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Topic: Homotopy equivalent

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	PlanetMath: homotopy equivalence
		This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence.
		For topological spaces, homotopy equivalence is an equivalence relation.
		This is version 10 of homotopy equivalence, born on 2002-01-23, modified 2003-07-10.
	planetmath.org /encyclopedia/HomotopyEquivalence.html (101 words)

	Poincare.htm
		He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology.
		Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants.
		Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three.
	www.cse.ohio-state.edu /~brinkmei/math/Poincare.htm (426 words)

	Open problems in homotopy theory of Lie groups
		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.
		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 (Site not responding. Last check: 2007-10-31)
		Simplicial Homotopy Theory, by P.G. Goerss and J.F. Jardine...
		Homotopy and homology groups in general topology, by Katsuya Eda...
		Spin(n) IS NOT HOMOTOPY NILPOTENT FOR n 7.
	www.scienceoxygen.com /math/688.html (247 words)

	Encyclopedia: Homotopy type (Site not responding. Last check: 2007-10-31)
		In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.
		Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map id
		Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
	www.nationmaster.com /encyclopedia/Homotopy-type (1602 words)

	Homotopy Article, Homotopy Information (Site not responding. Last check: 2007-10-31)
		An outstanding use of homotopy is the definition of homotopygroups and cohomotopy groups, important invariants in algebraic topology.
		Formally, a homotopy between two continuous functions f and g from a topological space X to atopological space Y is defined to be a continuous function H : X × [0,1] → Y fromthe product of the space X with the unit interval [0,1] to Y such that, for all points x inX, H(x,0)=f(x) and H(x,1)=g(x).
		Clearly, every homeomorphism is a homotopy equivalence, but theconverse is not true: a solid disk is not homeomorphic to a single point.
	www.anoca.org /spaces/homotopic/homotopy.html (867 words)

	PlanetMath: fundamental group
		It can also be shown that two homotopically equivalent path-connected spaces have isomorphic fundamental groups.
		Homotopy groups generalize the concept of the fundamental group to higher dimensions.
		Cross-references: homotopy groups, homotopically equivalent, isomorphic, homeomorphic, topological invariant, category, category of pointed topological spaces, functor, torus, isomorphism, path-connected, isomorphic groups, structure, group, product, maps, classes, homotopy, basepoint, topological space, pointed topological space
	planetmath.org /encyclopedia/FundamentalGroup.html (254 words)

	Homotopy - Definition, explanation
		An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
		Clearly, every homeomorphism is a homotopy equivalence, but the converse is not true: a solid disk is not homeomorphic to a single point.
		A typical homotopy invariant is the fundamental group of a space, already mentioned earlier.
	www.calsky.com /lexikon/en/txt/h/ho/homotopy.php (1174 words)

	[No title]
		These incl* ude an internal description of homotopy* colimits in a cofibrantly generated model structure and a general notion of regularity, which amounts the assertion that an A-set X is a homotopy colimit of its cells.
		That homotopy theory arises in part from a "Swiss army knife" result (The- orem 46 in Section 3) which establishes a model structure for the category of A-presheaves in which some set S of monomorphisms become weak equiva- lences, and which depends on a suitable theory of intervals on the category of A-presheaves.
		LY of fibrant models is a naive homotopy equivalence, and this is equivalent to f being an (1, S)-equivalence.
	hopf.math.purdue.edu /Jardine/cat5.txt (13494 words)

	Homotopy
		is an equivalence relation on any collection of topological spaces and one sometimes speaks (loosely) of spaces in the same class as being homotopic.
		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)

	Homotopy Equivalence - Homotopy Type
		We have now introduced an equivalence relation on the class of all continuous functions, which allows us to think about two maps as the same if they can be continuously deformed into each other.
		For a given space X the equivalence class of X under homotopy equivalence is called the homotopy type of X.
		Homotopy type is a much weaker topological invariant than topological type, the class of a space under topological equivalence.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node13.html (497 words)

	Springer Online Reference Works
		expresses the theorem on the homotopy equivalence of an arbitrary topological manifold to a polyhedron.
		Of course, the existence of a lifting, for example, in (4) is a necessary condition for the existence of a PL-manifold homotopy equivalent to the Poincaré polyhedron
		homotopy equivalent to it exists if and only a lifting (4) exists.
	eom.springer.de /T/t093230.htm (888 words)

	Springer Online Reference Works
		are said to be homotopy equivalent, or spaces of the same homotopy type.
		Since any homotopy equivalence is a weak homotopy equivalence, it follows that homotopy-equivalent spaces are weakly homotopy equivalent.
		The fact that the homotopy type is completely determined by the resolution shows that any problem in homotopy theory reduces to some statement on the resolutions of the corresponding spaces.
	eom.springer.de /H/h047940.htm (1801 words)

	[No title]
		Hence there is a homotopy equivalence Yr ' m ji Sm {d} ____- S {pi }: i=1 Moreover, by naturality of the localization functors it is clear that Sm {d} is* * a loop space if and only if each factor on the right hand side of the above equation i* *s a loop space.
		The homotopy fibre of fl is easily seen to have the mod-p cohomology of an m-sphere and since it i* s p- complete, it is equivalent* to the p-completion of an m-sphere.
		Homotopy Uniqueness for the odd prime case Let p be an odd prime and let X be a classifying space for S2n-1{pr}.
	hopf.math.purdue.edu /Broto-Levi/snk.txt (8766 words)

	Surgery
		A necessary condition for this to occur is that the homotopy equivalence be simple.
		Simple or not, there is a first obstruction to further study called a normal invariant, which measures whether the homotopy equivalence is normally cobordant to the identity map on the target manifold.
		Simple or not, if the surgery obstruction for sugery up to ordinary homotopy vanishes, then the result of the coring and pasting operations on the normal cobordism is an h-cobordism between the two manifolds.
	www.albany.edu /~mark/surgery.htm (307 words)

	PlanetMath: homotopy invariance
		An important example of a homotopy invariant functor is the fundamental group
		Cross-references: groups, fundamental group, argument, isomorphic, implies, identity maps, homotopic, continuous maps, homotopy equivalent, induced, morphisms, homotopic maps, topological spaces, category, functor
		This is version 1 of homotopy invariance, born on 2004-06-15.
	planetmath.org /encyclopedia/HomotopyInvariance.html (74 words)

	[No title]
		The inverse equivalence is induced by the zeroth spac* *e functor 1.
		Null_=A are isomorphic to __=C and hence are homotopy equivalent to the contractible C. The* n by Quillen's Theorem A the inclusion is a homotopy* equivalence.
		BNull=BA oo____B=BA ______//BA This diagram shows that up to homotopy equivalence, the B of (5.2.4) is inde* ntified to the identity map of A. A fortiori is a fortiori a natural stable homotopy* eq* *uivalence.
	www.math.purdue.edu /research/atopology/Thomason/thomason_SymMon_equals_Spectra.txt (7868 words)

	Good Math, Bad Math : Homotopy
		Homotopy is a formal equivalent of homeomorphism for functions between topological spaces, rather than between the spaces themselves.
		A homotopy is a function h which associates every value in the unit interval [0,1] with a function from S to T.
		R^n is homotopy equivalent to a point though, and a mobuis band is homotopy equivalent to a circle.
	scienceblogs.com /goodmath/2007/03/homotopy_1.php (1580 words)

	Chain Equivalence (Site not responding. Last check: 2007-10-31)
		By definition, C is homotopy equivalent to D iff D is homotopy equivalent to C. The composition of two chain equivalences gives another chain equivalence.
		If C and D are homotopy equivalent via f and g, then fg produces a map that is homotopic to the identity.
		Assume C and E are homotopy equivalent, as demonstrated by the chain maps f and g.
	www.mathreference.com /mod-hom,cheq.html (288 words)

	Knowledge King - Homotopy (Site not responding. Last check: 2007-10-31)
		Given two spaces X and Y, we say they are homotopy equivalent if there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map id
		The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism h such that h moves K
		For, the homotopy classes actually form a homotopy group.
	www.knowledgeking.net /encyclopedia/h/ho/homotopy.html (639 words)

	Contractible space - tScholars.com (Site not responding. Last check: 2007-10-31)
		A contractible space is precisely one with the homotopy type of a point.
		Since a contractible space is homotopy equivalent to a point, all the homotopy groups of a contractible space are trivial.
		X is homotopy equivalent to a one-point space.
	www.tscholars.com /encyclopedia/Contractible_space (280 words)

	A uniqueness theorem for stable homotopy theory (Site not responding. Last check: 2007-10-31)
		We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of spectra.
		One sufficient condition is that the associated homotopy category is equivalent to the stable homotopy category as a triangulated category with an action of the ring of stable homotopy groups of spheres, $\pi^s$.
		In other words, the classical stable homotopy theory, with all of its higher order information, is determined by the homotopy category as a triangulated category with an action of $\pi^s$.
	www.math.uic.edu /~bshipley/unique.html (138 words)

	research
		Understanding the homotopy groups of finite complexes remains one of the central challenges of homotopy theory.
		In fact, it is a classical result that the rational homotopy groups have the structure of a graded Lie algebra and the loop space homology is a universal enveloping algebra on this Lie algebra.
		In this case the homotopy groups of the given space are completely determined by the homotopy groups of the atomic spaces.
	academic.csuohio.edu /bubenik_p/research/research.html (1079 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		Vladimir Ryazanov Institute of Applied Mathematics and Mechanics, Donetsk, Ukraine Topic: Generalized homotopy spaces Date: Thursday, May 15, 2003 Place: Room 619 of the Amado Building, the Technion Time: 10:30 AM Abstract: The theory of generalized manifolds leads to the characterization of factors for Euclidean spaces as generalized homotopy spaces.
		In all dimensions, including n=3, the generalized Poincare conjecture is equivalent to the topological characterization by Siebenmann for R^n among manifolds as proper homotopy equivalent to to R^n.
		We also prove that the one-point compactification of a generalized proper homotopy n-space is a generalized homotopy n-sphere.
	www.math.technion.ac.il /~techm/20030515103020030515rya (114 words)

	Lecture 2 (Site not responding. Last check: 2007-10-31)
		Define a retract and show that this is not the same as deformation retract since every space X retracts to a point.
		Define a homotopy relative to a subspace (or relative homotopy), homotopy equivalence, and homotopy type.
		Show that if X deformation retracts onto A (A subspace of X) then X is homotopy equivalent to A (hence homotopy equivalence is a generalization of deformation retraction).
	math.rice.edu /~shelly/18.904/lecture2.html (148 words)

	Surgery (Site not responding. Last check: 2007-10-31)
		For any of these conditions to hold, it is a necessary condition that the two manifolds be homotopy equivalent, so a surgery problem starts with a homotopy equivalence between them.
		One may then ask whether the homotopy equivalence in question could be homotopic to a homeomorphism or diffeomorphism.
		If the original homotopy equivalence was simple and if the simple homotopy surgery obstruction vanishes, then the homotopy equivalence is homotopic to a homeomorphism (or diffeomorphism, if the normal cobordism is smooth).
	math.albany.edu /~mark/surgery.htm (307 words)

	[No title]
		Date: Fri, 2 Dec 2005 16:43:13 -0600 (CST) Concerning the discussion of homotopy equivalence between BG and BG': It has been known for a while that two compact (but not necessarily connected) Lie groups G and G' are isomorphic if and only if BG is homotopy equivalent to BG'.
		Beware that compact Lie groups can be equivalent at all primes without being isomorphic (the first examples of this kind were pointed out by Dietrich Notbohm and Larry Smith, I think).
		Also beware that in general whether BG^_p and BG^_p are homotopy equivalent cannot be determined just by looking at cohomology together with Steenrod operations, though this does happen in certain cases.
	www.lehigh.edu /dmd1/public/www-data/jg123 (304 words)

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