In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for , the cup product
be an isomorphism.
The topology of these symplectic manifolds is severely constrained, for example their odd Betti numbers are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to William Thurston.[1]
^Thurston, William P. (1976). "Some simple examples of symplectic manifolds". Proceedings of the American Mathematical Society. 55 (2): 467. doi:10.2307/2041749. JSTOR 2041749. MR 0402764.
and 19 Related for: Lefschetz manifold information
In mathematics, a Lefschetzmanifold is a particular kind of symplectic manifold ( M 2 n , ω ) {\displaystyle (M^{2n},\omega )} , sharing a certain cohomological...
1929. The Lefschetz fixed-point theorem, now a basic result of topology, was developed by him in papers from 1923 to 1927, initially for manifolds. Later...
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was...
mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between...
conjecture is due to Lefschetz (1924). In fact, it predates the conjecture and provided some of Hodge's motivation. Theorem (Lefschetz theorem on (1,1)-classes)...
In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an...
(or compact Kähler manifolds), building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge-Riemann...
In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by Egbert Brieskorn (1966, 1966b), is the intersection of a small sphere around...
algebraic topology, using the Lefschetz fixed-point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on...
Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic...
work of René Thom, Frankel and Aldo Andreotti gave a new proof of the Lefschetz hyperplane theorem using Morse theory. The crux of the argument is the...
Hurewicz Egbert van Kampen Daniel Kan Hermann Künneth Ruth Lawrence Solomon Lefschetz Jean Leray Saunders Mac Lane Mark Mahowald J. Peter May Barry Mazur John...
complex manifolds is Picard–Lefschetz theory. To illustrate, consider a mountainous landscape surface M {\displaystyle M} (more generally, a manifold). If...
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map...
independently proven that any such manifold admits a parallel 4-form Ω {\displaystyle \Omega } .The long-awaited analog of strong Lefschetz theorem was published in...
the function field. In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use...
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear...
the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957). It was proved by Shing-Tung Yau (1977...
Global Information
![{\displaystyle \cup [\omega ^{k}]\colon H^{n-k}(M,\mathbb {R} )\to H^{n+k}(M,\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b22f03033ca2f5ae898046a87ef35b8799928057)