In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was...

Poincaré duality to such stratified spaces. There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander...

Solomon Lefschetz ForMemRS (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental...

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X...

extend Poincaré duality to that case too (see Lefschetz duality). In fact, the cap product with a fundamental class gives a stronger duality result saying...

a sphere having no sources or sinks. Poincaré–Lefschetz duality theorem: a version of Poincaré duality in geometric topology, applying to a manifold with...

Lagrange duality Langlands dual Lefschetz duality Local Tate duality Opposite category Poincaré duality Twisted Poincaré duality Poitou–Tate duality Pontryagin...

In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold ( M 2 n , ω ) {\displaystyle (M^{2n},\omega )} , sharing a certain cohomological...

Harer, John (2008-04-04). "Extending Persistence Using Poincaré and Lefschetz Duality". Foundations of Computational Mathematics. 9 (1): 79–103. doi:10...

Poincaré–Wirtinger inequality Poincaré–Lelong equation Poincaré lemma Poincaré-Lefschetz duality Poincaré–Lindstedt method Poincaré line bundle Poincaré map Poincaré...

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...

applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality. Deligne (1980) found...

algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic...

Solomon Lefschetz. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to Atiyah's memoir, Lefschetz and Hodge...

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...

theorem Hurewicz theorem Künneth theorem Lefschetz fixed-point theorem Leray–Hirsch theorem Poincaré duality theorem Seifert–van Kampen theorem Universal...

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...

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)...

O ( ⌊ D ⌋ ) . {\displaystyle {\mathcal {O}}(\lfloor D\rfloor ).} The Lefschetz hyperplane theorem implies that for a smooth complex projective variety...

Lurie (2019), and a future proof using a version of the Grothendieck-Lefschetz trace formula will be published in a second volume. Ono (1965) used the...

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...

theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by...

category Poincaré duality Fundamental class Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem...

duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in...

frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective plane Real projective space Segre...

proving the Kodaira and Nakano vanishing theorems, the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Hodge...

instanton invariants). Any compact symplectic manifold admits a symplectic Lefschetz pencil (Donaldson 1999). Donaldson's recent work centers on a problem...

Artin–Verdier duality. Broadly speaking, Artin–Verdier duality, a consequence of class field theory, is an arithmetic analogue of Poincaré duality, a duality for...

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...

parallel 4-form Ω {\displaystyle \Omega } . Bonan's later results include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms...