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Topology of one-dimensional systoles. (Topologie des systoles unidimensionnelles.) (French) Zbl 1236.53037

From the text: “For a closed \(m\)-dimensional not simply connected Riemannian manifold \((M,g)\), we denote by \(\text{sys}_1(M,g)\) the length of the shortest non contractible closed geodesic. This value is called the systole or 1-systole of \(M\) with respect to the metric \(g\). The main direction of research in geometry and topology on this number is the study of the numerical invariant \[ \sigma(M)=\inf_g\frac{\text{Vol}(M,g)}{\text{sys}_1(M,g)^m}, \] called systolic constant of \(M\), where \(g\) is running over the set of all smooth Riemannian metrics on \(M\). We can summarize the research activities on this subject in the following two questions. (1) Which topological conditions must be posed on \(M\) in order that \(\sigma(M)>0\)? (2) Provided that \(\sigma(M)>0\), can \(\sigma(M)\) be computed?
An answer to the first question was given by M. Gromov [J. Differ. Geom. 18, 1–147 (1983; Zbl 0515.53037)]. The main objective of this paper is to compare the systolic constants of two manifolds \(M_1\), \(M_2\) of equal dimensions and isomorphic fundamental groups \(\pi_1(M_i)\), \(i=1,2\). The underlying question is, whether \(\sigma(M)\) is uniquely determined by \(\Phi_\ast\left([M]_\mathbb{Z}\right)\in H_m\left(K(\pi_1(M),1);\mathbb{Z}\right)\), where \([M]_\mathbb{Z}\) is the fundamental class of \(M\) with respect to \(\mathbb{Z}\)-coefficients and \(\Phi:M\to K(\pi_1(M),1)\) is the characteristic map to the Eilenberg-MacLane space \(K(\pi_1(M),1)\). We obtain a positive answer in two special cases; namely, let \(M_1,M_2\) be closed and orientable, \(\pi_1(M_i)\to\pi\) isomorphisms with some group \(\pi\), and \[ \Phi_i:M_i\to K(\pi,1),\;i=1,2 \] the characteristic maps. Then, the condition \(\Phi_{1,\ast}\left([M_1]\right)=\Phi_{2,\ast}\left([M_2]\right)=a\in H_m(\pi,\mathbb{Z})\) implies \[ \sigma(M_1)=\sigma(M_2) \] if (1) \(\pi\) is finitely generated and \(a\) is of finite order or (2) \(\pi\) is finite containing \(k\) elements and \(a\) is of order \(k\).”

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55P20 Eilenberg-Mac Lane spaces
58D17 Manifolds of metrics (especially Riemannian)

Citations:

Zbl 0515.53037
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