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Topic: Homotopy

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	Homotopy - Wikipedia, the free encyclopedia
		An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
		Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence.
		Another useful property involving homotopy is the homotopy extension property, which characterises the extension of a homotopy between two functions from a subset of some set to the set itself.
	en.wikipedia.org /wiki/Homotopy (1065 words)

	Homotopy: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-09-17)
		An outstanding use of homotopy is the definition of homotopy groups[Click link for more facts about this topic] and cohomotopy groups[Click link for more facts about this topic], EHandler: no quick summary.
		In the mathematical field of topology a homeomorphism or topological isomorphism (from the greek words homeos = identical and morphe = shape) is a...
		Homotopy equivalence is important because in algebraic topology algebraic topology quick summary:
	www.absoluteastronomy.com /encyclopedia/h/ho/homotopy.htm (2217 words)

	Homotopy lifting property - Wikipedia, the free encyclopedia
		In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as the right lifting property) is a technical condition on a continuous function from a topological space E to another one, B.
		The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.
		This is the definition of fibration in the sense of Hurewicz, which is more restrictive than the fibration in the sense of Serre, for which homotopy lifting only for X a CW complex is required.
	en.wikipedia.org /wiki/Homotopy_lifting_property (343 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		Journal of Homotopy and Related Structures (E-ISSN 1512-2891) is the all-electronic refereed journal on homotopy and in the broad sense on all related structures of mathematical and physical sciences.
		Homotopy is a basic discipline of mathematics having fundamental and various applications to important fields of mathematics.
		Diverse algebraic, geometric, topological and categorical structures are closely related to homotopy and the influence of homotopy is found in many fundamental areas of mathematics such as general algebra, algebraic topology, algebraic geometry, category theory, differential geometry, computer science, K-theory, functional analysis, Galois theory and in physical sciences as well.
	www.emis.de /journals/JHRS/about.htm (253 words)

	PlanetMath: homotopy groups
		This is continuous because we required that the boundary go to a fixed point, and well defined up to homotopy.
		Homotopy groups are invariant under homotopy equivalence, and higher homotopy groups (
		This is version 9 of homotopy groups, born on 2002-02-02, modified 2003-08-13.
	planetmath.org /encyclopedia/HomotopyGroups.html (217 words)

	[No title]
		Definition 0.4.A generalized homotopy representation A is a finitely dominated based G-CW complex such that, for each subgroup H of G, AH is homotopy equiv- alent to a sphere Sn(H).
		A stable homotopy representation is a G-spectrum of the form -V 1 A, where V is a representation of G and A is a generalized homotopy representation.
		For any generalized homotopy repre- sentation A, there is a representation V such that A^SV is equivalent to a homo* topy representation B. Therefore every element of Pic(HoGS) can be represented as -W 1 B for some homotopy* representation B and representation W.
	hopf.math.purdue.edu /Fausk-Lewis-May/FLMApril20.txt (5456 words)

	Algebraic Topology: Homotopy
		A homotopy F from X × I to Y is called homotopy relative to A if for each a in A the map F(a,t) is constant (independent of t).
		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)

	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.
		For instance this has been used to show in [3] that every finite loop space is homotopy equivalent to a smooth parallelizable manifold.
	www.lehigh.edu /~dlj0/Grodal-problemsession.html (694 words)

	Homotopy Equivalence - Homotopy Type
		Homotopy type is a much weaker topological invariant than topological type, the class of a space under topological equivalence.
		The homotopy type of a space is easier to determine than its homeomorphism type.
		The study of homotopy types is the main theme of ``homotopy theory''.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node13.html (497 words)

	Geometry, topology and homotopy
		Following the idea of continuity, the fundamental concept in topology is that of homotopy, for spaces and maps; we do not need homotopy theory for this course but it is so important in pure mathematics and you can understand what it is about quite easily through some examples.
		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.
	www.ma.umist.ac.uk /kd/ma351/node1.html (721 words)

	New Contexts for Stable Homotopy Theory
		Stable homotopy theory is the ultimate context in which to perform the type of conversion from geometrical to algebraic data which Poincaré began.
		This exploitation of motivic homotopy theory in number theory and algebraic topology was spurred on by the programme, which also played an important role in spreading the developing body of knowledge.
		The subject of stable homotopy theory has been transformed in the last ten years by key technical advances making distant dreams into reality, but the fact that its methods have also been used in recent spectacular progress in motivic homotopy theory was a quite separate development.
	www.newton.cam.ac.uk /reports/0203/nst.html (2154 words)

	Homotopy of Maps
		Then the homotopy defined in the previous example can be divided by its norm to give a homotopy of f to g, i.e.
		Homotopy between two maps from X to Y, or ``absolute homotopy'' is a particular case of homotopy relative to a subspace A of X.
		Understanding homotopy classes of maps between two given spaces is in some sense the most fundamental problem in homotopy theory.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node12.html (572 words)

	PlanetMath: weak homotopy equivalence
		are then said to be weakly homotopy equivalent.
		Cross-references: map, isomorphic, homotopy groups, isomorphism, topological spaces, path-connected, between, continuous map
		This is version 4 of weak homotopy equivalence, born on 2003-02-07, modified 2003-02-07.
	planetmath.org /encyclopedia/WeakEquivalence.html (77 words)

	Homotopy (Site not responding. Last check: 2007-09-17)
		In topology, two continuous functions from one topological spaceto another are called homotopic if one can be "continuously deformed" intothe other, such a deformation being called a homotopy between thetwo functions.
		This allows to define the homotopy category: the objects are topological spaces, and the morphisms are homotopy classes of continuous maps.
		However, it is the intent and use rather than the form of for they give the people authority for certain beliefs and conceptions the abode of the dead, where the spirits lead much the same sort of of this realm from the Baluga [69] tale, in which the home of.
	www.termsdefined.net /ho/homotopy.html (711 words)

	Homotopy
		Homotopy is studied in detail in Dodson and Parker [
		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)

	Homotopic Functions and Homotopy Classes
		It may even be a little misleading, since the "blobs" in a true homotopy are not arbitrary sets; they are the continuous images of some other space r.
		The homotopy classes are the path connected components of s.
		A homotopy contracts t linearly onto its center, hence the point and the ball are homotopically equivalent.
	www.mathreference.com /at,topy.html (1515 words)

	Complex Cobordism and Stable Homotopy Groups of Spheres
		The history of computing homotopy groups is illustrated by a brief discussion of the Cartan-Serre method of killing homotopy groups and of its descendent, the classical Adams spectral sequence.
		In this case and in Bott's computation of the homotopy of $b\text{o}$, the $E\sb 2$ term is rather nice and the spectral sequence collapses.
		The computations for the homotopy of spheres are more difficult and useful techniques such as the May spectral sequence and the lambda algebra are introduced.
	www.math.rochester.edu /people/faculty/doug/mu.html (1161 words)

	Homotopy Methods for Solving Polynomial Systems, ISSAC 2005
		Homotopy continuation methods [Li, 2003] provide symbolic-numeric algorithms to solve polynomial systems.
		The homotopy connects the system we want to solve with a system which is easier to solve.
		The performance of the homotopy continuation solver is primarily determined by the selection of a homotopy which matches best the structure of the given polynomial system.
	www.math.uic.edu /~jan/issac05.html (393 words)

	Motivic Homotopy Theory Program (Site not responding. Last check: 2007-09-17)
		The motivic homotopy theory is the homotopy theory for algebraic varieties and, more generally, for Grothendieck's schemes which is based on the analogy between the affine line and the unit interval.
		Eventually, the motivic homotopy theory is expected to provide techniques which may help to solve problems in algebraic geomerty such as various "standard conjectures on algebraic cycles", Beilinson-Soule vanishing and rigidity conjectures, the Bloch-Kato conjecture etc.
		The triangulated categories of motives and the motivic stable homotopy categories are connected by pairs of adjoint functors.
	www.math.ias.edu /~vladimir/seminar.html (870 words)

	Problems on axiomatic stable homotopy theory
		In our memoir, we give a conjecture for the thick subcategories in a Noetherian stable homotopy category C--they should be in 1-1 correpondence with subsets of Spec pi_* S closed under specialization.
		This has been done for categories that are homotopy categories of model categories by Schwede, I think, and a lot is known in general.
		Suppose G is a self-equivalence of the stable homotopy category.
	claude.math.wesleyan.edu /~mhovey/problems/axiomatic.html (681 words)

	Modelling : Homotopy
		As a first argument, the previous figure shows a typical failure of Newton-Raphson's iteration and next figure the behaviour of homotopy in the same problem: the six circles must be tangent each other, and must be tangent to the triangle.
		On the contrary, homotopy converges much more often to the solution intuitively closest to the initial guess: bassins for homotopy are semi-algebraic sets (when the system to be solved is algebraic).
		Climbing complex homotopy: this method is relevant only in the field of complex numbers.
	www.emse.fr /~roelens/lisse/FRENCH/MODELISATION/homotopy.html (695 words)

	Fields Institute - Homotopy Theory Program 1995-96
		During 1995-96 the Fields Institute is sponsoring an emphasis year in homotopy theory.
		What follows is information concerning various aspects of the homotopy program which will be offered after the current January stable homotopy emphasis session ends.
		The major activities will be workshops in unstable homotopy theory during the week of May 27-31 and in rational homotopy theory during the week of June 3-7.
	www.fields.utoronto.ca /programs/scientific/95-96/homotopy (360 words)

	ABSTRACT HOMOTOPY AND SIMPLE HOMOTOPY THEORY
		The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g.
		Studying categorical versions of homotopy structure, such as cylinders and path space constructions, enables not only a unified development of many examples of known homotopy theories but also reveals the inner working of the classical spatial theory.
		This demonstrates the logical interdependence of properties (in particular the existence of certain Kan fillers in associated cubical sets) and results (Puppe sequences, Vogt's Iemma, Dold's theorem on fibre homotopy equivalences, and homotopy coherence theory).
	www.worldscibooks.com /mathematics/2215.html (196 words)

	Graduiertenkolleg Homotopy and Cohomology
		The strategy of Homotopy Theory is to establish this invariance for as many properties as possible, and then exploit the invariance to obtain a classification.
		As the direct approach is typically prohibitively difficult, one uses cohomology to calculate these homotopy groups by indirect means.
		The areas of Homotopy Theory in question are Stable Homotopy Theory and Elliptic Cohomology.
	www.math.uni-bonn.de /people/GRK1150 (413 words)

	55Q: Homotopy groups
		The "homotopy continuation methods" in numerical analysis and control are essentially unrelated to homotopy theory (but rather are more akin to analytic continuation in complex analysis.) One is, at best, using a linear homotopy between two constant maps into M_n(R).
		Tables of the homotopy groups of spheres [Hatcher].
		What is Bott periodicity (homotopy groups of SO(n) and related topics).
	www.math.niu.edu /~rusin/known-math/index/55QXX.html (247 words)

	Homotopy Theory - The MIT Press
		Presupposing a knowledge of the fundamental group and of algebraic topology as far as singular theory, it is designed to introduce the student to some of the more important concepts of homotopy theory.
		The book emphasizes (relative) CW-complexes, which the author believes to be the natural setting for obstruction theory, and follows the spirit of J. Whitehead's "combinatorial homotopy."
		Homotopy Theory will prove valuable to first- and second-year graduate students of mathematics and to mathematicians interested in this unique treatment of the subject.
	mitpress.mit.edu /catalog/item?ttype=2&tid=5294 (103 words)

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