On Euler characteristic and fundamental groups of compact manifolds
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Mathematische Annalen
On Euler characteristic and fundamental groups of compact manifolds Bing-Long Chen1 · Xiaokui Yang2 Received: 16 April 2020 / Revised: 23 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract → M be the universal covering and Let M be a compact Riemannian manifold, π : M ∗ ω be a smooth 2-form on M with π ω cohomologous to zero. Suppose the fundamental group π1 (M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, of linear (resp. bounded) growth we show that there exists a smooth 1-form η on M ∗ such that π ω = dη. As applications, we prove that on a compact Kähler manifold (M, ω) with π ∗ ω cohomologous to zero, if π1 (M) is CAT(0) or automatic (resp. hyperbolic), then M is Kähler non-elliptic (resp. Kähler hyperbolic) and the Euler characteristic (−1)
dimR M 2
χ (M) ≥ 0 (resp. > 0).
Contents 1 Introduction . . . . . . . . . . . . . . . 2 Hyperbolic fundamental groups . . . . . 3 General fundamental groups . . . . . . . 4 Quadratic radial isoperimetric inequality References . . . . . . . . . . . . . . . . . .
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Communicated by Ngaiming Mok.
B
Xiaokui Yang [email protected] Bing-Long Chen [email protected]
1
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
2
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
123
B. Chen, X. Yang
1 Introduction In differential geometry, there is a well-known conjecture due to H. Hopf (e.g. [32, Problem 10]): Conjecture 1.1 (Hopf) Let M be a compact, oriented and even dimensional Riemannian manifold of negative sectional curvature K < 0. Then the signed Euler n characteristic (−1) 2 χ (M) > 0, where n is the real dimension of M. For n = 4, Conjecture 1.1 was proven by Chern [8] (who attributed it to Milnor). Not much has been known in higher dimensions. This conjecture can not be established just by use of the Gauss–Bonnet–Chern formula (see [13,22]). Singer suggested that in view of the L 2 -index theorem an appropriate vanishing theorem for L 2 -harmonic forms on the universal covering of M would imply the conjecture (e.g. [10]). In the work [18], Gromov introduced the notion of Kähler hyperbolicity for Kähler manifolds which means the Kähler form on the universal cover is the exterior differential of some bounded 1-form. He established that the L 2 -cohomology groups of the universal covering of a Kähler hyperbolic manifold are not vanishing only in the middle dimension. Combining this result and the covering index theorem of Atiyah, Gromov n showed (−1) 2 χ (M) > 0 for a Kähler hyperbolic manifold M. One can also show that a compact Kähler manifol
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