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Topic: Fundamental group

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Homotopic

Torsion free

Principal bundle

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Topological space

First category

History of the separation axioms

Algebraic topology

Flag manifold

Brouwer fixed point theorem

Grassmannian

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Cohomology

Logarithm

	NationMaster - Encyclopedia: Fundamental group (Site not responding. Last check: 2007-11-07)
		Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.
		For example, the fundamental group of a graph G is a free group.
		The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle.
	www.nationmaster.com /encyclopedia/Fundamental-group (0 words)

	PlanetMath: fundamental group (Site not responding. Last check: 2007-11-07)
		In general, the fundamental group of a topological space depends upon the choice of basepoint.
		Homotopy groups generalize the concept of the fundamental group to higher dimensions.
		This is version 12 of fundamental group, born on 2001-11-14, modified 2006-10-07.
	planetmath.org /encyclopedia/FundamentalGroup.html (254 words)

	Fundamental group Encyclopedia (Site not responding. Last check: 2007-11-07)
		In mathematics, the fundamental group is one of the basic concepts of algebraic topology.
		Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.
		For studying "higher-dimensional holes", the homotopy groups are used.
	www.hallencyclopedia.com /topic/Fundamental_group.html (1283 words)

	The fundamental group
		The fundamental group of a space X is a canonical example of this situation.
		You are also asked to calculate the fundamental group of a product and of a wedge in terms of the fundamental groups of the factors.
		The outcome is that the fundamental group of
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node14.html (0 words)

	S.O.S. Mathematics CyberBoard :: View topic - Topology - Fundamental Group!
		If not then you are probably correct and the group is trivial (though this does not quite constitute a proof).
		That's what rubbed me the wrong way and caused me to doubt that the fundamental group is trivial.
		The fundamental group at any given base point is trivial, but you can't get (continuously) from one base point to another, so it would not make sense to talk about the fundamental group
	www.sosmath.com /CBB/viewtopic.php?t=22822&start=0&postdays=0&postorder=asc&highlight= (653 words)

	The Fundamental Group
		Hence f concatenated with its inverse produces the identity element of the group.
		In summary, the fundamental group does not depend on c.
		The difference between end points, which is the degree after all, is the difference in end points of the first lift plus the difference in end points of the second.
	www.mathreference.com /at,fung.html (0 words)

	Marketing Research Specialists \| Fundamental Research Group
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		The Fundamental Research Group is not only eminently competent, but very easy to work with and exceedingly articulate in making presentations to large and small audiences.
	funresearch.com /testimonial_gen.html (0 words)

	Fundamental Media Group
		Fundamental Media, and our subsidiaries' objectives are to assist our clients in generating assets.
		If you are looking for a media consultancy to provide both strategic planning and media buying in the investment banking, fund management and securities services industries, Fundamental Media will present your solution.
		Fundamental Media, through our 'Intelligence' product, has identified over 800 fund management relevant publications across Europe and Asia which claim to be read by institutional, retail and personal investors.
	www.fundamentalmedia.net (0 words)

	Definition of fundamental group - Merriam-Webster Online Dictionary
		: a set that is a subset of all paths defined on a set of points each pair of which is joined by a path and that is the quotient group of the group of all paths beginning and ending in a given point
		Learn more about "fundamental group" and related topics at Britannica.com
		See a map of "fundamental group" in the Visual Thesaurus
	www.webster.com /dictionary/fundamental+group (0 words)

	Fundamental Theory Group: Particle and Gravitational Physics Home Page (Site not responding. Last check: 2007-11-07)
		Syracuse University has a large group of researchers working on a diverse selection of topics in cosmology, gravitation and particle physics.
		We are always on the lookout for talented undergraduates, graduate students and postdocs to join the group and participate in original research into the fundamental nature of the matter and forces that govern our universe.
		Follow the links to see who we are, what we work on, and what goes on in our group.
	www.phy.syr.edu /FTGHome.htm (84 words)

	Group theory Summary
		Group theory is that branch of mathematics concerned with the study of groups.
		Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of automorphism groups.
		The Fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
	www.bookrags.com /Group_theory (3252 words)

	Kids.Net.Au - Encyclopedia > Fundamental group
		The fundamental group is one of the basic concepts of algebraic topology.
		Although the fundamental group in general depends on the choice of basepoint, it turns out that, up to isomorphism, this choice makes no difference if the space X is path-connected.
		An example of a space with a non-Abelian fundamental group is the complement of a trefoil knot[?] in R
	www.kids.net.au /encyclopedia-wiki/fu/Fundamental_group (817 words)

	Fundamental Research
		Phosphazene cyclic trimers with amino acid ester or glyceryl side groups were found to hydrolyze slowly to phosphate, ammonia, and amino acid or glycerol, and this provided the impetus for us to develop the corresponding high polymers for several biomedical applications.
		Our continuing work on the small molecule chemistry involves studies of the role of side group steric hindrance on the substitution chemistry of small molecule phosphazenes, on the sensitivity to hydrolysis of phosphazenes with biological important side groups, and on ring and chain conformations.
		The side groups in compound 6 are twisted in the solid state, and this essentially fills space that would be allotted to tunnels or cavities.
	research.chem.psu.edu /hragroup/fundamental.htm (1531 words)

	List of Publications
		Fundamental groups of complements of branch curves as solvable groups (with B. Moishezon), Israel Mathematical Conference Proceedings 9 (1995), 329-346.
		Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Amer.
		The fundamental group of a CP2-complement of a branch curve as an extension of a solvable group by a symmetric group, Math.
	www.cs.biu.ac.il /~teicher/publications.htm (1177 words)

	Group Theory at the Library of Math (Free Online Mathematics) (Site not responding. Last check: 2007-11-07)
		Basically a group is a set together with a single operation that satisfies certain properties: (1) there must be an identity element, (2) every element must have an inverse, and (3) the associative law must be obeyed.
		In this topic, many examples are given to explain the importance of permutation groups when the underlying set is a finite set of counting numbers; and the matrix form and cycle notation of such permutations are detailed so as to fully explore the groups of permutations of finite sets of counting numbers (called symmetric groups).
		Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element.
	libraryofmath.com /Group_Theory.html (1788 words)

	Good Math, Bad Math : Walking in Circles: Fundamental Groups
		That's basically what the objects in the fundamental group of a point are: the continuous loops that follow some path through the space, both starting and ending at the point.
		So the fundamental group isn't really all of the loops; it's a set of continuous loops such that every continuous loop is a continuous deformation of some object in the group.
		Groups also need some operation; the operation for loops in a topological space is basically concatenation: connect the end point of each loop to the start point of the other - which is itself another loop.
	scienceblogs.com /goodmath/2006/11/walking_in_circles_fundamental.php (1973 words)

	Why is group theory important?
		Different cryptosystems use different groups: the group of units in modular arithmetic, the group of rational points on elliptic curves over finite fields, and braid groups.
		This use of group theory derives not from the "symmetry" perspective, but from the efficiency or difficulty of carrying out certain computations in the groups.
		Group theory (symmetric groups, conjugations, commutators, and semi-direct products) is what you find under the hood of Rubik's cube.
	www.math.uconn.edu /~kconrad/math216/whygroups.html (1074 words)

	Fundamental group (Site not responding. Last check: 2007-11-07)
		In mathematics, the fundamental group is one of the basicconcepts of algebraic topology.
		It is a group associated with every point of a topological space and conveying information about the 1-dimensionalstructure of the space.
		Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism, this choice makes no difference if the spaceX is path-connected.
	www.therfcc.org /fundamental-group-35915.html (929 words)

	Algebraic Topology: Homotopy
		Then the fundamental group of X is generated by (the images of) the fundamental groups of A and B.
		3.9 The fundamental group of a topological group is Abelian
		Theorem The group operation on X induces a group operation on P(X;x) that coincides with the old group operation, and P(X;x) is commutative.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		An >important invariant is the fundamental group of the complement S3-K, which >we call the group of K'.
		This group is called the "fundamental group of X with basepoint " (or "the first homotopy group* of X with basepoint *").
		The group generated by x_1,...,x_n subject to the relations just written down is the knot group.
	www.math.niu.edu /~rusin/known-math/98/whatis_pi1 (447 words)

	Guiding Group Culture: An Often Overlooked Management Skill
		Code is the group's greatest asset because it is the main thing that it has to show for all of its work.
		This prevents group members from doing anything that harms code quality -- if they care about the code, they won't hack at it, shortchange it, wire it, etc.; they will always try to ensure that it is solid and working.
		When this occurs, groups usually decide that code reviews are not valuable, drop them from their development process and lose the potential benefits that can stem from a correctly implemented review.
	www.computerworld.com /printthis/2006/0,4814,108424,00.html (1069 words)

	Good Math, Bad Math : Groups and Topology
		I wrote a series of posts on group theory for GM/BM when it was at blogger; if you're interested in details, you might want to pop over there, and take a skim.
		Group theory is a branch of abstract algebra that focuses on the study of symmetry.
		Lie groups are incredibly important in analysis, and in the basic math of relativity.
	scienceblogs.com /goodmath/2006/11/groups_and_topology_1.php (1381 words)

	Groups
		The equivalence relation induced by homotopy starts to enter the realm of algebraic topology, which is a branch of mathematics that characterizes the structure of topological spaces in terms of algebraic objects, such as groups.
		The group in this case is called commutative or Abelian.
		The set of all 3D rotations is an example of a noncommutative group.
	msl.cs.uiuc.edu /planning/node142.html (178 words)

	Elliptic Curves and Modular Functions
		This group has a "fundamental domain" with the property that any point in the whole plane is a transformation of a point in the fundamental domain by an element of the group.
		For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
		Associating a group with a geometric object provides a very powerful way of studying the object, since the algebraic structure of the group has a close relation to geometric properties of the object.
	cgd.best.vwh.net /home/flt/flt05.htm (2993 words)

	The Knot Group (Site not responding. Last check: 2007-11-07)
		The fundamental group of M is called the knot group of K. Obviously this group does not depend on any particular projection of a knot and is thus a knot invariant.
		If a particular group G is a module of the group of a knot K, but not of a knot K', then K and K' are inequivalent.
		This group is a module of the group of the trefoil, but not of the group of the trivial knot.
	www.inst.bnl.gov /~wei/group.html (441 words)

	Amazon.com: Classical Topology and Combinatorial Group Theory: Books: John Stillwell (Site not responding. Last check: 2007-11-07)
		The homeomorphism problem and other fundamental problems are essentially algorithmic (i.e., given two spaces, decide whether they are different of the same) so unsolvability (non-existence of algorithms) is indeed a force to be reckoned with, so it is given its own chapter 9, naturally culminating with the unsolvability of the homeomorphism problem.
		The fundamental group is defined to be an equivalence class of maps, and with the exception of the circle, it is calculated using deformation retraction and the Seifert-Van Kampen theorem.
		Homology theory is presented in chapter 5 as an abelianization of the fundamental group, and the abelianization is shown to be independent of the presentation of the fundamental group.
	www.amazon.com /Classical-Topology-Combinatorial-Group-Theory/dp/0387979700 (2086 words)

	YouTube - The fundamental Group of the Torus is abelian
		YouTube - The fundamental Group of the Torus is abelian
		The fundamental Group of the Torus is abelian
		Since these two path generate the fundamental group of the torus this proves that this group is abelan.
	www.youtube.com /watch?v=nLcr-DWVEto (184 words)

	[No title]
		The main result is that the fundamental group of a second countable, connected, locally path connected one-dimensional metric space is free if and only if it is countable, if and only if the space has a universal cover.
		Informally, this is the group one gets by replacing the unit interval in the notions of arc and homotopy by arbitrary compact, connected ordered topological spaces (which are called big arcs).
		The main results of this paper are that a translation proper solvable group of finite virtual cohomological dimension is metabelian-by-finite, and that a translation discrete solvable group of finite virtual cohomological dimension, m, is a finite extension of the free abelian group of rank m.
	www.math.byu.edu /~conner/research (544 words)

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