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Topic: First homotopy group

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	Fundamental group - Wikipedia, the free encyclopedia
		Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.
		For example, the fundamental group of a graph G is a free group.
		The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle.
	en.wikipedia.org /wiki/Fundamental_group (1138 words)

	Poincare.htm
		Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants.
		Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces.
		His first major contribution to number theory was made in 1901 with work on the Diophantine problem of finding the points with rational coordinates on a curve f(x, y) = 0, where the coefficients of f are rational numbers.
	www.cse.ohio-state.edu /~brinkmei/math/Poincare.htm (426 words)

	Foundations of Algebraic Topology
		the first homotopy group for a sphere is the trivial group
		The chain groups of a polyhedron are not very interesting, being isomorphic to n-tuples of the coefficient group, where n is the number of simplexes of a particular dimension.
		The group of boundaries is a subgroup of the group of cyles.
	www2.sjsu.edu /faculty/watkins/algtop.htm (1167 words)

	Anyon - Wikipedia, the free encyclopedia
		In mathematics and physics, an anyon is a type of projective representation of a Lie group.
		The topological reason behind the phenomenon is this: the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is Z (infinite cyclic).
		The relevant part here is that the spatial rotation group is SO(2), which has an infinite first homotopy group.
	en.wikipedia.org /wiki/Anyon (388 words)

	PlanetMath: fundamental group
		a group structure and we define the fundamental group of
		The fundamental group of a topological space is an example of a homotopy group.
		This is version 8 of fundamental group, born on 2001-11-14, modified 2006-04-03.
	planetmath.org /encyclopedia/FundamentalGroup.html (162 words)

	The Knot Group (Site not responding. Last check: 2007-10-23)
		The set of equivalence classes under loop multiplication is called the fundamental (or first homotopy) group of M. Let M be the complement of a knot K in the three-sphere.
		The fundamental group of M is called the knot group of K. Obviously this group does not depend on any particular projection of a knot and is thus a knot invariant.
		This group is a module of the group of the trefoil, but not of the group of the trivial knot.
	www.inst.bnl.gov /~wei/group.html (441 words)

	Homotopy (Site not responding. Last check: 2007-10-23)
		An outstanding of homotopy is the definition of homotopy and cohomotopy groups important invariants in algebraic topology.
		Clearly every homeomorphism is a homotopy equivalence but the is not true: a solid disk is homeomorphic to a single point.
		A typical homotopy invariant is the fundamental group of a space already mentioned earlier.
	www.freeglossary.com /Homotopy (917 words)

	Geometry, topology and homotopy
		Homotopy arguments have led to some of the deepest theorems in all mathematics, particularly in the algebraic classification of topological spaces and in the solution of extension and lifting problems.
		Homotopy yields algebraic invariants for a topological space, the homotopy groups, which consist of homotopy classes of maps from spheres to the space.
		Continuous maps between spaces induce group homomorphisms between their homotopy groups; moreover, homotopic spaces have isomorphic groups and homotopic maps induce the same group homomorphisms.
	www.ma.umist.ac.uk /kd/ma351/node1.html (721 words)

	[No title]
		Well, cohomology only depends on homotopy type and in that category, a space is "determined by" its homotopy groups, so we ought to ask how the individual homotopy groups affect cohomology.
		I never understood the deep significance of group cohomology until I realized one could break it down into two independent steps: 1) To understand a group G, it's good to turn it into a space K(G,1): the space having fundamental group is G and vanishing higher homotopy groups.
		Right, it's sorta sneaky: a map between spaces that induces an isomorphism on homotopy groups must be a homotopy equivalence, but it ain't true that spaces with the same homotopy groups are homotopy equivalent.
	www.math.niu.edu /~rusin/known-math/00_incoming/group_coho (2076 words)

	[No title]
		The homotopy theoretic analogue of a compact Lie group is a p-c* ompact group, i.e a space X with finite mod-p cohomology and an loop structure g *iven by an equivalence of the form X ' BX.
		The motivation of this definition comes from the fact th* at, for a compact connected Lie group* G the maximal torus is self centralizing, and that therefore the centralizer of the maximal torus of a nonconnected compact L* ie group* is always a p-toral group.
		Hence ss2(BA) is a free Z^p-module and A a p-compact toral group.
	www.math.purdue.edu /research/atopology/Moller-Notbohm/centers.txt (12228 words)

	Science-tician? Poincaré (Site not responding. Last check: 2007-10-23)
		Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.
		For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work.
		He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof.
	www.francesfarmersrevenge.com /stuff/science/poincare.htm (2916 words)

	Licenciatura en Matemáticas
		The first chapter considers Hermite´s interpolation, together with the convergence and minimization of the interpolation error.
		The first one completes some of the topological concepts not studied in previous subjects (separation axiomes and numberability, included in the Tietze´s extension theorem and the Urysolin´s theorem.
		The factorial function (Gamma) as a solution for a equation in linear difference, of first degree and variable coefficients.
	www.ual.es /Universidad/relint/ECTSMathemat.htm (2038 words)

	[No title]
		\end{lemma} \sp{\bf Proof.} First remark that the map $$ \psi^\beta_m: \U_m \to \M_m^\beta \ \, \ \ \psi_m^\beta (\beta_0,\tau,0) = (\beta,\tau, \rho_m(\beta)) $$ (where we used adapted coordinates based at $m$) is smooth from a neighbourhood $\U_m$ of $m$ in $\Lambda$ to a neighbourhood of $\sigma_\beta(m)$ in $\Lambda_\beta$.
		Let $X_\a$ be the vector field $X_\a = \sum_i c_i Y_i$ having periodic orbits $\gamma_\a$ on $\La$ with period one and in the homotopy class $\a \in \Z^s$.
		Fix $\alpha\in\pi_1(\Lambda)$; first of all we want to determine the vector field $X_\alpha$, or, more precisely, its linearization $Y_\alpha$ about the torus.
	www.ma.utexas.edu /mp_arc/papers/01-426 (4056 words)

	The fundamental group
		Now an interesting group will be constructed from the space of paths and the equivalence relation obtained by homotopy.
		In the last case, the fundamental group does not indicate that there is a ``twist'' in the space.
		An alternative to basing groups on homotopy is to derive them using homology, which is based on the structure of cell complexes instead of homotopy mappings.
	msl.cs.uiuc.edu /planning/node143.html (910 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		Informally, the fundamental (or first homotopy) group gives an account of the nature of ``holes'' in a topological space.
		There have therefore been attempts to adapt tools originating from algebraic topology, such as homotopy groups, to theories of job scheduling and of distributed computing.
		As will be shown, the transition from tuples of cooperating processes to their homotopy cpo-s is functorial, preserves information about the ``holes'' and abstracts from inessential details.
	www.cis.ksu.edu /Department/Seminars/stefan.html (280 words)

	Universität Osnabrück - Institut für Mathematik - Algebraic Topology/Differential Topology
		In such cases the algebraic structures in the spaces are handed down to those structures that live on the homology and homotopy groups of these spaces.
		Many constructions in topology can be interpreted as (homotopy) limites or colimites of topological diagrams so that both a unifying handling as well as, per conclusions by analogy, an expansion of the results are possible.
		Based on the works of Drinfeld Tuarev identified a connection between quantum groups, the theory of knots, braids and ribbons which in turn is in relationship to the theory of monoidal categories with braid and ribbon structures.
	www.mathematik.uni-osnabrueck.de /en/1750.htm (617 words)

	[No title]
		The first subsection is concerned with the suspension spec* trum S[BU] = 1 (BU+) of the classifying space for the infinite unitary group, and its interpretation * (follow- ing [36]) as a Hopf algebra object in the category of spectra.
		Galhot of group objects of some sort, over Q. 2.2.2 In arithmetic geometry there is currently great interest in a groupscheme Galmot= Foddn Gm which is conjectured to be (isomorphic to) the motivic Galois group of a certain Tannakian category, that of mixed Tate motives over Z [10, 12].
		The composition (the first arrow is the projection of the H-spa* *ce splitting, and the second corresponds to the abelianization of a graded free Lie algebra).
	hopf.math.purdue.edu /Morava/Rosendal.txt (6109 words)

	Physics 230 Course Information
		First term: Introduction to supersymmetry, including the minimal supersymmetric extension of the standard model, supersymmetric grand unified theories, extended supersymmetry, supergravity, and supersymmetric theories in higher dimensions.
		The braid group and its one-dimensional unitary representations.
		Magnetic charge as an element of the first homotopy group of the (unbroken) gauge group.
	theory.caltech.edu /people/preskill/ph230 (1525 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		Given a convex, bounded domain R in R^n, there is an intimate relationship between a conformal change of Riemannian metric on the domain and the change in the Dirichlet to Neumann map for the Laplacian acting on functions.
		Consider two concrete manifestations of this: First, there is a uniqueness result for the inverse problem of Electrical Impedance Tomography (EIT), which states that knowledge of the Dirichlet to Neumann (i.e.
		first homotopy) group in terms of generators and relations.
	www.math.unc.edu /Faculty/jds/S99 (642 words)

	UR Math: Topology group
		Algebraic topology specifically problems related to homotopy groups of spheres, the Adams-Novikov spectral sequence, and its connections with number theory.
		Homotopy theory, cohomology of groups, group actions on spaces and connections between group theory and algebraic topology.
		Stable homotopy groups, stunted projective spaces, the Adams Spectral Sequence and the root invariant (also known as the Mahowald invariant).
	www.math.rochester.edu /research/topology (216 words)

	Algebraic Topology (Site not responding. Last check: 2007-10-23)
		In the first half of this century many mathematicians defined homology for more and more extended classes of topological spaces.
		Thus, for instance singular homology was first defined by Lefschetz in 1933.
		Thus, higher homotopy groups were defined by Hurewicz in 1935 and their properties were developed.
	www.maths.lth.se /matematiklu/personal/jaak/Alg-Top.html (324 words)

	The Fundamental Group
		Map the first half of the circle onto the first loop, and the second half of the circle onto the second loop.
		In the first expression, f and g are images of the first and second quarter of the circle, and h is the image of the second half of the circle.
		In the second expression, g and h are images of the third and fourth quarter of the circle, and f is the image of the first half of the circle.
	www.mathreference.com /at,fung.html (972 words)

	Existence of a Topological Group Structure (Site not responding. Last check: 2007-10-23)
		For example, we know that the 2-dimensional torus is the product of two copies of the unit circle of complex numbers and, hence, is a topological group.
		It is a fact that the first homotopy group of any topological group is commutative.
		Every connected polyhedron which can be given the structure of a topological group must have Euler characteristic 0.
	www.ualberta.ca /dept/math/gauss/fcm/topology/AlgbrcTop/TplgclGrpStrctr.htm (156 words)

	Search Results for Poincare
		Following his first trip, Poincare wrote two memoirs, one on the exploitation of the coal mines of Staatsbahn in Hungary, and the other on the metallurgy of tin in Banat in eastern Europe.
		First Poincare stated the well known fact that in a gas explosion, miners were burned upstream in the air flow while those downstream suffocated.
		R L Gomes, Ruy Luis On the first centenary of the birth of Henri Poincare (Portuguese), Gaz.
	www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Poincare&CONTEXT=1 (3428 words)

	Topology - Abstract Shape
		It is straightforward to show that the notion of homotopy is an equivalence relation (symmetric, reflexive and transitive) upon the set of all continuous functions from X to Y. More importantly, it is straightforward to show that homotopies are preserved under composition of functions.
		The group of equivalence classes of loops based at o in X is called the Fundamental Group of X at o.
		The fundamental group of the torus (doughnut or inner tube) is Z * Z, the free Abelian group on two generators.
	ourworld.cs.com /jamessfreeman16/Topology.htm (2436 words)

	The SocioWeb: Sociology Books » Handbook of K-Theory, 2 volume set
		The author shows various ways in which spectral sequences can be constructed, such as the use of long exact sequences in homotopy theory and by using filtrations of a spectrum (such as the familiar Postnikov tower of a space).
		This approach reflects the well-known strategy of studying the behavior of groups by relating them to the homotopy of a particular space (the mathematician Daniel Quillen used this idea to arrive at his definition of the higher K-groups).
		The advantage of A(X) according to Rosenberg is that there is essentially a linear map from it to the K-group of the first homotopy group ring, which in some cases is an equivalence.
	www.socioweb.com /sociology-books/book/354023019X (930 words)

	Blogger: Email Post to a Friend
		The resulting orbifold is a smooth Calabi-Yau manifold whose first homotopy group is "Z_3 x Z_3".
		This discrete group implies that the degeneracies typically look 9 times bigger on the covering space than what you get after you orbifold the covering space.
		It's been kind of fun when he showed the long sequences where you first calculate a whole rectangle of things that you don't really need, and only at the very end, you deduce the entry in the middle of the rectangle that you're interested in.
	www.blogger.com /email-post.g?blogID=8666091&postID=112976004171158116 (2240 words)

	Vignettes on automorphic and modular forms, representations, L-functions, and number theory
		We prove that spaces of square-integrable cuspforms on reductive groups (best adelized) decompose discretely, with finite multiplicities, by proving that the operators naturally induced by test functions on the group are compact.
		We want to prove that the singular homology of quotients X/Gamma is the group homology of Gamma, under some mild conditions on X (such as that X be a ball).
		We recap work of Hopf, Hurewicz, Eilenberg-MacLane: homology of spaces with vanishing higher homotopy is determined by the first homotopy group, giving a functor on groups.
	www.math.umn.edu /~garrett/m/v (1327 words)

	HOW THE PETRI-NETS ARE DERIVED FROM TREE-GENERATED ORDER . CONCEQUENCES TO THEIR IMPLEMENTATION .
		A homotopy transformation of a path is a continuous transformation of it that fixes the end points and is realizable in a continuous way inside the topological space.
		One is purely algebraic and associates flow-charts to groups and makes use of the fact that any group is the quotient of a free group.
		A universal covering tree of a graph G, covers any other covering graph of G. Remark 15 The flow-charts of programs are finite connected graphs with two pointed vertices called the beginning and the end and with all their branches being oriented.
	www.softlab.ntua.gr /~kyritsis/PapersInComputerScience/Hrm98Stf.htm (1683 words)

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