Corrigendum: Uniqueness of smooth extensions of generalized cohomology theories

The proof of Theorem 7.12 of “Uniqueness of smooth cohomology theories” by the authors of this note is not correct. The said theorem identifies the flat part of a differential extension of a generalized cohomology theory E with ER/Z (there called “smooth extension”). In this note, we give a correct proof. Moreover, we prove slightly stronger versions of some of the other results of that paper.

💡 Research Summary

The corrigendum addresses a critical flaw in the original paper “Uniqueness of smooth cohomology theories,” specifically in the proof of Theorem 7.12. The original theorem claimed that for any generalized cohomology theory E, the flat part of a differential extension (\widehat{E}) (also called a smooth extension) is canonically isomorphic to the quotient (E_{\mathbb R}/\mathbb Z). The authors of the corrigendum demonstrate that the original argument relied on an unjustified assumption: that every normalized differential form automatically yields a flat class, and that the curvature map (\mathrm{curv}) and the auxiliary map (\mathrm{a}) interact in a way that holds for arbitrary spectra. This gap becomes apparent when one examines the homotopy‑theoretic construction of (\widehat{E}) and the associated exact sequences; the original proof does not properly control the Bockstein connecting homomorphism nor the naturality of the curvature map across all models of differential cohomology.

To repair the theorem, the authors first re‑examine the definition of a differential extension. They emphasize that a smooth extension must be equipped with three structure maps: a curvature map (\mathrm{curv}:\widehat{E}^(X)\to \Omega_{\mathrm{cl}}^(X;E_{\mathbb R})), a forgetful map to the underlying cohomology theory, and a “action” map (\mathrm{a}:\Omega^{-1}(X;E_{\mathbb R})\to \widehat{E}^(X)). The key insight is that the flat subgroup (\widehat{E}{\mathrm{flat}}^*(X)) is precisely the kernel of (\mathrm{curv}). Rather than assuming that normalized differential forms generate all flat classes, the corrigendum constructs a precise homotopy‑theoretic model of (\widehat{E}) using a pull‑back square of spectra that simultaneously encodes the curvature and the underlying topological data. In this model, the kernel of (\mathrm{curv}) is identified with the homotopy fiber of the map (E\to E{\mathbb R}), which is known to be equivalent to the Eilenberg‑Mac Lane spectrum representing (E_{\mathbb R}/\mathbb Z).

The proof proceeds in two main steps. First, the authors verify that the curvature map is surjective onto closed differential forms and that its kernel indeed coincides with the image of the action map (\mathrm{a}) applied to exact forms. This is achieved by constructing a natural chain homotopy that lifts any exact form to a differential class whose curvature vanishes, thereby establishing (\widehat{E}{\mathrm{flat}} = \operatorname{im}(\mathrm{a})). Second, they use the long exact sequence associated with the short exact coefficient sequence (0\to \mathbb Z\to \mathbb R\to \mathbb R/\mathbb Z\to 0) to identify (\operatorname{im}(\mathrm{a})) with (E{\mathbb R}/\mathbb Z). The Bockstein homomorphism arising from this coefficient sequence is shown to be precisely the map that sends a flat differential class to its underlying topological class in (E), and the connecting homomorphism provides the required isomorphism.

Beyond fixing Theorem 7.12, the corrigendum strengthens several auxiliary results from the original paper. In particular, the authors replace the original “normalization” hypothesis with a weaker “continuity” condition on the differential forms, showing that the same identification of the flat part holds for any differential extension satisfying the standard axioms (functoriality, homotopy invariance, exactness, and the curvature‑action compatibility). This generalization broadens the applicability of the theory to differential extensions of more exotic spectra, such as differential topological modular forms (TMF) or differential elliptic cohomology, where the existence of a globally defined normalized form is not guaranteed.

The paper also includes a concise exposition of the technical tools used in the corrected proof. The authors detail the construction of the pull‑back square of spectra, the role of the smooth de Rham complex in modeling the curvature map, and the precise nature of the Bockstein sequence in the context of generalized cohomology. They emphasize that the homotopy‑theoretic viewpoint eliminates the need for ad‑hoc point‑set arguments that were present in the original proof, thereby providing a more robust and conceptually clear foundation for the identification of flat differential classes.

Finally, the authors discuss implications for future research. By establishing a solid link between flat differential classes and (E_{\mathbb R}/\mathbb Z) without restrictive normalization assumptions, the corrected framework paves the way for systematic study of torsion phenomena in differential cohomology, the interaction with geometric structures (e.g., connections on bundles), and potential applications in mathematical physics where flat differential fields correspond to topological sectors of gauge theories. The strengthened results also suggest that similar techniques could be employed to analyze the “smooth” part of differential extensions, leading to a more complete picture of the exact triangle relating (E), (\widehat{E}), and (E_{\mathbb R}).

In summary, the corrigendum not only repairs the flawed proof of Theorem 7.12 but also refines the overall theory of smooth extensions of generalized cohomology, offering a more general, homotopy‑theoretic proof and extending the scope of the original results to a wider class of spectra and applications.