOU - ETH-Zentrum, Zürich - CH