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Uniqueness of differential characters and differential K-theory via homological algebra

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Abstract

Simons and Sullivan constructed a model of differential K-theory, and showed that the differential K-theory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential K-theory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential K-theory groups are uniquely determined by the Simons–Sullivan diagram up to an isomorphism compatible with the diagonal arrows of the hexagon diagram. We state a necessary and sufficient condition for an affirmative answer to the full question. This approach further yields an alternative proof of a weaker version of Simons and Sullivan’s results concerning axiomatization of differential characters. We further obtain a uniqueness result for generalised differential cohomology groups. The proofs here are based on a recent work of Pawar.

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Notes

  1. Our convention of degrees differs from the one in original paper [9] where this group is called \({\hat{H}}^{k-1}(M;{\mathbb {R}}/{\mathbb {Z}})\).

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Acknowledgements

I wish to thank my supervisor Dr. Rishikesh Vaidya for discussions and support. I am thankful to Jitendra Rathore for valuable discussions. Prof. Dennis Sullivan kindly shared his feedback and comments on the pre-print version of this article arXiv:2005.02056, I feel thankful to him. I wish to profusely thank an anonymous expert for several comments which helped improve the results herein, especially for pointing out that the method herein applies to the case of generalised differential cohomology theories as well. An anonymous reviewer gave valuable suggestions which helped me improve the presentation, and pointed out some typos. Part of this work was carried out while I was financially supported by the Council of Scientific and Industrial Research under the CSIR-SRF(NET) scheme.

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  1. Department of Physics, Birla Institute of Technology and Science, Pilani, Pilani, Rajasthan, 333031, India

    Ishan Mata

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  1. Ishan Mata
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Mata, I. Uniqueness of differential characters and differential K-theory via homological algebra. J. Homotopy Relat. Struct. 16, 225–243 (2021). https://doi.org/10.1007/s40062-021-00278-4

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