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Relative systoles of relative-essential 2-complexes. (English) Zbl 1228.53056

Let \(X\) be a finite connected 2-complex, and \(\phi\) be a homomorphism from \(\pi_1X\) to some finitely presented group \(G\). We say that \(X\) is called \(\phi\)-essential if the classifying map \(X\to K(G,1)\) cannot be homotoped into the 1-skeleton. Given a piecewise smooth Riemannian metric on \(X\), the \(\phi\)-relative systole of \(X\) is the least length of a loop in \(X\) whose free homotopy class is mapped by \(\phi\) to a nontrivial class. When \(\phi\) is the identity, one recovers the usual notion of (absolute) systole.
The authors prove an inequality for relative systoles. The main consequence for absolute systoles is the following result:
Corollary 1.9 If \(M\) is a closed Riemannian 3-manifold whose fundamental group is finite of even order, then the systole of \(M\) is no greater than 24 times its volume.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57N65 Algebraic topology of manifolds
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