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Equivariant eta forms and equivariant differential K-theory

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Abstract

In this paper, for a compact Lie group action, we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equivariant case and introduce the equivariant version of the Dai-Zhang higher spectral flow for arbitrary-dimensional fibers. Using these results, we construct a new analytic model of the equivariant differential K-theory for compact manifolds when the group action has finite stabilizers only, which modifies the Bunke-Schick model of the differential K-theory. This model could also be regarded as an analytic model of the differential K-theory for compact orbifolds. Especially, we answer a question proposed by Bunke and Schick (2009) about the well-definedness of the push-forward map.

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Acknowledgements

This work was supported by Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 18dz2271000), Natural Science Foundation of Shanghai (Grant No. 20ZR1416700) and National Natural Science Foundation of China (Grant No. 11931007). The author thanks Professors Xiaonan Ma and Ulrich Bunke for helpful discussions. The author thanks Shu Shen and Guoyuan Chen for the conversations. The author also thanks the anonymous referees for various helpful suggestions that improved the clarity and quality of the paper.

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  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

    Bo Liu

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Liu, B. Equivariant eta forms and equivariant differential K-theory. Sci. China Math. 64, 2159–2206 (2021). https://doi.org/10.1007/s11425-020-1852-5

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