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Topic: Chain (algebraic topology)


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  Chain   (Site not responding. Last check: 2007-11-05)
Chain (mathematics) In algebraic topology, a simplicial k- chain is a formal linear combination of k-simplices.
Chain reaction A chain reaction is a reaction in which one of the agents necessary to the reaction is itself produced by...
Hawaiian-Emperor seamount chain The Hawaiian-Emperor seamount chain is composed of the 1963, geologist Tuzo Wilson hypot...
www.brainyencyclopedia.com /topics/chain.html   (1488 words)

  
 Chain. Who is Chain? What is Chain? Where is Chain? Definition of Chain. Meaning of Chain.
A chain is a reliable machine component, which transmits power by means of tensile forces, and is used primarily for power transmission and conveyance systems.
In algebraic topology, a chain is a formal combination of simplices.
A chain is a measurement of length equivalent to 22 yards (20.12 metres), which is one tenth of a furlong or one eightieth of a mile.
www.knowledgerush.com /kr/encyclopedia/Chain   (184 words)

  
 Matematik   (Site not responding. Last check: 2007-11-05)
Widang modern ngeunaan differential geometry jeung algebraic geometry ngalegakeun geometri ka arah anu rada beda: geometri differensial nekenkeun konsep fungsi, fiber bundles, derivatives, smoothness jeung arah, sedengkeun aljabar geometri naliti wangun geometri anu dijieun tina jawaban sasaruaan (persamaan) sakumpulan polynomial.
Topology ngaitkeun ulikan rohangan jeung ulikan parobahan ku alatan nekenkeun kana konsep continuity.
Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear algebra -- Fractal geometry
sundanese.encyclopedia.st /Matematik   (1851 words)

  
 Chains -- Slide 24   (Site not responding. Last check: 2007-11-05)
The chain models formalism can be viewed as a generalization of the theory of electrical circuits.
Like circuit theory, the theory of chain models is based on concepts from algebraic topology, which provides a rigorous mathematical foundation.
As is the case with electrical circuits, building a chain model results in a mathematical model as well -- a set of equations that can be used to analyze the behavior of the system.
www.cs.cornell.edu /Simlab/slides/chains/slide24.html   (294 words)

  
 Algebraic topology -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-05)
An older name for the subject was (Click link for more info and facts about combinatorial topology) combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
Namely, any free group G may be realized as the fundamental group of a (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) graph X.
The most celebrated geometric open problem in algebraic topology is the (Click link for more info and facts about Poincaré conjecture) Poincaré conjecture, which may have been resolved by (Click link for more info and facts about Grigori Perelman) Grigori Perelman.
www.absoluteastronomy.com /encyclopedia/a/al/algebraic_topology.htm   (638 words)

  
 Mathematics 261: Algebraic Topology I   (Site not responding. Last check: 2007-11-05)
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/spring04/math261.html   (215 words)

  
 ScienceDaily Books : A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics)   (Site not responding. Last check: 2007-11-05)
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.
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.
One of the reasons that Algebraic Topology is difficult to learn is that often the more general constructions (which are algebraic) are difficult to motivate visually.
www.sciencedaily.com /cgi-bin/apf4/amazon_products_feed.cgi?Operation=ItemLookup&ItemId=0226511839   (1791 words)

  
 Articles - Mathematics   (Site not responding. Last check: 2007-11-05)
The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations.
Group theory investigates the concept of symmetry abstractly; topology, the greatest growth area in the twentieth century, has a focus on the concept of continuity.
Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry
www.totalorange.com /articles/Mathematics   (2239 words)

  
 [No title]
In the first flowering of stable algebraic topology, with t* *he introduction of cobordism and K-theory, the solidly established theory of fiber bundles was absolutely central to the translation of problems in geometric topo* *logy to problems in stable algebraic topology.
Stable and unstable homotopy groups Another important precursor of stable algebraic topology was a substantial in- crease in the understanding of the relationship between stable and unstable hom* *o- topy groups and of certain fundamental exact sequences relating homotopy groups in different dimensions.
As a matter of algebra, there is a copy of the polynomial algebra generated by certain elements that deserve to be denoted k sitting inside R*, and the images of the k under j are the Adams operations.
hopf.math.purdue.edu /May/history.txt   (14492 words)

  
 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.
For more information, see the article about Algebraic topology.
This page was last modified 06:31, 30 July 2005.
en.wikipedia.org /wiki/Category:Algebraic_topology   (54 words)

  
 Research in Geometry & Algebraic Topology   (Site not responding. Last check: 2007-11-05)
Algebraic Topology has developed important machinery such as cohomology theories including ordinary cohomology, K -theory, cobordism and elliptic cohomology.
There have also been significant interactions with many areas of Algebra, and indeed much of Algebraic Topology can be viewed as `applied algebra' as well as being a major source of innovative algebraic ideas.
Baker and A. Lazarev, On the Adams Spectral Sequence for R -modules, Algebraic and Geometric Topology 1 (2001), 173-99.
www.maths.gla.ac.uk /research/groups/geoalgtop   (954 words)

  
 Order vs. Algebraic Topology   (Site not responding. Last check: 2007-11-05)
In this section we present the basics of algebraic topology and how they can be used to derive fixed point results for ordered sets.
In particular Theorem 4.25 is a homological analogue of Theorem 3.5, leading to a possible similar reduction algorithm as discussed in section 3.2 in order to compute the homology for some ordered sets.
Most of the (standard) concepts from algebraic topology have been taken from [ 124 ], chapter 4, sections 1-4.
www.csi.uottawa.ca /ordal/papers/schroder/node15.html   (176 words)

  
 MATH 734. Algebraic Topology
A Concise Course in Algebraic Topology by J. Peter May, Chicago Lectures in Math., Univ. of Chicago Press, 1999, for $20.
Recast the discussion of subdivision in Bredon, section 17, by showing that barycentric subdivision gives a natural transformation from the singular chain functor to itself, and using the Acyclic Models Theorem to show that this natural transformation is always a chain homotopy equivalence.
Prove that T is a chain homotopy equivalence for finite polyhedra, using the fact that a finite polyhedron has a finite CW structure for which the simplicial chains are just the cellular chains, and the simplicial boundary map coincides with the cellular boundary map.
www.math.umd.edu /~jmr/734   (1480 words)

  
 Algebraic topology preprints 2002
Higher dimensional algebra frees mathematics from the restriction to a purely linear notation, in order to improve the modelling of geometry and so obtain more understanding and more modes of computation.
We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras.
Part of the interest of these results is that the family of categories equivalent to that of crossed complexes can be regarded as a foundation for a non-abelian approach to algebraic topology and the cohomology of groups.
www.informatics.bangor.ac.uk /public/mathematics/research/preprints/02/algtop02.html   (1069 words)

  
 Euler-Poincare Formula in Algebraic Topology
One of these vectors is called a chain and the set of them form a mathematical group under the operation of component-wise addition with the identity being the vector of zeroes and the inverse of a vector being the vector of the negatives of the components.
A chain may be represented as a formal sum of the form
Within the group of chains on m-simplexes there are special chains that have an empty boundary; i.e.
www.applet-magic.com /eulerpoincare.htm   (704 words)

  
 [No title]
Aguilar/S. Gitler/C. Prieto: Algebraic topology from a homotopical viewpoint.
The authors have succeeded in presenting practically a complete theory of ends of all kinds and their interrelations which are relevant to the topology of high-dimensional manifolds.
Without any doubt, this book, the text of which evolved from the long-standing experience of a leading researcher and teacher in the field, is a highly valuable enhancement of the standard literature in algebraic topology." (W. Kleinert in Zentralblatt) Karl Heinz Mayer: Algebraische Topologie.
felix.unife.it /Root/d-Mathematics/d-Geometry/d-Algebraic-topology/b-Algebraic-topology   (388 words)

  
 Curriculum Vitae for John Rognes   (Site not responding. Last check: 2007-11-05)
Lecturing, Algebraic Topology (MA 362), University of Oslo, fall 1992, fall 1993, fall 1999 and spring 2002.
Lecturing, Algebraic Topology II (MA 422), University of Oslo, spring 1993 (The image of J in algebraic K-theory), spring 1995 (Characteristic classes), spring 1997 (Homotopy theory), spring 1998 (Manifold models for A-theory), spring 1999 (Elliptic curves and chromatic homotopy theory), fall 2001 (Elliptic cohomology and topological modular forms) and fall 2003 (Morava K-theory).
``Algebraic K-theory of the two-adic integers,'' talks at Algebraic K-theory and Homotopy Theory meeting, Oberwolfach, November 1995, at Stanford University, January 1996, at Northwestern University, April 1996, at the University of Chicago, May 1996, at M.I.T., May 1996, and at Algebraic K-theory meeting, Oberwolfach, June 1996.
folk.uio.no /rognes/cv.html   (2586 words)

  
 OUP: Simplicial and Operad Methods in Algebraic Topology:   (Site not responding. Last check: 2007-11-05)
The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated.
The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure.
Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems.
www.oup.co.uk /isbn/0-8218-2170-9   (365 words)

  
 University of Chicago Algebraic Topology Seminar   (Site not responding. Last check: 2007-11-05)
The algebraic topology seminar is held in Eckhart Hall room 203, on Tuesdays at 4:30PM, unless otherwise specified.
Algebraic Topology : Jets and operads in homotopy calculus of functors
A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator".
www.math.uchicago.edu /~jg/topcal.cgi   (808 words)

  
 Morse Theory (L24)   (Site not responding. Last check: 2007-11-05)
Morse Theory, developed by Marston Morse as a calculus of variations for paths in a manifold, became a powerful tool in the study of the topology of manifolds.
The basic idea is that the gradient flow of a generic function determines a decomposition of the manifold into cells, and the ``attaching maps'' can be deduced from local information about the function.
Basic algebraic topology (homology of a space, chain complexes, the notion of homotopy) and and differential topology (manifolds, vector fields and flows) are needed, as well as (fairly) basic Riemannian geometry.
www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node37.html   (210 words)

  
 OUP: Lecture Notes in Algebraic Topology: Davis   (Site not responding. Last check: 2007-11-05)
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating.
It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.
There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem.
www.oup.co.uk /isbn/0-8218-2160-1   (485 words)

  
 A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) - Hotel Resource Book Store   (Site not responding. Last check: 2007-11-05)
I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.
Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology.
There are many excellent and elementary introductions to Algebraic Topology of...
www.hotelresource.com /bookstore/asinsearch_0226511839.html   (305 words)

  
 Algebraic Topology (M24)   (Site not responding. Last check: 2007-11-05)
Algebraic topology permeates all of modern pure mathematics.
I shall assume that you are familiar with basic analytic topology (topological spaces, compactness, connectedness etc).
There are many books on algebraic topology, most of which would be perfectly reasonable for accompanying at least parts of the course.
www.maths.cam.ac.uk /CASM/courses/descriptions/node19.html   (198 words)

  
 The Lefschetz Number
If K is a P -chain complex and f is an order-preserving map on P, then f maps a chain to itself, which naturally means that f has a fixed point.
Moreover all fixed point theorems for ordered sets that are derived using algebraic topology are fixed clique or even fixed simplex theorems.
Combinatorial fixed clique theorems for finite graphs that are analogous to fixed point theorems for ordered sets have been derived in [ 120 ] (due to the fact that the definitions of graph endomorphisms and order-preserving maps are formally very similar this is not too hard).
www.csi.uottawa.ca /ordal/papers/schroder/node18.html   (367 words)

  
 AMS Summer 1999 Research Conference in Algebraic Topology Abstracts   (Site not responding. Last check: 2007-11-05)
Spectral algebra or the algebra of spectra is the study of algebra in the context of stable homotopy theory.
If M has dimension n greater than or equal to 5, then M admits a positive scalar curvature metric if and only if an obstruction in R_n vanishes; moreover, the group R_{n+1} acts freely and transitively on the set of concordance classes of such metrics (provided of course this set is non-empty).
So far, a purely algebraic definition of the groups R_n is sadly lacking; we discuss an "assembly map" with target R_n that is closely related to the topological K-theory assembly map.
www.math.wayne.edu /~rrb/Summer99/abstracts6.html   (439 words)

  
 Citations: Marcel Dekker Inc - Agoston (ResearchIndex)   (Site not responding. Last check: 2007-11-05)
Arguably the most important property of a topological polyhedron is its combinatorial boundary which is itself a lower dimensional polyhedron and can be obtained by a pure algebraic computation using the concept of chains.
Boundary of a three dimensional solid must be a 2 chain whose boundary is 0, i.e.
....and their formal properties, including validity, are rooted in algebraic topology [14, 27] Algebraic topology is also the proper setting for formulating BR deformation.
citeseer.ist.psu.edu /context/1495438/0   (446 words)

  
 Directory - Science: Math: Publications: Subject Preprint Archives
Hopf Topology Archive   ·  iweb  ·  cached  · Algebraic topology and related areas.
Linear Algebraic Groups and Related Structures   ·  cached  · Including Azumaya Algebras, Algebras with Involutions, Brauer Groups, Quadratic and Hermitean Forms, Witt Rings, Lie and Jordan Algebras, Homogeneous Varieties.
Topology Atlas Preprints   ·  cached  · Mostly general topology.
www.incywincy.com /default?p=187598   (306 words)

  
 Chain   (Site not responding. Last check: 2007-11-05)
conveyance systems, similar to a conveyor belt, as in flat chains and pintle chains;
a mathematical term used in algebraic topology ;
a measure of length within the Imperial system of measurement, the chain equal to 22 yards, formerly used in England and giving rise to
www.brainyencyclopedia.com /encyclopedia/c/ch/chain.html   (189 words)

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