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Topic: Homotopy groups of spheres

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	Homotopy groups of spheres - Wikipedia, the free encyclopedia
		Homotopy groups of spheres is a branch of mathematics that attempts to understand the different ways spheres of various dimensions can be wrapped around each other.
		An n-dimensional sphere, n-sphere or hypersphere, is a generalization of the familiar circle and sphere and is defined as all the points in a space of n+1 dimensions that are a fixed distance from a center point.
		The group that is formed is therefore the group of integers, known as the infinite cyclic group, and denoted Z.
	en.wikipedia.org /wiki/Homotopy_groups_of_spheres (1120 words)

	Bott periodicity theorem - Wikipedia, the free encyclopedia
		In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
		The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated).
		The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished.
	en.wikipedia.org /wiki/Bott_periodicity_theorem (734 words)

	MA 422, Elliptic curves and chromatic homotopy theory (Site not responding. Last check: 2007-10-25)
		The subject of the present course is the next family of elements in the stable homotopy groups of spheres, namely the chromatic type 2 elements.
		Homotopy of KU, E(n), ku, BP; the Bockstein spectral sequence for KO, ko; the homotopy fixed point spectral sequence for KO, ko; subalgebras E(n), A(n), E of the Steenrod algebra A. Lecture 21:
		On the homotopy of EO The Bockstein and Adams-Novikov spectral sequences for EO
	www.math.uio.no /~rognes/kurs/ma422v99/summary.html (886 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		It was killed by David > Anick's proof that the problem of computing the homotopy groups of > spheres is of the NP-complete type, which essentially means that > algorithms are useless for this purpose.
		To stick with the homotopy groups of spheres (without claiming that I submit one of those questions...) -- Find infinite families of homotopy groups of spheres with "nice definitions".
		Incidentally, I do not claim that interesting infinite families of homotopy groups should coincide with ones that have simple subrecursive definitions, and God forbid that I should suggest that the beauty of Adams' result is that it allows this kind of formalist interpretation.
	www.lehigh.edu /~dmd1/tb120 (564 words)

	[No title]
		Spheres are pretty simple spaces, so one might at first guess there is some simple answer to this question for all m and k.
		The relationship between homotopy groups of spheres and higher- dimensional knot theory is a wonderful thing.
		Normally we treat the integers as the free group on one generator, or else the free commutative group on one generator.
	www.math.niu.edu /~rusin/known-math/97/homotopy.spheres (2758 words)

	[No title]
		Relations between the homotopy groups of spheres a* nd that of a space X can be set up by constructing a map from a sphere* or its loops to X or its loops, or vice versa.
		In the last case, the homotopy fibre of the map OE is homotopy equivalent to the 7-skeleton of the double loop space 2S5.
		For instance, the homotopy group ß11(P 3(2)) = Z=2 14 Z=4 2 is obtained by taking the direct sum of the corresponding homotopy groups of many factors in a decomposition of the triple loop space of P 3(2).
	www.math.purdue.edu /research/atopology/WuJ/mod2Moore2-2.txt (12952 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.
		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.
		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)

	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)

	V1-periodic homotopy groups, by Donald M. Davis
		The v_1-periodic homotopy groups of a space are roughly a localized version of the portion of the p-primary homotopy groups detected by K-theory.
		The v_1-periodic homotopy groups of spheres were calculated by Mahowald (p=2) and Thompson (p odd).
		This implies that the v_1-periodic homotopy groups of spherically resolved spaces can be determined whenever their UNSS can be calculated.
	at.yorku.ca /z/a/a/b/07.htm (174 words)

	Jie Wu's Home Page
		Homotopy Groups: Talked at NUS Summer School, 9:30-10:30, May 25, 2006.
		If you are not in the area of homotopy theory, you may think about these problems from your views.
		A table of the homotopy groups of the suspensions of the projective plane.
	www.math.nus.edu.sg /~matwujie (384 words)

	Compositions in the v1-periodic homotopy groups of spheres (Site not responding. Last check: 2007-10-25)
		Compositions in the v1-periodic homotopy groups of spheres
		We determine the image of the composite p_j o p_k in v1-periodic homotopy v1^{-1}pi_{n+8i+8j-2}(S^n).
		The method is to use Adams operations to compute the 1-line of an unstable homotopy spectral sequence constructed by Bendersky and Thompson.
	www.lehigh.edu /dmd1/public/www-data/comp.html (87 words)

	[No title]
		The group of homotopy equivalences of products of spheres and of Lie groups Martin Arkowitz and Jeffrey Strom Abstract We investigate the group E# (X) of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups.
		The group E# (X) is a natural subgroup of the group E(X) of all self homotopy equivalences of X. There are essentially two types of results on E(X) and E# (X* ): (1) properties of these groups* for large classes of spaces, and (2) detailed calcul* ations of the group* structure for specific spaces.
		In this section we determine the structu* re of the abelian group* Z# (P) in terms of the homotopy groups of spheres.
	hopf.math.purdue.edu /Arkowitz-Strom/Equivalences.txt (5336 words)

	Christopher French's Research
		For one, group actions arise naturally both in physics and in mathematics, and understanding these actions can be used to help compute a variety of invariants in both fields.
		The homotopy groups of SF are the stable homotopy groups of spheres, one of the most fundamental objects of study in homotopy theory.
		When a finite group G acts on a manifold M, the quotient M/G is an orbifold, and one can associate to M/G an orbifold string genus in physics and an orbifold elliptic genus in mathematics.
	www.math.grin.edu /~frenchc/Research (435 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)

	OUP: UK General Catalogue
		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.com /uk/catalogue/?ci=9780821829677 (414 words)

	[No title]
		Once we know aej O aek on the smallest sphere on which it is defined, its* * value on larger spheres is easily determined from the well-known effect of the suspen* sion homomorphism on the unstable v1-periodic homotopy* groups of spheres.
		In order to make the E2-group and the homotopy group have the same orde* r, there must be a compensating d3-differential on the generator of the 1-line gro *up.
		Results for multiples of the generators are not usually implied by result* s for the COMPOSITIONS IN HOMOTOPY* OF SPHERES 19 generators because the multiples are defined on smaller spheres than are the ge* nera- tors, and the double suspension homomorphism is often not injective on the unst able homotopy* groups where these compositions lie.
	www.math.purdue.edu /research/atopology/Bendersky-DavisD/comp1.txt (5257 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.
		The homotopy groups can stabilize sooner, as we saw for n = 2, but never later, and often they stabilize right at k = n+2.
		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)

	Computing v 1 -periodic homotopy groups of spheres and some compact Lie groups (ResearchIndex)
		2 periodic homotopy groups of exceptional Lie groups: torsion-..
		1 theory and unstable homotopy groups with an application to B..
		1 Group representations and the Adams spectral sequence (context) - Milgram - 1972
	citeseer.ist.psu.edu /160665.html (816 words)

	page4
		What is the first element in the homotopy groups of spheres which cannot be represented by a polynomial map of affine quadrics?
		Which elements in the stable homotopy groups of spheres are representable by stable framings on a hypersurfaces?
		For any given positive integer k is there an element in the stable homotopy groups of spheres which cannot be represented by a stable framing on a manifold embedded in codimension k?
	www.ma.man.ac.uk /~reg/page4.html (538 words)

	Arbeitsgemeinschaft Bochum-Bonn-Düsseldorf-Wuppertal
		Winter semester 05/06: Exotic spheres and the Kervaire invariant 1 problem
		During the winter semester 05/06, the Arbeitsgemeinschaft focuses on the work of Kervaire-Milnor on the classification of smooth structures on spheres and its relation, thanks to the work of Browder, to the Adams spectral sequence computing the stable homotopy groups of spheres.
		Homotopy spheres and stable homotopy groups of spheres
	www.math.uni-bonn.de /people/topology/agbbdw.html (162 words)

	Jerry (Site not responding. Last check: 2007-10-25)
		Beginning from intuitive ideas of deforming curves in a space, we build up to the homotopy groups of spheres.
		Amongst all the maps from any sphere to itself, there are some that are polynomials.
		The question is explained; these maps are related to representatives of the elements of a homotopy group of the sphere.
	www.ma.man.ac.uk /~mhorsham/seminars/jerry.html (121 words)

	-Periodic Homotopy Groups of (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Abstract: this paper we calculate the 2-primary v 1 -periodic homotopy groups of the symplectic groups Sp(n).
		One corollary is that some homotopy group of Sp(n) (Update)
		0.2: Computing v 1 -periodic homotopy groups of spheres and some..
	citeseer.ist.psu.edu /41525.html (237 words)

	Unstable homotopy groups of spheres
		From the bottom of this page you can download Mathematica programs that know many results about the unstable homotopy groups of spheres up to the 19-stem.
		Most of the information is taken from Toda's book "Composition methods in homotopy groups of spheres"; a few additional facts are proved in an accompanying note or quoted from elsewhere.
		I have tried to use a fairly general framework so that other computations in unstable homotopy can be included later if people are interested.
	neil-strickland.staff.shef.ac.uk /toda (317 words)

	Citebase - The symmetric action on secondary homotopy groups (Site not responding. Last check: 2007-10-25)
		We show that the symmetric track group, which is an extension of the symmetric group associated to the second Stiefel- Withney class, acts as a crossed module on the secondary homotopy group of a pointed space.
		An application is given to cup-one products in unstable homotopy groups of spheres, generalizing a formula of Barratt-Jones-Mahowald.
		Users are cautioned not to use it for academic evaluation yet.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0604030 (116 words)

	Amazon.ca: Composition Methods in Homotopy Groups of Spheres: Books (Site not responding. Last check: 2007-10-25)
		Amazon.ca: Composition Methods in Homotopy Groups of Spheres: Books
		Be the first person to review this item.
		Top of Page : Composition Methods in Homotopy Groups of Spheres
	www.amazon.ca /exec/obidos/ASIN/0691095868 (181 words)

	Atlas: Buildings, elliptic curves, and the stable homotopy groups of spheres by Mark Behrens
		Atlas: Buildings, elliptic curves, and the stable homotopy groups of spheres by Mark Behrens
		I will describe a dense arithmetic subgroup of the second Morava stabilizer group using isogenies of elliptic curves.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqd-64.
	atlas-conferences.com /cgi-bin/abstract/caqd-64 (110 words)

	id:A048648 - OEIS Search Results
		Order of n-th stable homotopy group of the sphere.
		Known to be finite for all positive n.
		D. Fuks, "Spheres, homotopy groups of the", Encyclopaedia of Mathematics, Vol.
	www.research.att.com /~njas/sequences/A048648 (129 words)

	DOCUMENTA MATHEMATICA, Extra Vol. ICM II (1998), 465-472 (Site not responding. Last check: 2007-10-25)
		This talk will describe recent advances in getting a global picture of the homotopy groups of spheres.
		These results begin with the work of Adams on the homotopy determined by K-theory.
		Keywords and Phrases: homotopy groups, spheres, periodicity in homotopy
	www.math.uiuc.edu /documenta/xvol-icm/06/Mahowald.MAN.html (64 words)

	List KWIC PACS and MSC+ZDM E-N lexical connection
		homotopy groups of wedges, joins, and simple spaces
		homotopy of topological groups and related structures # homology and
		homotopy theory # relations between equivariant and nonequivariant
	www.math.unipd.it /~biblio/kwic/msc-pacs/pml_11_042.htm (1074 words)

	AMCA: Buildings, elliptic curves, and the stable homotopy groups of spheres by Mark Behrens (Site not responding. Last check: 2007-10-25)
		AMCA: Buildings, elliptic curves, and the stable homotopy groups of spheres by Mark Behrens
		This arithmetic group acts on the building for GL
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/q/d/64.htm (108 words)

	Find in a Library: Complex cobordism and stable homotopy groups of spheres
		Complex cobordism and stable homotopy groups of spheres
		To find this item in a library, enter a postal code, state, province, or country in the field above.
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/95ff31b62b148c8ba19afeb4da09e526.html (55 words)

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