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Mapping theorem in topology
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
Lefschetz_fixed-point_theorem
Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle
Brouwer_fixed-point_theorem
Condition for a mathematical function to map some value to itself
space Kakutani fixed-point theorem Kleene fixed-point theorem Knaster–Tarski theorem Lefschetz fixed-point theorem Nielsen fixed-point theorem Poincaré–Birkhoff
Fixed-point_theorem
Theorem about complex manifolds
Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a
Holomorphic Lefschetz fixed-point formula
Holomorphic_Lefschetz_fixed-point_formula
Fixed-point theorem for smooth manifolds
Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth
Atiyah–Bott fixed-point theorem
Atiyah–Bott_fixed-point_theorem
Russian-born American mathematician (1884–1972)
in 1925 and the American Philosophical Society in 1929. The Lefschetz fixed-point theorem, now a basic result of topology, was developed by him in papers
Solomon_Lefschetz
Element mapped to itself by a mathematical function
have a fixed point, but it doesn't describe how to find the fixed point. The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic
Fixed_point_(mathematics)
Topological duality
introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous
Lefschetz_duality
Theorem in differential topology
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
Hairy_ball_theorem
Method for dividing a simplicial complex
instance in Lefschetz's fixed-point theorem. The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This
Barycentric_subdivision
Concept in Nielsen theory
zero when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed-point theorem. A. Katok and B. Hasselblatt(1995), Introduction
Fixed-point_index
Mathematical result in differential geometry
generalizations of the Lefschetz fixed-point theorem, with terms coming from fixed-point submanifolds of the group G. See also: equivariant index theorem. Atiyah (1976)
Atiyah–Singer_index_theorem
infinite-dimensional spaces, topological degree theory, Jordan separation theorem, Lefschetz fixed-point theorem) Morse theory and Lusternik–Schnirelmann category theory
Nonlinear_functional_analysis
Representation of mathematical space
instance in Lefschetz's fixed-point theorem. The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This
Triangulation_(topology)
theorem (algebraic topology) Lefschetz fixed-point theorem (fixed points, algebraic topology) Lefschetz–Hopf theorem (topology) Leray–Hirsch theorem (algebraic
List_of_theorems
Topics referred to by the same term
Grothendieck trace formula, an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology, used to express the Hasse–Weil zeta function
Trace_formula
specific case of the K-theory of a stack.) A version of the Lefschetz fixed-point theorem holds in the setting of equivariant (algebraic) K-theory. Let
Equivariant algebraic K-theory
Equivariant_algebraic_K-theory
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map
Lefschetz_zeta_function
Mathematical branch
known as the Nielsen fixed-point theorem: Any map f has at least N(f) fixed points. Because of its definition in terms of the fixed-point index, the Nielsen
Nielsen_theory
algebraic geometry, a branch of mathematics, the Lefschetz theorem on (1,1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic
Lefschetz theorem on (1,1)-classes
Lefschetz_theorem_on_(1,1)-classes
Expresses the number of points of a variety over a finite field
Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology. One application of the Grothendieck trace
Grothendieck_trace_formula
British-Lebanese mathematician (1929–2019)
his work in developing K-theory, a generalized Lefschetz fixed-point theorem and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly
Michael_Atiyah
Hungarian-American mathematician (1923-2005)
fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site
Raoul_Bott
Branch of mathematics
theorem Freudenthal suspension theorem Hurewicz theorem Künneth theorem Lefschetz fixed-point theorem Leray–Hirsch theorem Poincaré duality theorem Seifert–van
Algebraic_topology
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
function with the diagonal may be computed using homology via the Lefschetz fixed-point theorem; the self-intersection of the diagonal is the special case of
Diagonal
Rational function of the form (az + b)/(cz + d)
characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more
Möbius_transformation
Continuous mappings can be approximated by ones that are piecewise simple
simplicial approximation theorem is used.) Here is another more substantial but typical application. (Lefschetz fixed point theorem) For a compact manifold
Simplicial approximation theorem
Simplicial_approximation_theorem
On generating functions from counting points on algebraic varieties over finite fields
fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological
Weil_conjectures
Theorem in algebraic geometry
{\displaystyle \operatorname {fix} (\varphi )} is finite, then by the Lefschetz fixed-point theorem, | fix ( φ ) | = 1 − 2 tr ( h ( φ ) ) + 1 = 2 − 2 tr (
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
and have Frobenius mappings acting in such a way that the Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
mappings with finitely many fixed points is the Lefschetz-Hopf theorem. Since every vector field induces a flow on manifolds and fixed points of small flows
Poincaré–Hopf_theorem
Generalization of matrix trace
algebro-geometric version of the Atiyah–Bott fixed point formula, an extension of the Lefschetz fixed point formula. Ponto & Shulman (2014, Def. 2.2) Dold
Categorical_trace
Theorem in mathematical logic
compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important
Compactness_theorem
Series whose partial sums eventually only have a fixed number of terms after cancellation
occurs in the derivation of a probability density function; Lefschetz fixed-point theorem, where a telescoping sum arises in algebraic topology; Homology
Telescoping_series
Generalization of a fixed point
the Lefschetz coincidence theorem, which is typically known only in its special case formulation for fixed points. Coincidence points, like fixed points
Coincidence_point
Sheaf cohomology on the étale site
and to prove general results such as Poincaré duality and the Lefschetz fixed-point theorem in this context. Grothendieck originally developed étale cohomology
Étale_cohomology
Algebraic topology uses abstract algebra to study topological spaces
Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous
List of algebraic topology topics
List_of_algebraic_topology_topics
Theorem about complex manifolds
matrix of the holomorphic tangent bundle Atiyah–Bott fixed-point theorem Holomorphic Lefschetz fixed-point formula Bott, Raoul (1967), "Vector fields and characteristic
Bott_residue_formula
German mathematician (1859–1919)
automorphisms theorem. This work anticipates a number of later theories, such as the general theory of algebraic correspondences, Hecke operators, and Lefschetz fixed-point
Adolf_Hurwitz
collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because
Hyperplane_section
Mathematics glossary
space of formal group laws. Lefschetz 1. Solomon Lefschetz 2. The Lefschetz fixed-point theorem says: given a finite simplicial complex K and its geometric
Glossary of algebraic topology
Glossary_of_algebraic_topology
Fundamental result in the branch of mathematics known as character theory
the Lefschetz fixed-point theorem). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably
Brauer's theorem on induced characters
Brauer's_theorem_on_induced_characters
Generalized notion of counting curve intersections
intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed-point theorem in quantitative form. Serre
Intersection_number
Projective variety that is also an algebraic group
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
Abelian_variety
Mathematical manifold theory
singular homology. Separately, a 1927 paper of Solomon Lefschetz used topological methods to reprove theorems of Riemann. In modern language, if ω1 and ω2 are
Hodge_theory
Manifold with Riemannian, complex and symplectic structure
Nakano vanishing theorems, the Lefschetz hyperplane theorem, Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Hodge index theorem. On a Riemannian
Kähler_manifold
Hungarian and American mathematician and physicist (1903–1957)
functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator
John_von_Neumann
About polynomials in several variables
reduced to K = C {\displaystyle \mathbb {K} =\mathbb {C} } using the Lefschetz principle. Further, if F {\displaystyle F} is injective, it can be shown
Jacobian_conjecture
American mathematician
demonstrating the existence of wild involutions on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere. This meant that the original
R._H._Bing
homology of a (real) surface of genus g. A classical result is the Picard–Lefschetz formula, detailing how the monodromy round the singular fiber acts on
Vanishing_cycle
Mathematics award
first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier referred to the plaque as a "quantized Fields Medal". Accounts
Fields_Medal
French mathematician, physicist and engineer (1854–1912)
Poincaré–Lefschetz duality theorem: a version of Poincaré duality in geometric topology, applying to a manifold with boundary Poincaré separation theorem: gives
Henri_Poincaré
American mathematician
becomes relevant in the context of Lefschetz's theorem, by considering a Morse function given by the distance to a fixed point. The second-order analysis at
Theodore_Frankel
hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group H 2 ( S ) {\displaystyle
Algebraic_cycle
Analysis of datasets using techniques from topology
Herbert; Harer, John (2008-04-04). "Extending Persistence Using Poincaré and Lefschetz Duality". Foundations of Computational Mathematics. 9 (1): 79–103. doi:10
Topological_data_analysis
American mathematician (1927–2016)
Thesis The Topological Fixed Point Theory and Its Applications in Functional Analysis (1948) Doctoral advisor Solomon Lefschetz Witold Hurewicz Doctoral
Felix_Browder
American mathematician (1880–1960)
relativity. He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille
Oswald_Veblen
Concept in model theory
structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language
Transfer_principle
Counterintuitive mathematical object
strong as a Kähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems on sheaf cohomology break down in every possible way. In Pathologies
Pathological_(mathematics)
geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field conjectured
Behrend's_trace_formula
Behavior in a nonlinear system
stationary point of the system, i.e. a point p {\displaystyle p} where V ′ ( p ) = 0 {\displaystyle V'(p)=0} . The Bendixson–Dulac theorem and the Poincaré–Bendixson
Limit_cycle
Algebraic structure with addition, multiplication, and division
The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. Moreover, any fixed statement
Field_(mathematics)
Theorem in abstract algebra
and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Dutch mathematician (1942–2010)
ISBN 978-0-8176-8107-4, MR 1362544 Duistermaat, J. J. (2011), The heat kernel Lefschetz fixed point formula for the Spinc dirac operator, Boston: Birkhäuser, ISBN 978-0-8176-8247-7;
Hans_Duistermaat
Vector bundle of rank 1
non-vanishing global sections at a given point. (As in the case when this procedure constructs a Lefschetz pencil.) In fact, it is possible for a bundle
Line_bundle
Discrete (i.e., incremental) version of infinitesimal calculus
These two points of view are related to each other by the fundamental theorem of discrete calculus. The study of the concepts of change starts with their
Discrete_calculus
Algebraic geometry analog of a principal bundle in algebraic topology
"Torsors Made Easy". math.ucr.edu. Retrieved 2022-11-22. Behrend, K. The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. PhD dissertation
Torsor_(algebraic_geometry)
Concept in algebraic geometry
the supports of all the effective divisors in the system. Consider the Lefschetz pencil p : X → P 1 {\displaystyle p:{\mathfrak {X}}\to \mathbb {P} ^{1}}
Linear_system_of_divisors
the basic formulae of the general theory.) It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2
Local_zeta_function
1973 Kaminker & Miller 1985 Atiyah, M. F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics, 86 (2):
Signature_operator
Algebraic structure used in topology
precursors to cohomology. In the mid-1920s, J. W. Alexander and Solomon Lefschetz founded intersection theory of cycles on manifolds. On a closed oriented
Cohomology
surfaces) Jean-Louis Koszul, Théorèmes de points fixes pour les groupes élémentaires, d'après Borel (fixed-point theorems) Jean-Pierre Serre, Structure
Séminaire Nicolas Bourbaki (1960–1969)
Séminaire_Nicolas_Bourbaki_(1960–1969)
American mathematician (1935–2020)
the Graham–Rothschild theorem in the Ramsey theory of parameter words and Graham's number derived from it, the Graham–Pollak theorem and Graham's pebbling
Ronald_Graham
American mathematician
(2010), pp. 163–188, ISSN 0022-040X [arXiv:0808.0667] [abs] M Stern, Fixed point theorems from a de Rham perspective, Asian Journal of Mathematics, vol. 13
Mark_Stern
96) pencil A 1-dimensional linear system. See pencil (mathematics) and Lefschetz pencil. pentad A set of 5 points pentahedron A union of 5 planes, in particular
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Latvian-American mathematician (1914–1993)
wrote several major retrospectives of flows, pseudoanalytic functions, fixed point methods, Riemann surface theory prior to his work on moduli, and the
Lipman_Bers
Algebra of formal sums
to the axiom of choice) can be found in Serge Lang's Algebra. Solomon Lefschetz and Irving Kaplansky argue that using the well-ordering principle in place
Free_abelian_group
History of maths
geometry Charles Weibel; History of homological algebra Peter Johnstone; The point of pointless topology Stasheff, Jim (January 21, 1996). "The Pre-History
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
American annual mathematics conference
Potential theory for nonlinear PDE's John Pardon (Stanford): Existence of Lefschetz vibrations on Stein/Weinstein domains Raanan Schul (Stony Brook): Qualitative
Geometry_Festival
incontinence invented the expandable tampon Solomon Lefschetz (1884–1972), known for his topological fixed-point theorem Stanley Lombardo, classics professor and
List of University of Kansas people
List_of_University_of_Kansas_people
Mathematical conjecture
structure on H 3 ( X ) {\displaystyle H^{3}(X)} . Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is H 3 ( X ) {\displaystyle
Mirror_symmetry_conjecture
Mathematical classification
Complexification and Symplectization based on analogies between Picard–Lefschetz theory which he interprets as the Complexified version of Morse theory
ADE_classification
travel, tourism, insurance
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
Girl/Female
Hindu, Indian, Marathi
Directed; Fixed
Boy/Male
Indian, Sanskrit
Well Fixed
Boy/Male
Shakespearean
King Henry IV, Part 1 and 2' Edward Poins, an irregular humorist.
Surname or Lastname
English, Scottish, French, and Catalan
English, Scottish, French, and Catalan : topographic name for
someone who lived near a bridge, Middle English, Old French, Catalan
pont (Latin pons, genitive pontis).Catalan : habitational name from any of the numerous places named
with Pont.Dutch : variant of
Pond 2.A Pont from the Lorraine region of France is documented in Quebec City in
1640; Pont appears to be a secondary surname to
Girl/Female
Tamil
Bindushri | பீநà¯à®¤à¯à®·à¯à®°à¯€Â
Point
Bindushri | பீநà¯à®¤à¯à®·à¯à®°à¯€Â
Boy/Male
Hindu, Indian, Kannada, Telugu
Fixed
Girl/Female
Hindu
Fixed
Girl/Female
Tamil
Fixed
Girl/Female
Bengali, Indian, Kannada, Marathi
Firmly Fixed
Surname or Lastname
English and French
English and French : probably an altered form of French Pons, a habitational name from places so named in Bourgogne and Franche-Comté.
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Fixed
Girl/Female
Gujarati, Indian
Firmly Fixed
Girl/Female
Norse
Point.
Boy/Male
Indian, Sanskrit
Firmly Fixed
Girl/Female
Tamil
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Firmly fixed
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Boy/Male
Indian, Sanskrit
Fixed
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya
Firmly Fixed
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : from the medieval personal name Ponc(h)e, Pons (see Ponce).English (of Norman origin) : habitational name from Ponts in La Manche and Seine-Maritime, Normandy, from Latin pontes ‘bridges’ (see Pont).English (of Norman origin) : nickname for a fop or dandy, from points ‘laces for hose’ (see Pointer 1).
Boy/Male
Indian, Sanskrit
Firmly Fixed
Girl/Female
Tamil
Fixed
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
LEFSCHETZ FIXED-POINT-THEOREM
a.
Alt. of Point-devise
n.
To indicate or discover by a fixed look, as game.
n.
One of the points of the compass (see Points of the compass, below); also, the difference between two points of the compass; as, to fall off a point.
n.
A short piece of cordage used in reefing sails. See Reef point, under Reef.
n.
Lace wrought the needle; as, point de Venise; Brussels point. See Point lace, below.
n.
A movement executed with the saber or foil; as, tierce point.
n.
To supply with punctuation marks; to punctuate; as, to point a composition.
n.
Whatever serves to mark progress, rank, or relative position, or to indicate a transition from one state or position to another, degree; step; stage; hence, position or condition attained; as, a point of elevation, or of depression; the stock fell off five points; he won by tenpoints.
adv.
In a point-blank manner.
adv.
Alt. of Point-devise
v. i.
To direct the point of something, as of a finger, for the purpose of designating an object, and attracting attention to it; -- with at.
a.
Repaired by foxing; as, foxed boots.
n.
To give a point to; to sharpen; to cut, forge, grind, or file to an acute end; as, to point a dart, or a pencil. Used also figuratively; as, to point a moral.
n.
Printed letters; the impression taken from type, as to excellence, form, size, etc.; as, small print; large print; this line is in print.
n.
To direct toward an abject; to aim; as, to point a gun at a wolf, or a cannon at a fort.
n.
A fixed conventional place for reference, or zero of reckoning, in the heavens, usually the intersection of two or more great circles of the sphere, and named specifically in each case according to the position intended; as, the equinoctial points; the solstitial points; the nodal points; vertical points, etc. See Equinoctial Nodal.
n.
The attitude assumed by a pointer dog when he finds game; as, the dog came to a point. See Pointer.
n.
A core print. See under Core.
v. i.
To indicate the presence of game by fixed and steady look, as certain hunting dogs do.
n.
To mark (as Hebrew) with vowel points.
travel, tourism, insurance