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Topic: Contractible space

	PlanetMath: contractible
		A topological space is said to be contractible if it is homotopy equivalent to a point.
		Equivalently, the space is contractible if a constant map is homotopic to the identity map.
		This is version 5 of contractible, born on 2002-01-23, modified 2007-02-15.
	planetmath.org /encyclopedia/Contractible.html (82 words)

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		If you have a lot of menu links and not a lot of space, this is the script to turn to.
	www.dynamicdrive.com /dynamicindex1 (777 words)

	Topology glossary
		Two sets A and B in a space are functionally separated if there is a continuous function from the space into the interval [0,1] with the property that A is mapped to 0 and B is mapped to 1.
		A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
		A Hausdorff space is Tychonoff if whenever C is a closed set and p is a point not in C, then C and p are functionally separated.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Local_base.html (1004 words)

	Topology glossary - Wikipedia, the free encyclopedia
		An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.
		A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
		A pseudometric space (M, d) is a set M equipped with a function d : M × M → R satisfying all the conditions of a metric space, except possibly the identity of indiscernibles.
	en.wikipedia.org /wiki/Topology_glossary (4720 words)

	[No title]
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
		loop space Given a topological space X, its loop space is the topological space of all continuous functions from a circle to X. Loop spaces are important examples of new topological spaces formed from old ones, as well as examples of infinite-dimensional spaces in mathematics.
		PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	Wikinfo \| Topological space
		The category of all topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics.
		A space carries the trivial topology if all points are "lumped together" in the sense that there are only two open sets, the empty set and the whole space.
		A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.
	www.wikinfo.org /wiki.php?title=Topological_space (2014 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.
		Let X be a topological space that is the union of two path-connected subspaces A and B, where the intersection of A and B is nonempty and path-connected.
		A Hopf space is a space in which the proof given above for the statement that the fundamental group of a topological group is Abelian still works.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	Topology - Abstract Shape
		Since subsets of a topological space are themselves topological spaces under their relative topology, any definition or discussion involving subsets applies to the whole space.
		Notice that if a space can be decomposed into the disjoint union of two closed subsets, then, since closed subsets include all their limit points, this intuitively means the space is composed of two pieces "which do not touch".
		Contractible spaces are simply connected but the converse is not true.
	ourworld.cs.com /jamessfreeman16/Topology.htm (2436 words)

	[No title]
		Introduction and Statement of Results The growth of torsion in loop space homology is known to be rather hard to co* *ntrol in general.
		The motivation for this paper comes from considering spaces of the form BG^p,* * where G is a finite p-perfect group and by (-)^pwe denote the Fp-completion functor of * *Bousfield and Kan [4].
		We specialize to rationally contractible spaces with a reduced integral homol* ogy exponent and attempt to understand the growth of torsion in their loop space* homology.
	www.math.purdue.edu /research/atopology/Levi/lsht.txt (5248 words)

	Contractible space - Wikipedia, the free encyclopedia
		Intuitively, a contractible space is one that can be continuously shrunk to a point.
		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.
	en.wikipedia.org /wiki/Contractible_space (255 words)

	Quotient space - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		The quotient space X/~ is then homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).
		The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
	en.wikipedia.org /wiki/Quotient_space (975 words)

	Springer Online Reference Works
		By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings.
		Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose.
		First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension.
	eom.springer.de /h/h047860.htm (1160 words)

	Research Interests - Hanspeter Fischer (Site not responding. Last check: 2007-10-23)
		The concept of non-positive curvature can be generalized to geodesic metric spaces by demanding that distances on a geodesic triangle be no larger than those on a comparison triangle in the Euclidean plane.
		A group that acts properly discontinuously and cocompactly by isometries on a non-positively curved geodesic space is called non-positively curved and the space at infinity is called a boundary of the group.
		For example, the shape and local connectivity of the boundary, connectivity at infinity of the underlying space and cohomological properties of the group are being explored.
	www.math.byu.edu /~fischer/resume/research.html (601 words)

	Dirac (Technical Notes)
		A topological space is described as being simply connected if every loop it contains can be shrunk to a single point; a loop with this property is called contractible[1].
		For example, ordinary Euclidean space is simply connected, and so is the surface of a sphere, but the surface of a doughnut isn't.
		Two paths between points A and B in a topological space are called homotopic if they can be continuously deformed into each other, or equivalently, if the loop you get by travelling along the first path from A to B and then along the second one from B back to A is contractible.
	gregegan.customer.netspace.net.au /APPLETS/21/DiracNotes.html (1132 words)

	Glossary (Site not responding. Last check: 2007-10-23)
		a topological space is compact if every collection of open sets that covers the space has a finite subset that also covers the space.
		The components of a topological space are its maximal connected subspaces.
		a separable space is one that has a countable dense subset, that is a countable subset whose closure is the whole space.
	mcraefamily.com /MathHelp/BasicSetTopologyGlossary.htm (2717 words)

	The classifying space BG of a topological group G
		The quotient space BG = EG/G is thus the space of all unordered n-tuples of distinct points in an infinite-dimensional Hilbert space.
		If G is any topological group, there is a topological space BG with a basepoint such that the space of loops in BG starting and ending at this point is homotopy equivalent to G. This space BG is unique up to homotopy equivalence.
		Take any countable-dimensional Hilbert space H and let U(H) be the group of unitary operators on H. Just like the unit sphere in this Hilbert space is contractible, it turns out that U(H) is contractible if we give it the norm topology or the strong topology.
	www.lns.cornell.edu /spr/2002-06/msg0041985.html (3280 words)

	Homotopy Equivalence - Homotopy Type
		A space which is homotopy equivalent to a single point space is called contractible.
		For a given space X the equivalence class of X under homotopy equivalence is called the homotopy type of X.
		Namely, two spaces that are not of the same homotopy type can never have the same topological type.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node13.html (497 words)

	Homotopy
		Topological spaces are enormously varied and homeomorphisms in general give much too fine a classification to be useful.
		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
	www.ma.umist.ac.uk /kd/knots/node4.html (404 words)

	[No title]
		Then S^3 - X is contractible, and moreover (S^3 - X) x R is homeomorphic to R^4, but S^3 - X is not homeomorphic to R^3.
		Both these sorts of examples the contractible manifolds are distinguished from R^n because they are not simply-connected at infinity.
		If you assume that your open subset is the interior of a compact manifold, with boundary a sphere, and the sphere is nicely embedded, then the Schoenfliess theorem (in dimensions other than 4) implies that the compact manifold is a disc, and its interior is R^n.
	www.math.niu.edu /~rusin/known-math/95/contractible (731 words)

	The International Journal of Robotics Research
		We present a new roadmap for a rod-shaped robot operating in a three-dimensional workspace, whose configuration space is diffeomorphic to R3 × S2.
		One of the challenges in defining the roadmap revolves around a homotopy theory result, which asserts that there cannot be a one-dimensional deformation retract of a non-contractible space with dimension greater than two.
		Instead, we define an exact cellular decomposition on the free configuration space and a deformation retract in each cell (each cell is contractible).
	www.ijrr.org /contents/24_05/abstract/343.html (290 words)

	AboutHap (Site not responding. Last check: 2007-10-23)
		The homology of a group G can also be defined as the homology of an orbit space X/G where X is any contractible space admitting a fixed-point free action of G. Viewed in this way the homology of certain groups is easily calculated.
		The sum of the lengths of the boundaries of the 8-dimensional generators is 2813 without simplifications, 186 with simplifications and 264 in the mod 2 case.
		The vertices of the contractible G-space X correspond to the elements of G, and the 1-skeleton can be viewed as the Cayley graph of G with respect to the generators in the associated presentation.
	hamilton.nuigalway.ie /Hap/www/SideLinks/About/aboutTopology.html (710 words)

	Lecture 2 (Site not responding. Last check: 2007-10-23)
		Define a retract and show that this is not the same as deformation retract since every space X retracts to a point.
		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).
		If there is time you can show why the spaces in either example 0.8 or 0.9 (p.
	math.rice.edu /~shelly/18.904/lecture2.html (148 words)

	David Peckinpaugh ~ "Consciousness: Space and Content" (Site not responding. Last check: 2007-10-23)
		The spatial dimension of consciousness is that space or context, great or small, within which the content of consciousness shall arise and dissipate.
		While thingness is the multiplicity which floats, rides, bubbles up (so as too arise and dissipate) in the pure space of consciousness, the pure space of consciousness is that which gives Ground to, and within which, thingness is allowed to come and go.
		As space opens up and flowers through mere noting/seeing, beyond the attached identification with objects, so, too, this same experience of a flowering spaciousness is contracted as soon as one wills a magnetic allegiance with an object (be it a tangible object or a fantasy figment object).
	lightmind.com /library/essays/monk-04.html (1724 words)

	UCSD Topology/Geometry Seminars, Winter 2004 (Site not responding. Last check: 2007-10-23)
		I will give a general framework to study the topology of infinite dimensional groups via their actions on contractible spaces known as Buildings.
		I describe the topology of (the classifying space of) the symplectomorphism groups of a family of symplectic 4-manifolds.
		We also study the space of compatible complex structures on these symplectic manifolds and outline a proof showing that this space is contractible.
	math.ucsd.edu /~justin/topseminarW04.html (1020 words)

	The Fundamental Group (Site not responding. Last check: 2007-10-23)
		For example, the fundamental group of a contractible space is the 1-element group.
		The theorem of Seifert and Van Kampen, which expresses the fundamental group of the union of two spaces in terms of the fundamental groups of the two individual spaces and their intersection.
		The natural epimorphism of the fundamental group of a path connected space to the first homology group of that space.
	www.ualberta.ca /dept/math/gauss/fcm/topology/AlgbrcTop/FndmntlGrp/FndmntlGrp.htm (320 words)

	3.3 Fermat’s principle and Morse theory in globally hyperbolic spacetimes
		(The loop space of a connected topological space is the space of all continuous curves joining any two fixed points.) On this Hilbert manifold, the energy functional is always bounded from below, and its critical points are exactly the geodesics between the given end-points.
		A critical point (geodesic) is non-degenerate if the two end-points are not conjugate to each other, and its Morse index is the number of conjugate points in the interior, counted with multiplicity (“Morse index theorem”).
		) to the loop space of the spacetime manifold or, equivalently, to the loop space of a Cauchy surface.
	relativity.livingreviews.org /Articles/lrr-2004-9/articlesu12.html (1205 words)

	Localization in equivariant cohomology
		Just as singular cohomology is a functor from the category of topological spaces to the category of rings, so when a group
		It is well known that such a space is the total space of the universal
		In case the vector field on the manifold is generated by a circle action, the localization theorem specializes to Bott's Chern number formulas [41] of the Sixties, thus providing an alternative explanation for the Chern number formulas.
	www.math.harvard.edu /history/bott/bottbio/node21.html (271 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		If you are willing to assume that the point X is contractible to is non-degenerate, then you can show that you get a pointed contracting homotopy, and you're all set.
		Let H : X x I --> X be a homotopy from the identity map on X to the constant map to a point x in X. Let y and z be two points in X. I'll show that the space X(y,z) of paths from y to z is contractible.
		So when t = 0, this is just the path c, and when t = 1, it is the composite path b'a (no time is spent on the constant homotoped path).
	www.math.niu.edu /~rusin/known-math/01_incoming/contractibility (351 words)

	Classifying Spaces Made Easy
		It follows that each space E(n) is an infinite loop space: a space of loops in a space of loops in a space of loops in...
		By this I mean a space with a continuous binary operation satisfying all the usual laws for an abelian group up to homotopy, where these homotopies satisfy all the nice laws you can imagine up to homotopy, and so on ad infinitum.
		The quotient space BG = EG/G is thus the space of all nordered n-tuples of distinct points in an infinite-dimensional Hilbert space.
	math.ucr.edu /home/baez/calgary/BG.html (6059 words)

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