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CM 1 113

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

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