In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map of compact manifolds $f : X \to Y$ by post-composition with $f!$ in KK-theory (and thus generalizes the construction to KK-classes). Later Connes with Skandalis, and then Hilsum and Skandalis generalized this to KK-theory witnesses for pushforward along maps of foliations, hence for (the groupoid convolution algebras of) their homotopy Lie groupoids. More recently Rouse and Wang further generalized this to the twisted case. These references that I am aware of are listed here.

This work suggests an evident question:

To which extent can we make sense of K-orientability of functors $f \colon (\mathcal{X},\phi_1) \to (\mathcal{Y}, \phi_2)$ between differentiable stacks equipped with twisting circle 2-bundles/bundle gerbes $\phi_i$, such that there are such push-forward KK-witnesses

$f ! \in KK ( C^\ast(\mathcal{X}, \phi_1), C^\ast(\mathcal{Y}, \phi_2) )$

in the KK-classes between the corresponding twisted groupoid convolution algebras?

Is there any work in this direction, beyond the above references?

For instance with such a more general formulation it would be interesting to see if given a map between Lie groupoids with twists which is not K-orientable, that there is a weakly equivalent replacement with other twists that is, thus accounting for twist-change along non-K-orientable maps.

I am aware of the work of Brodzki, Mathai, Rosenberg, Szabo (also listed behind the above link) which gives a general but purely algebraic formulation. So maybe alternatively I am asking if there is any work on how to detect on maps of twisted differentiable stacks the corresponding algebraic conditions under the operation of forming twisted groupoid convolution algebras.


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