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Topic: Homotopy groups

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Fundamental group

Algebraic topology

Topological space

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Diophantine equation

	Homotopy - Wikipedia, the free encyclopedia
		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.
		An example of a homotopy invariant is the fundamental group of a space, already mentioned earlier.
	en.wikipedia.org /wiki/Homotopy (1019 words)

	Homotopy groups of spheres - Wikipedia, the free encyclopedia
		In mathematics, the homotopy groups of spheres are the groups
		The case k = n is always the infinite cyclic group of the integers Z by a theorem of Heinz Hopf, with mappings classified by their degree.
		As the homotopy groups of spheres turn out to be very difficult to compute, algebraic topologists searched for ways to simplify the problem.
	en.wikipedia.org /wiki/Homotopy_groups_of_spheres (342 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 (215 words)

	Homotopy group (Site not responding. Last check: 2007-11-06)
		In mathematics, homotopy groups are defined in algebraic topology, to classify homotopy classes of mappings of spheres into some topological space.
		In particular, the equivalence clases are given by homotopies that are constant on the basepoint of the sphere.
		The idea of composition in the fundamental group is that of following the first path and the second in succession, or, equivalently, setting their two domains together.
	www.sciencedaily.com /encyclopedia/homotopy_group (505 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Geometrically, this is formulated as an equivalence statement between the algebraic K-theory spectrum of the field and the so-called ``homotopy fixed set'' of the action of the absolute Galois group on the algebraic K-theory spectrum of the algebraic closure.
		However, the counterexamples are all in low dimensions, and the conjecture can be formulated as the statement that the homotopy groups of the homotopy fixed set described above agree with the K-groups of the field in high dimensions, where "high" means higher than the cohomological dimension of the absolute Galois group of the field.
		Equivariant stable homotopy theory has been a key ingredient in the description of various homotopy fixed point spectra under finite group actions, in particular in the Atiyah-Segal completion theorem and in the affirmative resolution of Segal's Burnside ring conjecture.
	www.aimath.org /projects/equivar.html (1330 words)

	Homotopy groups of spheres -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the homotopy groups of spheres are the groups
		For example, if n ≥ 2 the n-sphere is both (Click link for more info and facts about connected) connected and (Click link for more info and facts about simply connected) simply connected.
		It is the case k > n that is of real importance, and historically it came as a great surprise that the corresponding homotopy groups could be nontrivial (in stark contrast to the behavior expected from results in (Click link for more info and facts about homology theory) homology theory).
	www.absoluteastronomy.com /encyclopedia/h/ho/homotopy_groups_of_spheres.htm (374 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].
		Fundamental group of the space of all unlabeled orthogonal frames in R^3.
	www.math.niu.edu /~rusin/known-math/index/55QXX.html (247 words)

	Complex Cobordism and Stable Homotopy Groups of Spheres (Site not responding. Last check: 2007-11-06)
		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.
		The applications of this and related techniques to the existence of infinite families of elements in the stable homotopy groups of spheres are then indicated.
		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.
	www.math.rochester.edu /people/faculty/drav/mu.html (1161 words)

	UM Mathematics: Faculty-Detail
		The first examples of such invariants are homotopy groups, which are groups of homotopy classes of maps from a sphere into a space.
		While homotopy groups are easy to define, they are usually very difficult to calculate and other invariants must be studied in the process.
		Recently, I wrote a paper on asymptotic estimates of families of elements in the stable homotopy groups of spheres.
	www.math.lsa.umich.edu /people/facultyDetail.php?id=85 (357 words)

	Gargnano 1999: List of talks
		Then the set of homotopy classes of orbit spaces of all free $G$-actions on all $(2n-1)$-homotopy spheres is finite and bounded by the order of the quotient group $\mbox{Aut}(Z/G)/\{\varphi^\ast;\varphi\in\mbox{Aut}G\}$, where $\mbox{Aut}$ is the automorphism group and $\var\phi^\ast$ is the induced automorphism on the cohomology $H^{2n}(G,Z)=Z/G$ by an automorphism $\varphi$ in the group $\mbox{Aut}G$.
		In this talk, however, we will show that the subgroup of elements which induce the identity on homotopy groups is a genus invariant in some cases.
		Abstract: Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S^2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension.
	www.matapp.unimib.it /convegni/garg99/talks.html (1315 words)

	v1-periodic homotopy groups of SO(n) (Site not responding. Last check: 2007-11-06)
		The method is to calculate the Bendersky-Thompson spectral sequence, a K-based unstable homotopy* spectral sequence, of Spin(n).
		The E2-term is an Ext group in a category of Adams modules.
		As the spectral sequence converges to the v1-periodic homotopy groups of the K-completion of a space, one important part of the proof is that the natural map from Spin(n) to its K-completion induces an isomorphism in v1-periodic homotopy groups.
	www.lehigh.edu /~dmd1/son.html (154 words)

	[No title]
		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.
		So: together with the homotopy groups, the Postnikov data says what the space X is really like, up to homotopy equivalence.
		Group cohomology was invented by Eilenberg and MacLane in the 1940s.
	www.math.niu.edu /~rusin/known-math/00_incoming/postnikov (1497 words)

	Braids, links, and homotopy groups (Site not responding. Last check: 2007-11-06)
		The purpose of this lecture is to describe joint work with Jie Wu on a connection between certain choices of free groups, pure braid groups, and the homotopy groups of the 2-sphere.
		Further connections of homotopy groups, Stirling numbers of the second kind, the modules Lie(n), and the classical Möbius inversion function are listed.
		Illustrations of links which arise by "closing" pure braids, and which represent elements in homotopy groups of the 2-sphere will be given.
	www.maths.abdn.ac.uk /~stc2001/abstracts/Cohen/Cohen.html (210 words)

	Abstracts
		Included is a family of two-relator groups that was introduced by I. Anshel in her thesis, where the Freiheitssatz was proved for those groups.
		This paper is concerned with the second homotopy groups (both absolute and relative) of unions of CW complexes.
		Sufficient homological conditions are given under which the homotopy modules of a union admit algebraic decompositions in terms of the homotopy properties of the summands.
	oregonstate.edu /~bogleyw/research/abstracts.html (1394 words)

	Research Page
		The group of homotopy equivalences for a connected sum of closed aspherical manifolds, Indiana Univ. Math.
		Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds (with Allen Hatcher), Geometry and Topology 1 (1997), 91-109.
		Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups (with Richard D. Canary), to appear in Mem.
	www.math.ou.edu /~dmccullough/research/publist.html (511 words)

	Physics Help and Math Help - Physics Forums - Question On Homotopy Groups
		I have this question on homotopy groups: Spacial infinity in two dimensional space is a unit circle S1 (topologically).
		homotopy class, is non zero, cannot be contracted to a point, is just the fact that if you take a piece of string, wind it around a newspaper some non zero number of times, and then tie it tight, it won't just fall off.
		02-08-2005 03:45 PM by the way here is the first topic in homotopy theory: why the identity map of the circle to itslef in the plane, is not homotopic to a constant map, as a map into the opunvtured plane, i.e.
	www.physicsforums.com /printthread.php?t=63060 (2798 words)

	Homotopy Groups, Thom Spaces, and the Oriented Cobordism Ring (Site not responding. Last check: 2007-11-06)
		Homotopy Groups, Thom Spaces, and the Oriented Cobordism Ring
		and pi_m (T), the mth homotopy group of T, for m large, where T is the "Thom space" associated to the canonical bundle over the oriented grassmannian.
		This is a mouthful, but all of these spaces will be defined, and the talk will be accessible to undergraduates.
	www.math.columbia.edu /~welji/seminar/102504.html (162 words)

	Homotopy groups
		Table of the homotopy groups of the suspensions of the (real) projective plane.
		There is a braid group action on G(n) induced by the canonical braid group action on free groups.
		The center of G(n), that is the n-th homotopy group of S
	www.math.nus.edu.sg /~matwujie/homotopy_groups_sphere.html (170 words)

	Citebase - Zariski-van Kampen theorem for higher homotopy groups
		It provides a procedure for computation of the first non-trivial higher homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.
		[12] A. Libgober, Homotopy groups of the complements to singular hypersurfaces, II.
		Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0203019 (1046 words)

	From Representation Theory to Homotopy Groups (Site not responding. Last check: 2007-11-06)
		Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations.
		In this paper, we apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.
		Its determinant is closely related to the homotopy decomposition of E8 localized at each prime.
	www.lehigh.edu /~dmd1/e8.html (202 words)

	Tile Homotopy Groups (Site not responding. Last check: 2007-11-06)
		They show that their method is always at least as strong as any generalized coloring argument, and in some cases, is strictly stronger.
		They successfully apply their technique, which involves some understanding of specific finitely presented groups, to two tiling problems.
		Partly because of the difficulty in working with finitely presented groups, their technique has only been applied in a handful of cases.
	www.math.ucf.edu /~reid/Research/Tilehomotopy (156 words)

	Research (Site not responding. Last check: 2007-11-06)
		So, in the case of BZ/p-nullification, we generalize the description of Dwyer of PBG to groups G whose groups of components is not necessarily finite, and we obtain in particular that this space may not be the HZ[1/p]-localization of BG.
		With regard to cellularization, we prove that if G is a compact Lie groups CWBG is either the classifying space of a finite p-group, or it has an infinite of nonzero homotopy groups, that are all p-torsion.
		We also study the homotopy category of G-equivariant 1-dimensional CW-complexes, which is a theory of coactions whose cogroups are the 1-dimensional G-equivariant spherical objects.
	mat.uab.es /~ramonj/Research.htm (1036 words)

	AMS Summer 1999 Research Conference in Algebraic Topology Abstracts
		Broué's Abelian Defect Group Conjecture predicts that in many situations of interest in modular representation theory, the module categories of related blocks (i.e., direct factors of the group algebras) A and B of group algebras should have equivalent derived categories.
		Spectral algebra or the algebra of spectra is the study of algebra in the context of stable homotopy theory.
		1) A complete functorial homotopy decomposition of the n-fold self smash of a two-cell 2-local suspensions are specifically given, which includes all of the homology of homotopy indecomposable factors together with their multiplicities in the decomposition.
	www.math.wayne.edu /~rrb/Summer99/abstracts.html (1285 words)

	OUP: Complex Cobordism and Stable Homotopy Groups of Spheres: Ravenel (Site not responding. Last check: 2007-11-06)
		It remains the definitive reference on the stable homotopy groups of spheres.
		The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres.
		Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups.
	www.oup.co.uk /isbn/0-8218-2967-X (444 words)

	Homotopy groups of the combinatorial Grassmannian - Anderson (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Homotopy groups of the combinatorial Grassmannian - Anderson (ResearchIndex)
		Abstract: We prove that the homotopy groups of the oriented matroid Grassmannian MacP(k; n) are stable as n !
		37 Homotopy properties of the poset of non-trivial p- subgroups..
	citeseer.ist.psu.edu /128387.html (451 words)

	Abstracts
		The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups.
		In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple.
		For this to hold, it is sufficient that the fundamental groups of all path-connected components of $X$ and $Y$ be inverse limits of nilpotent groups.
	www.ual.es /~jlrodri/abstract.htm (1772 words)

	[No title]
		http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/Equivalences The group of homotopy equivalences of products of spheres and of Lie groups Martin Arkowitz and Jeffrey Strom AMS Classifications 55P10, 55P60, 55S37 Dartmouth College, Hanover, NH 03755 Martin.Arkowitz@Dartmouth.edu Jeffrey.Strom@Dartmouth.edu Abstract We investigate the group E_#(X) of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups.
		We also show that E_#(X)_(p) has a central series whose successive quotients are pi_n(X_(p)), which are direct sums of homotopy groups of p-local spheres.
		We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over S to the category of pro-spaces.
	claude.math.wesleyan.edu /~mhovey/archive/letter120 (627 words)

	A brief introduction to homotopy groups
		Such a combining of maps has the form of a group operation among the equivalence classes, with the class of the trivial map forming the unit element.
		If the target of mapping is a Lie group, or a representation of a Lie group then a theorem states that this group is Abelian.
		which are group manifolds as fields at infinity take usually values from a group.
	www.physics.uc.edu /suranyi/conf-lectures/node4.html (205 words)

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