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Topic: Homotopy class

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	Characteristic class - Wikipedia, the free encyclopedia
		Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance.
		When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the Stiefel-Whitney class, the Chern class, and the Pontryagin classes) were reflections of the classical linear groups and their maximal torus structure.
		Characteristic classes were later found for foliations of manifolds; they have (in a modified sense, for foliations with some allowed singularities) a classifying space theory in homotopy theory.
	en.wikipedia.org /wiki/Characteristic_class (755 words)

	Homotopy group - Wikipedia, the free encyclopedia
		In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
		Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but the converse is not true.
		In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere.
	en.wikipedia.org /wiki/Homotopy_group (778 words)

	Equivalence class - Wikipedia, the free encyclopedia
		The equivalence classes are known as right cosets of H in G; one of them is H itself.
		The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.
		In natural language processing, an equivalence class is a set of all references to a single person, place, thing, or event, either real or conceptual.
	en.wikipedia.org /wiki/Equivalence_class (866 words)

	Regular homotopy - Wikipedia, the free encyclopedia
		In particular, the homotopy must go through immersions and extend continuously to a homotopy of the tangent bundle.
		Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them.
		The Whitney-Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if their Gauss maps have the same winding number.
	en.wikipedia.org /wiki/Regular_homotopy (197 words)

	The Homotopy Classes of S1 into s
		In the context of the fundamental group, a homotopy, and the resulting homotopy classes, are quite constrained.
		The loop classes at c are the loop classes for all of s.
		The set of group elements that are conjugates of f is called the conjugacy class of f, and the homotopy classes of s are the conjugacy classes of π
	www.mathreference.com /at,hclass.html (1140 words)

	PlanetMath: fundamental groupoid
		morphisms are homotopy classes of paths “rel endpoints” that is
		It is easily checked that the above defined category is indeed a groupoid with the inverse of (a morphism represented by) a path being (the homotopy class of) the “reverse” path.
		This functor is not really homotopy invariant but it is “homotopy invariant up to homotopy” in the sense that the following holds.
	planetmath.org /encyclopedia/FundamentalGroupoid.html (203 words)

	[No title]
		Assuming that C is both left and right proper, the class of cofibrations having obstruction theories is the smallest class of cofibrations containing all cofibrations with weakly contractible target and closed under retract, weak equivalence, and cobase change.
		The class of cofibrations that have an obstruction theory (resp., fibrant obstruction theory, cofibrant fibrant obstruction theory) is closed under re- tract.
		This class coincides with the class of cofibra- tions that have a fibrant obstruction theory.
	jdc.math.uwo.ca /papers/obstruction.txt (7257 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-30)
		Thus, a homotopy of mappings is a specialization to the space of mappings of the general concept of "being connected by a continuous path" .
		This description of a homotopy is sometimes qualified as free, in distinction from "relative homotopyrelative" or "bound homotopybound" homotopies, which arise upon fixing a class
		This close connection between the problem of homotopy and the problem of extension is the reason why they are considered together under the heading of so-called homotopy theory.
	eom.springer.de /H/h047920.htm (453 words)

	Manifold - Wikipedia, the free encyclopedia
		A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps.
		Indeed several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds.
		However, they are of central interest in algebraic topology, especially in homotopy theory, where such dimensional defects are acceptable.
	en.wikipedia.org /wiki/Manifold (5766 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-30)
		One of the generalizations of the one-dimensional cohomology group; a concept which is, in a certain sense, dual to that of homotopy group.
		The value at a point of this theory is the same as the stable homotopy groups of spheres.
		As for homotopy groups, the cohomotopy groups cannot be explicitly calculated even in the simplest cases, and this severely restricts the possibility of practical application of the above functors.
	eom.springer.de /c/c023180.htm (371 words)

	[No title]
		The homotopy fibre F of f is an A-cellular spac* *e.
		We have considered C.(A) the smallest class of po* inted spaces closed under arbitrary pointed_hocolim, and homotopy* equivalences, which contai* ns the space A. Notice that if we consider classes* closed under arbitrary non-pointed* _hocolim we get only two classes* the empty class and the class of all unpointed space.
		A-Homotopy Theory and the Construction of CWA X In this section we describe some initial elements of A-homotopy theory.
	hopf.math.purdue.edu /Farjoun/cellularspaces.txt (8901 words)

	Algebraic Topology: Homotopy
		Given two spaces X,Y, and a map f from X to Y, let [f] denote the homotopy class of f, that is, the set of all maps from X to Y homotopic to f.
		Under suitable assumptions homotopy classes are precisely the path-components in the space C(X,Y) of continuous functions from X to Y.
		It follows that if f is a homotopy equivalence from X to Y (that is, a map such that there is a map g from Y to X such that both fg and gf are nullhomotopic), then f* is an isomorphism.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	PlanetMath: K-theory
		group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the
		is the homotopy class of their direct sum:
		Alternatively, one can consider equivalence classes of projections up to unitary transformations.
	planetmath.org /encyclopedia/KTheory.html (197 words)

	Homotopic Functions and Homotopy Classes
		The homotopy classes are the path connected components of s.
		Its one and only homotopy class is represented by the trivial function from d onto the center of the star.
		In any case, the homotopy classes of the circle are the homotopy classes of the punctured plane.
	www.mathreference.com /at,topy.html (1693 words)

	[No title]
		It is clear that a trivial projective class is strong, and that the pullback of a strong projective class is strong.
		The cycles, boundaries and homology classes in A(P, X) are called P -cycles, P -boundaries and P -homology classes in X. Note that a P -cycle is the same as a P -element of Zn X, but that not every P -element of Bn X is necessarily a P -boundary.
		We are concerned with two projective classes on the category A. The first is the categori- cal projective class C whose projectives are summands of free modules, whose exact sequences are the usual exact sequences, and whose epimorphisms are the surjections.
	jdc.math.uwo.ca /papers/relative.txt (10317 words)

	[No title]
		In the jargon of homotopy theory, we say two functions from some space to some other space are "homotopic" if we can continuously deform the first function to the second.
		Second, there is a nice formula for when the homotopy groups settle down as we jack up the dimension: pi_{k+n}(S^k) is independent of k as long as k >= n+2 The homotopy groups can stabilize sooner, as we saw for n = 2, but never later, and often they stabilize right at k = n+2.
		The relationship between homotopy groups of spheres and higher- dimensional knot theory is a wonderful thing.
	www.math.niu.edu /~rusin/known-math/97/homotopy.spheres (2758 words)

	Math 8306-07: Class Outlines
		1/31/05: Fibrations: the homotopy lifting property (HLP), the pullback of a fibration is a fibration, a covering space is a fibration with a unique homotopy lifting property (HLP), the dual of a cofibration is a fibration.
		Attempting to use the barycentric subdivision to show the subcomplex is homotopy equivalent to the singular chain complex.
		Brush-up on using the algebraic homotopy to see that the singular homology of a convex set is trivial in all degrees but zero, in which the homology is G. [Hatcher: pp.
	www.math.umn.edu /~voronov/8306/outline.html (1288 words)

	Brian White's Research Papers
		Let F be the set of maps g that are homotopic to f under lipschitz homotopies H:[0,1] x M --> X such that H(t,x)=f(x) for x in the boundary of M.
		on the homotopy class of the restriction of f to the [p]-dimensional skeleton of (some triangulation of) M.
		Homotopy classes in sobolev spaces and the existence of energy minimizing maps
	math.stanford.edu /~white/biblio.htm (3414 words)

	Math 215b Home Page
		Vector bundles modulo isomorphism over (a reasonable space) B are the same as homotopy classes of maps from B to the classifying space (the infinite Grassmannian).
		Introduced the Chern classes of a complex vector bundle, their basic properties, and their interpetation in terms of obstructions when the base is a CW complex.
		Showed that (in the category of smooth compact oriented manifolds without boundary) the Thom class of the normal bundle (which is diffeomorphic to a tubular neighborhood) of a submanifold is Poincare dual to the fundamental class of the submanifold.
	math.berkeley.edu /~hutching/teach/215b-2004 (2010 words)

	Polytime Algorithm for the Shortest Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane - Grigoriev, ...
		We use the representation of homotopy classes in a way that is as general as the classical one.
		It consists in representing generators of a free group which describes the classes of homotopy by disjoint cuts [GS97] homeomorphic to...
		Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane.
	citeseer.ist.psu.edu /325062.html (609 words)

	week102
		So what we mean is that there's only one homotopy class of ways to map a sphere to a sphere of higher dimension.
		There is a beautiful way to compute an integer called the "Hopf invariant" that keeps track of the homotopy class of a map from the 3-sphere to the 2-sphere.
		Using this correspondence, the "free k-tuply monoidal n-groupoid on one object" corresponds to the homotopy (k+n)-type of the k-sphere.
	math.ucr.edu /home/baez/week102.html (3281 words)

	Polytime Algorithm for the Shortest Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane
		It consists in representing generators of a free group which describes the classes of homotopy by disjoint cuts homeomorphic to rays.
		We show that given such a system of generators and a word representing a homotopy class, one can contruct the shortest path of this class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts.
		The homotopy class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path.
	www.univ-paris12.fr /lacl/lundi/slissenko.html (235 words)

	Abstracts
		homotopy class of closed loops in M the length of a shortest geodesic
		Farb and Mosher have developed a notion of convex co-compactness for subgroups of the mapping class group of a surface analogous to the notion of the same name in the theory of Kleinian groups.
		The new construction provides a broader class of groups exhibiting the same behavior, as well as new geometric information about the action of these groups on Harvey's complex of curves.
	www.math.utah.edu /~bromberg/wtc/Abstracts.html (753 words)

	[No title]
		The class is shown to be related to the Z2-equivariant Chern class on Atiyah's KR-t* *heory.
		The class *0(2q) is "primitive", i.e., it is of the form ae P q0-1 2q 1.
		This gives a characteristic class for real bundles which is a mixture of Z and Z2 classes an* *d satisfies Whitney duality.
	hopf.math.purdue.edu /Lawson-Lima-Filho-Michelsohn/alg-cycles1.txt (9025 words)

	The fundamental group
		The second and probably most important feature of sets of homotopy classes of maps is their homotopy invariance.
		Now that we have a homotopy between the maps r and l we are ready to prove associativity of multiplication in the fundamental group.
		] is represented by the class of the composition
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node14.html (1607 words)

	Symbolic Dynamics and the Braid Group
		The free (unbased) homotopy classes are encoded by the order and orientation with which the projection onto the shape sphere winds around the excluded collision points.
		the homotopy class encoded by a word) remains unchanged under cyclic permutations.
		not contractible) homotopy class in the braid group must wind around the punctures on the equator.
	merganser.math.gvsu.edu /david/reed03/projects/salomne/simulator/node7.html (270 words)

	Gauge Theory and Topology Seminar (Site not responding. Last check: 2007-10-30)
		In analogy to the Pontryagin-Thom construction in the case of proper maps between finite dimensional real vector spaces, one can associate to the monopole map an equivariant stable homotopy class of maps between spheres.
		This homotopy class is a differentiable invariant of $X$.
		In fact, this stable homotopy refinement can be used to explain, why and how chamber structures naturally arise in SW-theory.
	www.maths.tcd.ie /pub/Seminars/archive.00-01/2000.12.08.GTT.html (285 words)

	Open problems in homotopy theory of Lie groups
		These objects are homotopy theoretic analogs of compact Lie groups, but with all the structure concentrated at a single prime p.
		A major open problem in the homotopy theory of Lie groups is the classification of 2-compact groups, analogous to the classification of compact Lie groups.
		It has also been used in [1] to produce an exotic finite loop space which is not rationally homotopy equivalent to any compact Lie group, resolving an old conjecture in the negative.
	www.lehigh.edu /~dlj0/Grodal-problemsession.html (694 words)

	Homotopy
		Algebraic topology involves the classification of topological spaces in terms of algebraic objects (groups, rings) that are invariant under usefully large classes of homeomorphisms.
		The spaces in the homotopy equivalence class determined by a singleton space are called contractible; we often use
		It turns out that equivalences up to homotopy are sufficient for easy proof of a wide range of important results in topology and analysis, (like fixed point theorems, extension and lifting theorems, fundamental theorem of algebra, hairy ball theorem...) as may be seen in [
	www.ma.umist.ac.uk /kd/knots/node4.html (404 words)

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