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Topic: Algebraic topology

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	Allen Hatcher's Homepage
		This is the first in a series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time.
		This is intended to be a readable introduction to spectral sequences, with emphasis on their applications to algebraic topology.
		This is an improved version of a paper published in Topology in 1976 with the title: "Homeomorphisms of sufficiently large P^2-irreducible 3-manifolds".
	www.math.cornell.edu /~hatcher (901 words)

	What is Algebraic Topology?
		Modern algebraic topology is the study of the global properties of spaces by means of algebra.
		Algebraic topology is concerned with the whole surface and points to the obvious fact that the surface of a sphere is a finite area with no boundary and the flat plane does not have this property.
		Algebraic topology includes but is not confined to the study of spaces of dimensions only two or three.
	www.math.rochester.edu /people/faculty/jnei/algtop.html (1161 words)

	PlanetMath: algebraic geometry (Site not responding. Last check: )
		Algebraic geometry is the study of algebraic objects using geometrical tools.
		By algebraic objects, we mean objects such as the collection of solutions to a list of polynomial equations in some ring.
		Algebraic groups are essentially matrix group schemes, and as such allow the tools of algebraic geometry to be applied to their study.
	planetmath.org /encyclopedia/AlgebraicGeometry.html (2498 words)

	Algebraic topology - Wikipedia, the free encyclopedia
		Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
		The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces.
		In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here.
	en.wikipedia.org /wiki/Algebraic_topology (664 words)

	[No title]
		This text, developed from a first-year graduate course in algebraic topology, is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory.
		Although algebraic topology can be considered, by and large, as a creation of the 20th century, it has a long pre-history.
		From a position of minor importance, as compared as compared with the traditional areas of analysis and algebra, its concepts came to exert a profound influence, and it is now commonplace that a mathematical problem is "solved" by reducing it to a homology-theoretic one.
	www.lycos.com /info/algebraic-topology--theories.html (438 words)

	Introduction to Math Cinvestav
		Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings of mathematics.
		Modern algebraic topology is the study of the global properties of spaces by means of algebra.
		Algebraic topology includes but is not confined to the study of spaces of dimensions only two or three.
	www.math.cinvestav.mx /Topology.htm (953 words)

	[No title]
		Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html).
		An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together.
		Lie algebra An algebraic structure on a vector space which describes multiplication of elements of a Lie group which are very close to the identity (infinitesimal transformations).
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	55: Algebraic topology
		Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants reflect some of the topological structure of the spaces.
		Apart from homology groups and their kin, the principal algebraic tool used in topology is the set of homotopy groups of a space, and related concepts; in particular this includes the fundamental group (pi_1(X)) of a space.
		The tools of algebraic topology, when developed in isolation or for applications to other fields such as ring theory, give rise to homological algebra and category theory; this is the proper framework for comparing different algebraic tools.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	Research in Algebraic Topology
		Geometry and Algebraic Topology play major roles throughout Mathematics and its applications, with geometric and topological ideas often being indispensable.
		Algebraic Topology has developed important machinery such as cohomology theories including ordinary cohomology, K-theory, cobordism and elliptic cohomology.
		Dr A.J. Baker works on stable homotopy theory, bordism theory, the topology of classifying spaces and Thom spectra and the structure of periodic cohomology theories, in particular applying formal group theory and number theory.
	www.maths.gla.ac.uk /research/groups/algebraic_topology (457 words)

	Alex Suciu
		I mainly study the topology and combinatorics of hyperplane arrangements.
		On the homotopy Lie algebra of an arrangement (with Graham Denham), to appear in Michigan Math.
		I am currently organizing, together with Graham Denham, a Workshop and Special Session on Arrangements and Configuration Spaces, to be held at Northeastern and at the AMS Meeting in Durham, NH, in April, 2006.
	www.math.neu.edu /%7Esuciu/mathindex.html (427 words)

	MAT 539 -- Algebraic Topology -- Spring 2003
		Thus prior exposure to basic point set topology, homotopy, fundamental group, covering spaces is assumed, as well as a reasonable acquaintance with differentiable manifolds and maps, differential forms, the Poincaré Lemma, integration and volume on manifolds, Stokes' Theorem.
		Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu, GTM 82, Springer Verlag 1982.
		The guiding principle of the book is to use differential forms and in fact the de Rham theory of differential forms as a prototype of all cohomology thus enabling an easier access to the machineries of algebraic topology in the realm of smooth manifolds.
	www.math.sunysb.edu /~sorin/539 (708 words)

	Mathematics 261: Algebraic Topology I (Site not responding. Last check: )
		This course is an introduction to algebraic topology.
		Algebraic topology studies topological spaces by associating to them algebraic invariants.
		The principal algebraic invariants considered in this course are the fundamental group (also known as the first homotopy group) and the homology groups.
	www.math.duke.edu /graduate/courses/spring02/math261.html (215 words)

	[No title]
		Algebraic Topology preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives.
		The connections between algebraic topology, homological algebra and areas of "pure" algebra, number theory and algebraic geometry have remained very strong.
		One overall theme of this book is the use in algebraic topology of some higher categorical structures, which allow for the application of higher dimensional nonabelian methods to certain local-to-global problems.
	www.lycos.com /info/algebraic-topology--miscellaneous.html (308 words)

	Open problems in algebraic topology
		This is completely ridiculous, since the methods and ideas of algebraic topology have broad application to other areas of mathematics--witness Voevosdky's recent Fields Medal caliber work.
		We as algebraic topologists must bear part of the responsibility for this marginalization, and we must attempt to improve the situation.
		The most obvious method is to work on problems that arise externally to algebraic topology but for which the methods of algebraic topology may be helpful.
	claude.math.wesleyan.edu /~mhovey/problems (437 words)

	The Math Forum - Math Library - Algebraic Topology
		The use of these algebraic tools calls attention to some types of topological spaces which are well modeled by the algebra; fibre bundles and related spaces are included here...
		Algebraic Topology preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives.
		Category theory is a recent branch of mathematics originating in algebraic topology, but rapidly establishing connections with algebra, logic, algebraic and differential geometry, and most recently computer science.
	www.mathforum.org /library/topics/alg_topol (1435 words)

	The Math Forum - Math Library - Algebraic Topology
		The use of these algebraic tools calls attention to some types of topological spaces which are well modeled by the algebra; fibre bundles and related spaces are included here...
		A paper describing research interests in algebra at Bangor, which have to a large extent been motivated by problems in algebraic topology and homological algebra.
		Category theory is a recent branch of mathematics originating in algebraic topology, but rapidly establishing connections with algebra, logic, algebraic and differential geometry, and most recently computer science.
	mathforum.org /library/topics/alg_topol (1435 words)

	Algebraic Topology (Site not responding. Last check: )
		Although algebraic topology can be considered, by and large, as a creation of the 20th century, it has a long pre-history.
		Algebraic topology evolved slowly during the course of the 19th century.
		Emil Artin - Hel Braun, Vorlesungen über algebraische Topologie.
	www.maths.lth.se /matematiklu/personal/jaak/Alg-Top.html (324 words)

	May, J. P.: A Concise Course in Algebraic Topology
		Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups.
		Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields.
		The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.
	www.press.uchicago.edu /cgi-bin/hfs.cgi/00/13911.ctl (209 words)

	Algebra and Algebraic Topology Home Page
		The research in algebra at Bangor has to a large extent been motivated by problems in algebraic topology and homological algebra.
		The connections between algebraic topology, homological algebra and areas of "pure" algebra, number theory and algebraic geometry have remained very strong.
		The theory of crossed modules of groups started with their occurrence in algebraic topology, but they also occur naturally in the study of automorphisms of groups and in combinatorial group theory with the free crossed module generated by a presentation of a group.
	www.informatics.bangor.ac.uk /public/mathematics/research/algtop/algtop2.html (1545 words)

	Algebraic Topology (Site not responding. Last check: )
		Algebraic topology is the study of topological spaces using tools of an algebraic nature, such as homology groups, cohomology groups and homotopy groups.
		This is a first course in algebraic topology which will introduce the invariants mentioned above, explain their basic properties and develop both methods of computation and geometric intuition.
		Text: The text for the course is Algebraic Topology: A first course, by Marvin J. Greenberg and John R. Harper.
	jdc.math.uwo.ca /algtop (383 words)

	Unit Description: Algebraic topology (Site not responding. Last check: )
		The aim of the unit is to give an introduction to algebraic topology with an emphasis on homotopy and simplicial complexes and a brief introduction to homology.
		In the section on homology, the geometric methods are in the main sketched whilst the algebraic results have rigorous proofs.
		The aim is to give the students a glimpse of modern geometric methods and the power of twentieth-century algebraization of mathematics.
	www.maths.bris.ac.uk /undergrad/unitinfo/current/l4_units/alg_top.html (379 words)

	Algebraic Topology (M24) (Site not responding. Last check: )
		Algebraic topology begins with vague ideas such as the `number of holes' in a space.
		Basic point-set topology is an essential prerequisite (topological spaces, compactness, connectedness, quotient spaces).
		We will begin with a quick summary of results on the fundamental group and covering spaces (from Part II Algebraic Topology).
	www.maths.cam.ac.uk /CASM/courses/descriptions/node21.html (185 words)

	Paul Pearson, Northwestern Department of Mathematics
		Topology is the study of spaces, and algebra is the study of structure.
		Algebraic topology is based on the idea that we can associate algebraic structures to topological spaces in such a way that the algebraic structure becomes an invariant of the topological space.
		Algebraic Topology, Vector Bundles and K-Theory, Spectral Sequences, and Topology of 3-Manifolds by
	www.math.northwestern.edu /%7Epearsonp/topology.html (538 words)

	Algebraic Topology II (Site not responding. Last check: )
		Acquaintance with manifolds and basic algebraic topology (fundamental group and singular homology) would be helpful, but not essential.
		This is an introductory course in the use of differential forms in algebraic topology.
		It is recommended not only for students interested in topology, but also for those interested in eventually working in differential geometry, analysis on manifolds, algebraic geometry from a trancendental viewpoint, and the geometric analysis of noncompact and singular spaces.
	www.math.duke.edu /~saper/Instruction/math262.S98/262.html (358 words)

	Amazon.com: Algebraic Topology: Books: Edwin H. Spanier (Site not responding. Last check: )
		This book is a highly advanced and very formal treatment of algebraic topology and meant for researchers who already have considerable background in the subject.
		After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1.
		This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively).
	www.amazon.com /Algebraic-Topology-Edwin-H-Spanier/dp/0387944265 (2147 words)

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