On Euler characteristic and fundamental groups of compact manifolds

Bing-Long Chen Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China Xiaokui Yang Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Differential Geometry mathscidoc:2206.10006

Mathematische Annalen, 381, 1723–1743, 2020.9
Let M be a compact Riemannian manifold, π : \tilde{M} → M be the universal covering and ω be a smooth 2-form on M with π^∗ω cohomologous to zero. Suppose the fundamental group π_1(M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form η on \tilde{M} of linear (resp. bounded) growth such that π^∗ω = dη. As applications, we prove that on a compact Kahler manifold (M,ω) with π^∗ω cohomologous to zero, if π_1(M) is CAT(0) or automatic (resp. hyperbolic), then M is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic (−1)^{dim_RM/2χ(M) ≥ 0 (resp. >0).
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@inproceedings{bing-long2020on,
  title={On Euler characteristic and fundamental groups of compact manifolds},
  author={Bing-Long Chen, and Xiaokui Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622154552520091447},
  booktitle={Mathematische Annalen},
  volume={381},
  pages={1723–1743},
  year={2020},
}
Bing-Long Chen, and Xiaokui Yang. On Euler characteristic and fundamental groups of compact manifolds. 2020. Vol. 381. In Mathematische Annalen. pp.1723–1743. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622154552520091447.
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