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Given a correspondence $X_{\mathrm{in}} \stackrel{i_{\mathrm{in}}}{\leftarrow} X \stackrel{i_{\mathrm{out}}}{\to} X_{out}$ of suitable "spaces" of sorts, and given a (generalized) cohomology theory $E$, one can ask if there are $E$-orientations such that the pull-push operation $(i_{\mathrm{out}})_! \circ (i_{in})^\ast$ exists as a map from the (twisted) $E$-cohomology of $X_{in}$ to the (twisted) $E$-cohomology of $E_{\mathrm{out}}$.

But moreover, under suitable ambient conditions, correspondences can be composed by pullback, and then one can ask if one can find compatible or consistent choices of orientations for all relevant correspondences such that this pull-push construction extends to a functor from "spaces" with correspondences between them to $E$-cohomology groups (or better: $E$-module spectra) and the natural maps between those.

A famous example where this has been done (for ordinary cohomology $E = H\mathbb{C}$) is the realization of string topology operations as a topological quantum field theory obtained by pull-pushing ordinary (co-)homology through the correspondences obtained by mapping 2-dimensional cobordisms into the given base space. (This is due to Cohen, Godin, Kupers, see the pointers on the nLab here).

Also Gromov-Witten theory constructed by integration against virtual fundamental classes could probably be listed here as an example.

Another example in genuinely generalized cohomology, where functorial pull-push has been claimed is the pull-push of (twisted) K-theory classes through suitable correspondences of moduli stacks of flat principal connections. This is the topic of the preprint

  • Daniel S. Freed, Michael J. Hopkins, Constantin Teleman, Consistent Orientation of Moduli Spaces (arXiv:0711.1909)

There a definition of what a "consistent" (functorial under composition of correspondences of moduli stacks) K-orientation should be is given and studied.

But the argument that this definition does what it is supposed to do is not given, it seems. On p. 19 it says that "The details of this argument will be given on another occasion".

So in conclusion I am aware of maybe 2.5 examples in the literature where pull-push in (generalized) cohomology has been turned into a functor by a suitable choice of globally compatible orientations.

My question therefore is: Has anyone thought more about this? Are there more known examples in the literature? What else is known?

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