Stable systolic category of manifolds and the cup-length

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. Mathematics Subject Classification (2000)11R52 · 53C23 · 16K20 Congruence towers and the 4/3 boundHurwitz surfaces are an important and famous family of Riemann surfaces.We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. 1 A Hurwitz surface X by definition attains the upper bound of 84(g X − 1) for the order |Aut(X )| of the holomorphic automorphism group of X , where g X is its genus. In [25], we proved a systolic boundIn the literature, the term "Hurwitz quaternion order" has been used both in the sense used in the present text, and in the sense of the unique maximal order of Hamilton's rational quaternions (−1, −1) Q .

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“…Recent developments in systolic geometry include [1][2][3][4][5][7][8][9][10][11][12]15,16,18,17,20,21,23,24,26,28,29,[31][32][33][34]38]. …”

Section: Congruence Towers and The 4/3 Boundmentioning

confidence: 99%

Hurwitz quaternion order and arithmetic Riemann surfaces

2011

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“…Remark 2.1. Additional recent developments in systolic geomety include [1,3,4,5,6,7,8,9,10,13,14,15,16,19,22,23,25,26,28,30,31,33,34,35,36]. [17], but still using the Kuratowski imbedding in L ∞ , was recently developed by L. Ambrosio and the second-named author [1].…”

Section: Relative Systolesmentioning

confidence: 99%

“…Remark 2.1. Additional recent developments in systolic geomety include [1,3,4,5,6,7,8,9,10,13,14,15,16,19,22,23,25,26,28,30,31,33,34,35,36].…”

Section: Relative Systolesmentioning

confidence: 99%

Relative systoles of relative-essential 2–complexes

Katz

Sabourau

et al. 2011

Algebr. Geom. Topol.

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We prove a systolic inequality for a -relative systole of a -essential 2-complex X , where W 1 .X / ! G is a homomorphism to a finitely presented group G . Thus, we show that universally for any -essential Riemannian 2-complex X , and any G , the following inequality is satisfied: sys.X; / 2 Ä 8Area.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 †, we have sys. †/ 3 Ä 24Vol. †/.53C23, 57M20; 57N65 Relative systolesWe prove a systolic inequality for a -relative systole of a -essential 2-complex X , where W 1 .X / ! G is a homomorphism to a finitely presented group G . Thus, we show that universally for any -essential Riemannian 2-complex X , and any G , we have sys.X; / 2 Ä 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 †, we have sys. †/ 3 Ä 24 Vol. †/. To state the results more precisely, we need the following definition.Let X be a finite connected 2-complex. Let W 1 .X / ! G be a group homomorphism.Recall that induces a classifying map (defined up to homotopy) X ! K.G; 1/.Definition 1.1 The complex X is called -essential if the classifying map X ! K.G; 1/ cannot be homotoped into the 1-skeleton of K.G; 1/. Definition 1.2 Given a piecewise smooth Riemannian metric on X , the -relative systole of X , denoted sys.X; /, is the least length of a loop of X whose free homotopy class is mapped by to a nontrivial class. When is the identity homomorphism of the fundamental group, the relative systole is simply called the systole, and denoted sys.X /.Definition 1.3 The -systolic area .X / of X is defined as.X /;where the infimum is over all -essential piecewise Riemannian finite connected 2-complexes X , and homomorphisms with values in G .In the present text, we prove a systolic inequality for the -relative systole of aessential 2-complex X . More precisely, in the spirit of Guth's text [18], we prove a stronger, local version of such an inequality, for almost extremal complexes with minimal first Betti number. Namely, if X has a minimal first Betti number among all -essential piecewise Riemannian 2-complexes satisfying .X / Ä .G/ C " for an " > 0, then the area of a suitable disk of X is comparable to the area of a Euclidean disk of the same radius, in the sense of the following result.Theorem 1.4 Let " > 0. Suppose X has a minimal first Betti number among allessential piecewise Riemannian 2-complexes satisfying .X / Ä .G/ C ". Then each ball centered at a point x on a -systolic loop in X satisfies the area lower boundwhenever r satisfies " 1=3 Ä r Ä 1 2 sys.X; /.A more detailed statement appears in Proposition 8.2. The theorem immediately implies the following systolic inequality.Corollary 1.5 Every finitely presented group G satisfiesso that every piecewise Riemannian -essential 2-complex X satisfies the inequalityAlgebraic & Geometric Topology, Volume 11 (2011) Relative systoles of relativ...

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“…Note that the finite-dimensional approximation is used in an analytic proof of Gromov's systolic inequality by Ambrosio and the second-named author in [1]. Recent publications in systolic geometry include the articles [2][3][4][5][6][7][8][9]13,15,16,18,19,21,22,27,28].…”

Section: Theorem 11 Let M Be a Compact Riemannian Manifold Without Bmentioning

confidence: 99%

Bi-Lipschitz approximation by finite-dimensional imbeddings

Katz

2010

Geom Dedicata

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Gromov's celebrated systolic inequality from '83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a (1 + )-bi-Lipschitz (onto its image), finite-dimensional imbedding for every > 0, using the first variation formula and the mean value theorem. Mathematics Subject Classification (2000)Primary 53C23 · Secondary 26E35 Metric imbeddings and Gromov's theoremIn '83, Gromov proved the systolic inequality for essential manifolds M. Namely, consider the least length (systole, denoted "sys") of a non-contractible loop in a closed Riemannian manifold M. Then the systole is bounded above in terms of the volume of M, if M satisfies the topological hypothesis of being essential (for instance, if π i (M) = 0 for i ≥ 2). See Guth [14] for a helpful overview.A key technique in Gromov's seminal text [11] is the Kuratowski imbedding. Thus, Gromov imbeds a Riemannian manifold M into the space

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Stable systolic category of manifolds and the cup-length

Cited by 7 publications

References 14 publications

Hurwitz quaternion order and arithmetic Riemann surfaces

Hurwitz quaternion order and arithmetic Riemann surfaces

Relative systoles of relative-essential 2–complexes

Bi-Lipschitz approximation by finite-dimensional imbeddings

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