In her article *Higher string topology operations*, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes d}) \otimes H_\ast(LM)^{\otimes n} \to H_\ast(LM)^{\otimes m}$, where $Mod(S)$ is the mapping class group $\pi_0(Diff^+(S;\partial S))$.

As an example she gives claims that the generator of $H_1(BMod(Cylinder))$ (the twisting is trivial here) gives the BV-operator, where of course the generator of $H_1(BMod(Cylinder))$ corresponds to the generator of $Mod(Cylinder)$ corresponding to the Dehn twist around one of the boundary components. My first question is: how does it follow from Godin's construction that this generator acts as the BV-operator?

In general, the mapping class group of a surface with boundary is generated by a finite number of Dehn twists (e.g. *A primer on mapping class groups* v.4.02, page 131). My second question is: Do all of these have an action in string topology similar to a BV-operator?

Finally, in many cases (e.g. the pair of pants) $BMod(S)$ is an H-space, being the classifying space of an abelian group (thanks to Chris Schommer-Pries for pointing out a mistake here originally). This means there is an induced product in homology, and also a shifted product on twisted homology if the twistings is trivial (section 4.5 of Godin's article tells us this is the case is at most one boundary component is completely free). My third question is: how does this product interact with the string topology operations?

This post imported from StackExchange MathOverflow at 2014-09-20 22:43 (UCT), posted by SE-user skupers