21. Category: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. http://en.wikipedia.org/wiki/Category:Algebraic_topology

Category:Algebraic topology From Wikipedia, the free encyclopedia Jump to: navigation search Wikimedia Commons has media related to: Algebraic topology Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces The main article for this category is Algebraic topology Subcategories This category has the following 13 subcategories, out of 13 total. C Characteristic classes (10 P) Cohomology theories (27 P) D Digital topology (4 P) H Homological algebra (6 C, 88 P) Homology theory (1 C, 38 P) H cont. Homotopy theory (1 C, 92 P) I Intersection theory (9 P) K K-theory (1 C, 22 P) Knot theory (125 P, 1 F) S Surgery theory (27 P) T Topological graph theory (1 C, 28 P) Topological methods of algebraic geometry (1 C, 30 P) Topology of Lie groups (1 C, 7 P) Pages in category "Algebraic topology" The following 181 pages are in this category, out of 181 total. This list may not reflect recent changes ( learn more Algebraic topology List of algebraic topology topics House with two rooms ... 3-sphere A Abstract polytope Abstract simplicial complex Acyclic model Acyclic space ... A∞-operad B Barycentric subdivision Betti number Bockstein homomorphism Borsuk–Ulam theorem ... Brown–Peterson cohomology C CW complex Calculus of functors Cap product Cartan model ... Cup product D Degree (mathematics) Degree of a continuous mapping Delta set Direct limit of groups ... Duoprism E Eilenberg–Ganea conjecture Eilenberg–MacLane space E cont.

22. Algebraic Topology - Wiki.GIS.com Wiki.GIS.com is a communitygenerated, GIS-centric encyclopedia that serves as a repository for factual, unbiased GIS content. Wiki.GIS.com involves the GIS community in an http://wiki.gis.com/wiki/index.php/Algebraic_topology

Algebraic topology From Wiki.GIS.com Jump to: navigation search For the topology of pointwise convergence, see Algebraic topology (object) Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces . The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, the converse, using topology to solve algebraic problems, is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Contents The method of algebraic invariants Setting in category theory Results on homology Applications of algebraic topology ... edit The method of algebraic invariants An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex). 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 (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove.

23. Algebraic Topology Algebraic topology Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants http://www.fact-index.com/a/al/algebraic_topology.html

Main Page See live article Alphabetical index Algebraic topology Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further categorize or classify them. An older name for the subject was combinatorial topology , implying an emphasis on how a space X was contructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants: for example by mapping them to groups , which have a great deal of manageable structure, in a way that respects the relation of homeomorphism of spaces. Two major ways in which this can be done are through fundamental groups, or more general homotopy theory , and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space; but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation Homology and cohomology groups, on the other hand, are abelian, and in many important cases finitely generated. Finitely generated abelian groups can be completely classified and are particularly easy to work with.

24. Book:Algebraic Topology - TextbookRevolution May 22, 2009 Title Algebraic Topology. Author Allen Hatcher. Subjects Mathematics. Key words Algebra, Topology. Education Level http://textbookrevolution.org/index.php/Book:Algebraic_Topology

Log in / create account Home Want to contribute? Submit a new book ... History Book:Algebraic Topology From TextbookRevolution Jump to: navigation search edit Bibliographical Data Title: Algebraic Topology Author: Allen Hatcher Subjects: Mathematics Key words: Algebra, Topology Education Level: License: edit Abstract From the preface: In terms of prerequisites, the present book assumes the reader has some familiarity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identification spaces as they are sometimes called, which are quite important for algebraic topology. Available in a variety of PDF files, or in postscript forms. The electronic version is frequently updated to reflect corrections. No mirrors yet, pending author approval. Single paper or electronic copies for noncommercial personal use may be made without explicit permission from the author or publisher. All other rights are reserved. edit Download URL: http://www.math.cornell.edu/~hatcher/AT/ATpage.html Download link: Not Provided Flags: Duplicates: Retrieved from " http://textbookrevolution.org/index.php/Book:Algebraic_Topology

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28. Algebraic Topology - AoPSWiki May 1, 2009 Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate http://www.artofproblemsolving.com/Wiki/index.php/Algebraic_topology

Art of Problem Solving LOGIN/REGISTER Home Bookstore AoPS Curriculum ... Articles You are here: Resources AoPSWiki Search Wiki Page Tools Page Discussion View source History Personal tools Talk for this IP Log in Navigation Main Page Community portal Current events Recent changes ... Help Toolbox What links here Related changes Special pages Printable version ... Permanent link Algebraic topology From AoPSWiki Algebraic topology is the study of topology using methods from abstract algebra . In general, given a topological space , we can associate various algebraic objects, such as groups and rings Fundamental Groups Perhaps the simplest object of study in algebraic topology is the fundamental group . Let be a path-connected topological space, and let be any point. Now consider all possible "loops" on that start and end at , i.e. all continuous functions with . Call this collection . Now define an equivalence relation on by saying that if there is a continuous function with , and . We call a homotopy . Now define . That is, we equate any two elements of which are equivalent under Unsurprisingly, the fundamental group is a group. The

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30. Algebraic Topology - Wiktionary Oct 26, 2008 algebraic topology. Definition from Wiktionary, the free dictionary Retrieved from http//en.wiktionary.org/wiki/algebraic_topology http://en.wiktionary.org/wiki/algebraic_topology

algebraic topology Definition from Wiktionary, the free dictionary Jump to: navigation search edit English edit Noun algebraic topology uncountable That branch of topology that associates objects from abstract algebra to topological spaces edit External links Algebraic topology on Wikipedia. Wikipedia Retrieved from " http://en.wiktionary.org/wiki/algebraic_topology Category English nouns Personal tools New features Log in / create account Namespaces Entry Discussion Variants Views Read Edit History Actions Search Navigation Main Page Community portal Preferences Requested entries ... Contact us Toolbox What links here Related changes Upload file Special pages ... Permanent link In other languages Bahasa Indonesia This page was last modified on 19 October 2010, at 14:57. Text is available under the Creative Commons Attribution/Share-Alike License ; additional terms may apply. See for details. About Wiktionary

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33. Category:Algebraic Topology - ProofWiki Jan 12, 2009 Pages in category Algebraic Topology . The following 5 pages are in this category, out of 5 total. F http://www.proofwiki.org/wiki/Category:Algebraic_Topology

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37. Algebraic Topology Book A complete, downloadable, introductory text on Algebraic Topology, by Prof. Allen Hatcher, Cornell Univ. 3rd Ed. 553 pp. with illustrations. Available in pdf and postscript http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Algebraic Topology What's in the Book? To get an idea you can look at the Table of Contents and the Preface Printed Version The book was published by Cambridge University Press in 2002 in both paperback and hardback editions, but only the paperback version is currently available (ISBN 0-521-79540-0). This has gone through numerous reprintings, giving me the opportunity to correct various minor errors not yet weeded out from the earlier printings see farther down this page for a list of corrections. I have tried very hard to keep the price of the paperback version as low as possible, but it is gradually creeping upward and is now $37 in the US, after starting at $30 in 2002. Less expensive printings have been made for sale in China (Tsinghua University Press) and South Asia. A translation into Russian is underway. Electronic Version: By special arrangement with the publisher, an online version will continue to be available for free download here, subject to the terms in the . There are several different formats available: The whole book as a single rather large pdf file (3.5MB) of about 550 pages.

38. What Is Algebraic Topology? THE BEGINNINGS OF ALGEBRAIC TOPOLOGY. Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings http://www.math.rochester.edu/people/faculty/jnei/algtop.html

WHAT IS ALGEBRAIC TOPOLOGY? THE BEGINNINGS OF ALGEBRAIC TOPOLOGY Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings of mathematics. For example, if you want to determine the number of possible regular solids, you use something called the Euler characteristic which was originally invented to study a problem in graph theory called the Seven Bridges of Konigsberg. Can you cross the seven bridges without retracing your steps? No and the Euler characteristic tells you so. Later, Gauss defined the so-called linking number, a precise invariant which tells you whether two circles are linked. It is called an invariant because it remains the same even if we continuously deform the geometric object. Gauss also found a relationship between the total curvature of a surface and the Euler characteristic. All of these ideas are bound together by the central idea that continuous geometric phenomena can be understood by the use of discrete invariants. The winding number of a curve illustrates two important principles of algebraic topology. First, it assigns to a geometric odject, the closed curve, a discrete invariant, the winding number which is an integer. Second, when we deform the geometric object, the winding number does not change, hence, it is called an invariant of deformation or, synomynously, an invariant of homotopy.

39. 55: Algebraic Topology Encyclopedic reference for Algebraic Topology in Dave Rusin's Mathematical Atlas. Includes a brief history along with various links to textbooks, reference works, and http://www.math.niu.edu/~rusin/known-math/index/55-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 55: Algebraic topology Introduction Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants reflect some of the topological structure of the spaces. The use of these algebraic tools calls attention to some types of topological spaces which are well modeled by the algebra; fiber bundles and related spaces are included here, while complexes (CW-, simplicial-, ...) are treated in section 57. Finally, the use of the algebraic tools also calls attention to the aspects of a topological space which are well modeled by the algebra; this gives rise to homotopy theory. The algebraic tools used in topology include various (co)homology theories, homotopy groups, and groups of maps. These in turn have necessitated the development of more complex algebraic tools such as derived functors and spectral sequences; the machinery (mostly derived from homological algebra) is powerful if rather daunting. In all cases, the "naturality" of the construction implies that a map between spaces induces a map between the groups. Thus one can show that no maps of some sort can exist between two spaces (e.g. homeomorphisms) since no corresponding group homomorphisms can exists. That is, the groups and homomorphisms offer an algebraic "obstruction" to the existence of maps. Classic applications include the nonexistence of retractions of disks to their boundary and, as a consequence, the Brouwer Fixed-Point Theorem. (Obstruction theory is, more generally, the creation of algebraic invariants whose vanishing is necessary for the existence of certain topological maps. For example a function defined on a subspace Y of a space X defines an element of a homology group; that element is zero iff the function may be extended to all of X.)

40. Nantes 3-7 Sept 2001 UMR 6629 G.D.R.E.1110 Algebraic Topology Conference University of Nantes September 3 7 , 2001 The GDRE/CNRS 1110, the UMR 6629 CNRS/University of Nantes and the Topology Seminar Bonn http://www.math.sciences.univ-nantes.fr/~franjou/2001.html

http://www.math.sciences.univ-nantes.fr/~franjou/2001.html En francais last modified: July 19, 2001 UMR 6629 G.D.R.E.1110 Algebraic Topology Conference University of Nantes September 3 - 7 , 2001 Deadline for registration: July 6, 2001. Registration is now closed Check your registration: please check that your name appears in the List of registered participants . If you do not appear on the list, although you sent us registration, please re-send the form at NCAT@math.univ-nantes.fr Get updated information http://www.math.sciences.univ-nantes.fr/~franjou/2001pratique.html Scientific program Schedule of talks List of registered participants ... Maps SCIENTIFIC PROGRAM The scientific program concerns main research fields in Modern Homotopy Theory: Algebraic Homotopy Theory, Unstable Modules over the Steenrod algebra, MacLane Homology, Operads, Loop space homology, Cyclic Homology, Algebraic K-theory. Algebraic Topology coming from problems of Geometric Topology, e.g. Knot Theory, Low dimensional Manifolds, Foliations, will also be included. The Program will contain one hour keynote talks by invited speakers, three lectures on Knot Invariants

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