Relative systoles of relative-essential 2–complexes

We prove a systolic inequality for a $ϕ$ –relative systole of a $ϕ$ –essential $2$ –complex $X$ , where $ϕ : π_{1} (X) \to G$ is a homomorphism to a finitely presented group $G$ . Thus, we show that universally for any $ϕ$ –essential Riemannian $2$ –complex $X$ , and any $G$ , the following inequality is satisfied: $sys {(X, ϕ)}^{2} \leq 8 Area (X)$ . Combining our results with a method of L Guth, we obtain new quantitative results for certain $3$ –manifolds: in particular for the Poincaré homology sphere $Σ$ , we have $sys {(Σ)}^{3} \leq 24 Vol (Σ)$ .

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Keywords

coarea formula, cohomology of cyclic groups, essential complex, Grushko's theorem, Poincaré duality, systole, systolic ratio

Mathematical Subject Classification 2000

Primary: 53C23, 57M20

Secondary: 57N65

References

Publication

Received: 27 October 2009

Revised: 12 July 2010

Accepted: 2 October 2010

Published: 7 January 2011

Authors

Karin Usadi Katz
Department of Mathematics Bar Ilan University 52900 Ramat Gan Israel

Mikhail G Katz
Department of Mathematics Bar Ilan University 52900 Ramat Gan Israel
http://u.cs.biu.ac.il/~katzmik/sgtdirectory/katz.html
Stéphane Sabourau
Laboratoire de Mathématiques et Physique Théorique Université de Tours Parc de Grandmont 37200 Tours France
http://www.lmpt.univ-tours.fr/~sabourau/
Steven Shnider
Department of Mathematics Bar Ilan University 52900 Ramat Gan Israel

Shmuel Weinberger
Department of Mathematics University of Chicago 5734 S University Avenue Chicago IL 60637-1514 United States
http://www.math.uchicago.edu/~shmuel/

Volume 11, issue 1 (2011)

Karin Usadi Katz, Mikhail G Katz, Stéphane Sabourau, Steven Shnider and Shmuel Weinberger

Abstract