Lefschetz pencil
Lefschetz pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, in order to analyse the algebraic topology ... of a complex algebraic variety V, a Lefschetz pencil is something like a fibration over ... general hyperplane section will be smooth). A Lefschetz pencil restricts the nature of the ...
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Solomon Lefschetz
Solomon Lefschetz Solomon Lefschetz (3 September 1884-5 October 1972) was ... as allied to Morse theory, though a Lefschetz pencil of hyperplane sections is a more ... because hyperplanes intersect each other). The Picard-Lefschetz formula in the theory of vanishing cycles ... for his work in mathematical analysis. The Lefschetz fixed point theorem, now a basic ...
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Lefschetz hyperplane theorem
Lefschetz hyperplane theorem In mathematics, the Lefschetz hyperplane theorem states that a hyperplane section ... this direction were first given by Solomon Lefschetz in the early 1920s. They were subsequently ... classical setting. A modern approach deduces the Lefschetz results from the Andreotti-Frankel theorem, which ... and its long exact sequence, together with Lefschetz duality , can be applied to recover ...
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Lefschetz zeta function
Lefschetz zeta function In mathematics, the Lefschetz zeta-function is a tool used in ... math> where L(f n ) is the Lefschetz number of the nth iterate of f ... words θ → −θ. Then f has Lefschetz number 0, and f 2 is the identity map, which has Lefschetz number 2. Therefore we need exp( ...
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Lefschetz fixed-point theorem
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that ... of X. It is named after Solomon Lefschetz, who first stated it in 1926. The ... f if f(x) = x. Denote the Lefschetz number of f by $\backslash Lambda\_f.\backslash ,$Lefschetz fixed-point theorem states that if < ...
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Fixed point index
... chosen small sphere around x 0 . The Lefschetz-Hopf theorem The importance of the fixed ... largely due to its role in the Lefschetz-Hopf theorem, which states: $\backslash sum\_\{x\; ...\; of\; f,\; and\; L(f)\; is\; the$Lefschetz number of f. Since the quantity on ... when f has no fixed points, the Lefschetz-Hopf theorem trivially implies the Lefschetz fixed point theorem.
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Hairy ball theorem
... planet that would no longer be true. Lefschetz connection There is a closely-related argument from algebraic topology, using the Lefschetz fixed point theorem. Since the Betti numbers ... are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology of the ... to the identity. Therefore they all have Lefschetz number 2, also. Hence they are not without fixed points (which means Lefschetz number 0). Some more work would ...
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Atiyah–Bott fixed-point theorem
... point theorem is a general form of Lefschetz fixed-point theorem for smooth manifolds M ... to find the correct replacement for the Lefschetz number, which in the classical result is ... the identity mapping) in M�M, and the Lefschetz number thereby becomes an intersection number. The ... complex T. Such a T has its Lefschetz number L(T) which by definition is ...
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User:Charles Matthews/MathematicsL
... Lax-Maslov operator - Lebesgue lemma - Lebesgue spine - Lefschetz class - Lefschetz decomposition - Lefschetz hyperplane theorem - Lefschetz pencil - Lelong number - Lelong theorem - Lepage decomposition ...
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List of mathematics articles (L)
... Lecnac -- Lee Hwa Chung theorem -- Leech lattice -- Lefschetz fixed-point theorem -- Lefschetz hyperplane theorem -- Lefschetz pencil -- Lefschetz zeta function -- Left-hand side and right ...
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