Where results make sense
 About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

# Topic: Lefschetz fixed point theorem

###### In the News (Tue 18 Jun 13)

 Lefschetz fixed-point theorem - Wikipedia, the free encyclopedia In mathematics, the Lefschetz fixed-point theorem is a formula that counts the number of fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space. Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence points of mappings. en.wikipedia.org /wiki/Lefschetz_fixed-point_theorem   (552 words)

 Fixed-point theorem - Wikipedia, the free encyclopedia In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression. Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place. en.wikipedia.org /wiki/Fixed-point_theorem   (438 words)

 Atiyah–Singer index theorem - Wikipedia, the free encyclopedia s assigns to every point of the cotangent bundle a homomorphism of E (i.e., at each point in the cotangent bundle, E is a vector space and s is a matrix acting on that vector space), ellipticity is the requirement that s be invertible away from the zero section. The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. That is, to find a relative statement (this being a highly fashionable point of view in the 1960s), for which functoriality carried the main burden. en.wikipedia.org /wiki/Index_theory   (1275 words)

 Fixed Point Theorems Fixed points are of interest in themselves but they also provide a way to establish the existence of a solution to a set of equations. Point B represents the point A is mapped to. A physical example of a fixed point of a mapping is the center of a whirlpool in a cup of tea when it is stirred. www.applet-magic.com /fixed.htm   (1625 words)

 Springer Online Reference Works Lefschetz' fixed-point theorem, or the Lefschetz–Hopf theorem, is a theorem that makes it possible to express the number of fixed points of a continuous mapping in terms of its Lefschetz number. The hard Lefschetz theorem is a theorem about the existence of a Lefschetz decomposition of the cohomology of a complex Kähler manifold into primitive components. The Lefschetz decomposition commutes with the Hodge decomposition (cf. eom.springer.de /l/l058000.htm   (641 words)

 Recent Research The axiomatic characterization of the Leray-Schauder fixed point index (which is even more powerful) is also stated, and its application to issues concerning robustness of sets of equilibria is explained. Theorem 1 is a special case: every k-vertex tree of diameter 4 can be embedded in G. A more technical result, Theorem 2, is obtained by extending the main ideas in the proof of Theorem 1. Abstract: We present a treatment for mathematical economists of three topics in the theory of fixed points: (a) the Lefschetz fixed point index; (b) the Lefschetz fixed point theorem; (c) the theory of essential sets of fixed points. www.econ.umn.edu /~mclennan/Papers/papers.html   (978 words)

 SOLOMON LEFSCHETZ   (Site not responding. Last check: 2007-10-31) Lefschetz accepted a position at the University of Nebraska, then moved to the University of Kansas. One of Lefschetz's most widely used results, the Lefschetz fixed-point theorem, asserts that a map f from a "nice" compact space to itself has a fixed point (a point x such that f(x)=x) when a certain numerical invariant (the "Lefschetz number") of f is nonzero. For some spaces (for example, those having the same homology groups as a point) the Lefschetz number of any map f from the space to itself is nonzero, so the Lefschetz fixed-point theorem shows that these spaces have the fixed-point property (any map from the space to itself has a fixed point). www.usna.edu /Users/math/meh/lefschetz.html   (601 words)

 The Atiyah-Bott fixed point theorem In the Sixties Atiyah and Bott proved a far-reaching generalization of the Lefschetz fixed point theorem ([42], [44]). Specializing the Atiyah-Bott fixed point theorem to the de Rham complex, one recovers the classical Lefschetz fixed point theorem. When applied to other geometrically interesting elliptic complexes, Atiyah and Bott obtained new fixed point theorems, such as a holomorphic Lefschetz fixed point theorem in the complex analytic case and a signature formula in the Riemannian case. brauer.math.harvard.edu /history/bott/bottbio/node18.html   (366 words)

 Math 215a Home Page Homology with coefficients, Tor, and the universal coefficient theorem. Statement and sketch of proof of Lefschetz fixed point theorem for a finite simplicial complex. (11/1) Homology with coefficients, the universal coefficient theorem, and Tor. math.berkeley.edu /~hutching/teach/215a-2005/index.html   (2991 words)

 [No title]   (Site not responding. Last check: 2007-10-31) theorem of algebra via Lefschetz fixed point theorem Date: 19 Dec 2001 06:15:47 -0500 Newsgroups: sci.math.research A few years ago I found a proof of the fundamental theorem of algebra using the Lefschetz fixed point theorem. Since the general linear group of C is connected, A can be connected by a path to the identity I (this can be done explicitly by writing A as a product of elementary matrices and deforming these to I in the obvious way). It follows that the A is homotopic to I, and therefore its Lefschetz number on P coincides with the Euler characteristic of P which is nonzero. www.math.niu.edu /~rusin/known-math/01_incoming/FTA   (468 words)

 [UC Top] Fixed Point Theory- Generalized Lefschetz Number   (Site not responding. Last check: 2007-10-31) One interpretation of the Lefschetz Fixed Point theorem is that it provides an invariant which vanishes for self maps of polyhedra that are fixed point free. In general, the vanishing of this invariant, called the Lefschetz number or the index, does not guarantee that the self map is fixed point free. However, in the case that the space is simply connected, and satisfies a few additional assumptions, the Lefschetz number is zero exactly when the self map is homotopic to a fixed point free map. zaphod.uchicago.edu:8080 /pipermail/topology/2005q1/000324.html   (277 words)

 TMNA - Volume 20 Number 2   (Site not responding. Last check: 2007-10-31) The Lefschetz Fixed Point Theorem for compact absorbing contraction morphisms (CAC-morphisms) of retracts of open subsets in admissible spaces in the sense of Klee is proved. Moreover, the relative version of the Lefschetz Fixed Point Theorem and the Lefschetz Periodic Theorem are considered. This theorem is based on the concept of expceptional family of elements (EFE) for mapping and on the concept of (0,k)-epi mapping which is similar to the topological degree. www.mat.uni.torun.pl /~tmna/htmls/archives/vol-20-2.html   (620 words)

 [UC Top] fixed point theory   (Site not responding. Last check: 2007-10-31) The Lefschetz fixed point theorem gives an ("easy to compute") invariant that indicates when a self map of a polyhedron has at least one fixed point. This is really a consequence of the following statement of the theorem: Theorem: Let f be a self map of a polyhedron X, then the trace of the induced map on rational cohomology is equal to the index of f. While this theorem can sometimes indicate if a map has a fixed point, it does not give any indication of how many fixed points the map has. zaphod.uchicago.edu:8080 /pipermail/topology/2005q1/000303.html   (344 words)

 [No title]   (Site not responding. Last check: 2007-10-31) Hebrew University Basic Notions Seminar Title: Lefschetz' fixed point theorem Speaker: Ehud De-Shalit (HU) Date: Thursday, December 23rd, 17:15 Place: Ulam 2 of the Mathematics Building Abstract: Let f be a continuous mapping of a compact manifold M to itself. Lefschetz' fixed point theorem computes the number of fixed points of f in terms of its trace on cohomology. I shall then explain the arithmetic case and how Lefschetz' theorem is used to count the number of solutions to equations over finite fields. www.math.technion.ac.il /~techm/20041223171520041223des   (145 words)

 Pure Group Publications In 1986, using transfer techniques from stable homotopy and the Lefschetz fixed point theorem, Snaith discovered a canonical form - called Explicit Brauer Induction - for the famous induction theorem for finite-dimensional complex representations of finite groups which R. Brauer proved around 1946. The point, then, is that Explicit Brauer Induction can be a very useful tool in the study of mathematical objects which involve $R(G)$. Snaith applied the technique to study the Shintani correspondence for $GL_{2}{\bf F}_{q}$, the class-groups of integral group-rings, the construction of the conductor for a Galois representation when the residue field is inseparable (s problem of J-P. Serre from 1960) and the Chinburg conjectures in number theory. www.maths.soton.ac.uk /pure/researchabstract.phtml?keyword=representation   (264 words)

 Algorithms for the Fixed Point Property These theorems give a polynomial algorithm to decide if an ordered set has the fixed point property for some nice classes of ordered sets (height 1, width 2), and structural insights for other classes (chain-complete ordered sets with no infinite antichains, sets of (interval) dimension 2). Walker's relational fixed point property for which the analogous problem has a very satisfying solution also is discussed. Another variation on the retraction theme is the use of algebraic topology in deriving fixed point theorems initiated by Baclawski and Björner and continued for example by Constantin and Fournier. www.csi.uottawa.ca /ordal/papers/schroder/FINSURVE.html   (553 words)

 Penot: Fixed point theorems without convexity BROWN R.F. On the Lefschetz fixed point theorem. GOEBEL K. An elementary proof of the fixed point theorem of Browder and Kirk. KIRK W.A. Caristi's fixed point theorem and the theory of normal solvability. www.numdam.org /numdam-bin/recherche?h=nc&id=MSMF_1979__60__129_0&format=complete   (1010 words)

 research1   (Site not responding. Last check: 2007-10-31) Then "Fixed Point Theory" is not a specific location on the map of the continent, but instead it is a long highway. Proofs of the existence of fixed points and methods for finding them are important mathematical problems, since the solution of every equation f(x)=0 reduces, by transforming it to x-f(x)=x, to finding a fixed point of the mapping F=I-f, where I is the identity mapping. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an n-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. users.marshall.edu /~saveliev/Research/research11.html   (683 words)

 [No title]   (Site not responding. Last check: 2007-10-31) If a space is contractible, does a fixed point theorem hold, i.e. And, does a fixed point theorem hold for the projective plane? The Lefschetz fixed-point theorem answers some of these: it implies that if X is a path-connected compact polyhedron for which the homology groups H_n(X) are finite for n > 0, then every continuous f: X --> X has a fixed point. www.math.niu.edu /~rusin/known-math/96/lefschetz.fpt   (166 words)

 Handbook of Topological Fixed Point Theory   (Site not responding. Last check: 2007-10-31) The fixed point index of the Poincaré translation operator on differentiable manifolds FURS, M.P. On the existence of equilibria and fixed points of maps under constraints Topological fixed point theory and nonlinear differential equations www.booksmatter.com /b1402032218.htm   (75 words)

 [No title]   (Site not responding. Last check: 2007-10-31) As an answer to Peter McBurney's question, there are generalizations to Brouwer's fixed point theorem as the following. Therefore, by the Lefschetz fixed point theorem, f has a fixed point. qed There are generalizations of the Lefschetz fixed point theorem to situations similar to that of Brouwer's theorem of 1960. www.lehigh.edu /~dmd1/cp120.txt   (315 words)

 Topics: Fixed Point Theorems   (Site not responding. Last check: 2007-10-31) Motivation: If A is any differential operator, the existence of solutions of the equation A f = 0 is equivalent to the existence of fixed pts for A + I; We are interested in equations like df = 0 for the study of critical pts (> see morse theory, etc). R, any f: [a, b] → [a, b] must have at least one fixed point (the graph must cross at least once the line x = y). The thm states that this number can also be obtained as a sum of contributions of all the fixed pts of f. www.phy.olemiss.edu /~luca/Topics/f/fixed_point.html   (352 words)

 Proceedings of the American Mathematical Society A generalization of the Lefschetz fixed point theorem and detection of chaos Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic. www.ams.org /proc/2000-128-04/S0002-9939-99-05467-2/home.html   (300 words)

 Mark Goresky's Publications A decomposition theorem for the integral homology of a variety (with J. Carrell) Inv. Lefschetz fixed point formula for intersection homology (with R. MacPherson). Lefschetz numbers of Hecke correspondences (with R. MacPherson), in The Zeta Function of Picard Modular Surfaces, R. Langlands and D. www.math.ias.edu /~goresky/publ.html   (809 words)

 Not Even Wrong » Blog Archive » Witten Localization Before Witten’s work, it was well-known among mathematicians that such calculations could in many cases be reduced to a “localized” calculation about the fixed point set of the group action. This is related to the Atiyah-Bott version of the Lefschetz fixed point theorem they discovered in the mid-sixties, to general arguments about equivariant K-theory and fixed points due to Atiyah and Segal, as well as to the Duistermaat-Heckman theorem and generalizations due to Berline and Vergne. This is sometimes referred to as “non-abelian localization” since it applies directly to non-abelian group actions, whereas the earlier fixed point formulas typically looked at the fixed points of actions by abelian groups. www.math.columbia.edu /~woit/wordpress/?p=116   (572 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us