In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.[1] There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.
^Biographical Memoirs By National Research Council Staff (1992), p. 297.
In mathematics, Lefschetzduality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was...
Solomon Lefschetz ForMemRS (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental...
extend Poincaré duality to that case too (see Lefschetzduality). In fact, the cap product with a fundamental class gives a stronger duality result saying...
In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold ( M 2 n , ω ) {\displaystyle (M^{2n},\omega )} , sharing a certain cohomological...
applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality. Deligne (1980) found...
Harer, John (2008-04-04). "Extending Persistence Using Poincaré and LefschetzDuality". Foundations of Computational Mathematics. 9 (1): 79–103. doi:10...
surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now known as the Hodge...
duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in...
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...
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)...
is a point, the cycle map is required to be the inclusion Z ⊂ K. Weak Lefschetz axiom: For any smooth hyperplane section j: W ⊂ X (i.e. W = X ∩ H, H some...
cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann–Roch theorem for projective...
Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic...
sections at a given point. (As in the case when this procedure constructs a Lefschetz pencil.) In fact, it is possible for a bundle to have no non-zero global...
Artin–Verdier duality. Broadly speaking, Artin–Verdier duality, a consequence of class field theory, is an arithmetic analogue of Poincaré duality, a duality for...
influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. He was one of the first members of the Institute for Advanced...
methods in the study of abelian functions. Eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex...
category Poincaré duality Fundamental class Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem...
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