Volume 11, issue 1 (2011)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Relative systoles of relative-essential $2$–complexes

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

Algebraic & Geometric Topology 11 (2011) 197–217
Abstract

We prove a systolic inequality for a $\varphi$–relative systole of a $\varphi$–essential $2$–complex $X$, where $\varphi :{\pi }_{1}\left(X\right)\to G$ is a homomorphism to a finitely presented group $G$. Thus, we show that universally for any $\varphi$–essential Riemannian $2$–complex $X$, and any $G$, the following inequality is satisfied: $sys{\left(X,\varphi \right)}^{2}\le 8Area\left(X\right)$. 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 $\Sigma$, we have $sys{\left(\Sigma \right)}^{3}\le 24Vol\left(\Sigma \right)$.

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
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/