String topology for stacks

Understand

We establish the general machinery of string topology for differentiable stacks.

• This machinery allows us to treat on an equal footing free loops in stacks and hidden loops.
• In particular, we give a good notion of a free loop stack, and of a mapping stack $\map(Y,\XX)$, where $Y$ is a compact space and $\XX$ a topological stack, which is functorial both in $\XX$ and $Y$ and behaves well enough with respect to pushouts.
• We also construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products.