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Characteristic numbers, Jiang subgroup and non-positive curvature

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Abstract

By refining a result of Farrell and in terms of the Jiang subgroups of self-maps of smooth manifolds, we obtain a sufficient condition for the vanishing of the signature and the Hirzebruch’s \(\chi _y\)-genus. Along this line we show that the \(\chi _y\)-genus of a non-positively curved compact Kähler manifold vanishes when the center of its fundamental group is non-trivial, which realizes an expectation of Farrell’s in the Kähler setting. Moreover, in the latter case all the Chern numbers vanish whenever its complex dimension is no more than 4, which also provides some evidence towards a complex version’s Hopf Conjecture proposed by the author and Zheng.

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Acknowledgements

The author would like to express his sincere thanks to the referee for his/her very careful reading and useful suggestions, which significantly improve the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Ping Li

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Ping Li

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  1. Ping Li
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Correspondence to Ping Li.

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Dedicated to Professor Boju Jiang on the occasion of his 85th birthday.

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The author was partially supported by the National Natural Science Foundation of China (Grant No. 11722109) and the Fundamental Research Funds for the Central Universities.

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Li, P. Characteristic numbers, Jiang subgroup and non-positive curvature. Math. Z. 303, 1 (2023). https://doi.org/10.1007/s00209-022-03162-w

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