(1) We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. (2) We show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
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机译：（1）我们提供了一个封闭的非球面黎曼流形的同构性大小的上限，它仅取决于心脏的收缩和球的体积。 （2）我们证明，mod p Betti数的线性增长或扭转同源性的指数增长意味着闭合的非球面流形是“大的”。
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