Topic
Systolic category
About: Systolic category is a research topic. Over the lifetime, 13 publications have been published within this topic receiving 186 citations.
Papers
TL;DR: In this article, a Riemannian analogue of the Lusternik-Schnirelmann category, called the systolic category of M, is introduced, denoted catsys(M) and defined in terms of the existence of Systolic inequalities satisfied by every metric.
Abstract: We show that the geometry of a Riemannian manifold (M, ) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted catsys(M) and defined in terms of the existence of systolic inequalities satisfied by every metric , as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality catsysM ≤ catLSM is satisfied, which typically turns out to be an equality, e.g., in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to Massey products. The comparison with the value of catLS(M) leads us to prove or conjecture new systolic inequalities on M. © 2006 Wiley Periodicals, Inc.
54 citations
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TL;DR: In this article, a Riemannian analogue of the Lusternik-Schnirelmann category is introduced, called the systolic category of M, denoted cat(M) and defined in terms of the existence of Systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by Gromov.
Abstract: We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and defined in terms of the existence of systolic inequalities satisfied by every metric g, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality cat_{sys}(M) \le cat_{LS}(M) is satisfied, which typically turns out to be an equality, e.g. in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality, and that both categories are sensitive to Massey products. The comparison with the value of cat(M) leads us to prove or conjecture new systolic inequalities on M.
24 citations
TL;DR: In this article, a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2, was proved.
Abstract: We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole and the codimension 1 systole with coefficients in Z_2. As an application, we prove that Lusternik-Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our earlier result in the orientable case. Finally, we prove the homotopy invariance of systolic category.
22 citations
TL;DR: In this paper, a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole as well as the codimension-1 systole with coefficients in 2.
Abstract: We prove a new systolic volume lower bound for non-orientable n-manifolds, involving the stable 1-systole as well as the codimension-1 systole with coefficients in 2. As an application, we prove that Lusternik�Schnirelmann category and systolic category agree for non-orientable closed manifolds of dimension 3, extending our earlier result in the orientable case. Finally, we prove the homotopy invariance of systolic category
21 citations
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TL;DR: In this article, it was shown that manifolds of Lusternik-Schnirelmann category 2 necessarily have a free fundamental group, which is a conjecture of Gomez-Larranaga and Gonzalez-Acuna.
Abstract: We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category.
13 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2011 | 2 |
| 2010 | 1 |
| 2009 | 1 |
| 2008 | 5 |
| 2007 | 1 |
| 2006 | 1 |