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Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism classes of line bundles via the Chern class, and a connection on such a line bundle thus provides a refinement of the Chern class.

I am looking for a reference to a similar refinement of ordinary homology, which is in some sense Poincare dual to differential cohomology. Is this discussed in the literature?

My motivation comes from the fact that the Chern class of a line bundle $L$ is Poincare dual to the zero locus of a generic section of $L$. Thus, on the level of ordinary cohomology/homology there exist a nice duality phrased in the language of line bundles.

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Differential cohomology groups are computed as homotopy groups $\hat{\rm H}^n(M)=π_0(F_n(M))$, where $F_n\colon{\sf Man}^{\rm op}\to{\sf Sp}$ is a sheaf of spectra on the site of smooth manifolds. (See Bunke–Nikolaus–Völkl's Differential cohomology theories as sheaves of spectra for more details.)

Thus, Verdier duality (§5.5.5 in Lurie's Higher Algebra) suggests a very natural way to convert a differential cohomology theory into a differential homology theory: start with a sheaf of spectra $F_n$, convert it into a cosheaf of spectra $G_n$, then set $\hat{\rm H}_n(M):=π_0(G_n(M))$. This yields a covariant functor that can be naturally called a differential homology theory.

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  • $\begingroup$ This looks promising. Thanks! Do you know if the resulting homology theory satisfies similar exact sequences as the one defining differential cohomology? Is there a more "hands-on" description of the homology theory in terms of geometric data (e.g. sections of bundles)? $\endgroup$
    – Tobias Diez
    Commented Nov 23, 2020 at 0:06
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    $\begingroup$ @TobiasDiez: Yes, you get a similar hexagon diagram, by using arguments similar to the ones in Section 3 of Bunke–Nikolaus–Völkl. From there, you can also extract a hands-on description, which will basically amount to looking at compactly supported sections (e.g., compactly supported differential forms), so in a sense, we get a differential analogue of the Borel–Moore homology. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 23, 2020 at 1:27
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Check out the theory of smooth "currents", which intertwines the exterior derivative (on cohomology) to the boundary operator (on homology). For example, integration over a smooth surface submanifold $\Sigma$ (with possible boundary) defines a functional on 2-forms $\omega\mapsto\int_\Sigma\omega$ and its boundary defines a functional on 1-forms via Stokes' theorem.

Georges de Rham has a book on this, and I think it forms a homology theory.

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