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Zbl 1023.53025
Babenko, Ivan K.
Strong intersystolic freedom of closed manifolds and of polyhedrons. (Forte souplesse intersystolique de varieétés fermées et de polyèdres.) (French)
[J] Ann. Inst. Fourier 52, No.4, 1259-1284 (2002). ISSN 0373-0956; ISSN 1777-5310/e

In the 70's M. Berger initiated the study of a new Riemannian invariant, called ``systole'', $\text{sys}_{1}( g)$ defined as the minimum of the length of closed curves not homologous to zero. This definition has been generalized to many cases, for example to the $k$-systole, $\text{sys}_{k}(g)$, defined via areas of nontrivial cycles represented by maps of $k$-dimensional manifolds into a Riemannian manifold $( M,g) $. The important geometric meaning is played for a given $n$-dimensional Riemannian manifold by $$ \inf_{g}\frac{\text{vol}( g) }{\text{sys}_{k}( g) \cdot \text{sys}_{n-k}( g) }. $$ If this limit is $0$, the manifold is called systolically $( k,n-k)$-free. In his previous paper [Russ. Math. Surv. 55, 987-988 (2000; Zbl 1004.53031)] the author introduced the notions of homological systole $\text{sys}_{k}$ and $\text{stsys}_{k}$ via the volume of the $k$-dimensional class of homology with integer coefficients. The main goal is to prove that $$ \inf_{g}\frac{\text{vol}( g) }{\text{sys}_{k}( g) \cdot \text{sts}_{n-k}( g) }=0.$$
[Jan Kubarski (Łodz)]

MSC 2000:: *53C23 Global topological methods (a la Gromov)
58A25 Currents (global analysis)
57Q99 PL-topology

Keywords: Riemannian manifold; systole; systolically $(k,n-k)$-free; homological systole

Citations: Zbl 1004.53031

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