m at h . D G / 9 81 01 72 v 2 1 9 N ov 1 99 8 VOLUME OF RIEMANNIAN MANIFOLDS , GEOMETRIC INEQUALITIES , AND HOMOTOPY THEORY
نویسنده
چکیده
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-manifolds are 2-systolically free. More generally, let m be a positive even integer, and let n > m. Then all manifolds of dimension at most n are m-systolically free (modulo torsion) if all k-skeleta,m+1 ≤ k ≤ n, of the loop space Ω(Sm+1) are m-systolically free.
منابع مشابه
ar X iv : m at h / 98 10 17 2 v 1 [ m at h . D G ] 2 9 O ct 1 99 8 VOLUME OF RIEMANNIAN MANIFOLDS , GEOMETRIC INEQUALITIES , AND HOMOTOPY THEORY
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4.
متن کاملm at h . D G ] 1 9 N ov 1 99 8 VOLUME OF RIEMANNIAN MANIFOLDS , GEOMETRIC INEQUALITIES , AND HOMOTOPY THEORY
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-man...
متن کاملVolume of Riemannian manifolds , geometric inequalities , and homotopy theory
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-man...
متن کامل0 N ov 1 99 9 MORSE - NOVIKOV CRITICAL POINT THEORY , COHN LOCALIZATION AND DIRICHLET UNITS
In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The Main Theorem of ...
متن کامل