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Topic: Singular homology

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Homology (mathematics)

CW complex

Euler characteristic

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	PlanetMath: homology
		Homology is the general name for a number of functors from topological spaces to abelian groups (or more generally modules over a fixed ring).
		It is based on computing the homology groups of a simplicial complex (generally a finite one).
		From this definition, it is not at all clear that homology is at all computable.
	planetmath.org /encyclopedia/HomologyTopologicalSpace.html (525 words)

	Springer Online Reference Works
		Subsequently the concept of homology was generalized and its domain of application was extended by producing a number of homology theories for arbitrary spaces in which the concept of a complex is used everywhere, but in a situation which is more involved than that of triangulation.
		The definition of the Vietoris homology group for an arbitrary space is based on the study of complexes of coverings inscribed in each other (the so-called Vietoris complexes), the simplices of which are finite systems of points of the space which belong to the same element of the covering.
		The singular theory is the unique homology theory with a given coefficient group on the category of CW-complexes with the additivity property: The homology group of the topological sum of spaces is the direct sum of the homology groups of the terms.
	eom.springer.de /H/h047790.htm (1080 words)

	PlanetMath: cellular homology
		Theorem 1 The homology of the cellular complex is the same as the singular homology of the space.
		Cellular homology is tremendously useful for computations because the groups involved are finitely generated.
		This is version 3 of cellular homology, born on 2002-12-10, modified 2002-12-10.
	planetmath.org /encyclopedia/CellularHomology.html (126 words)

	Springer Online Reference Works
		However, the importance of the singular homology groups is not limited to this.
		However, although singular homology groups are defined for all topological spaces without any restriction, their application is only justified under such restrictions as local contractibility or homological local connectedness.
		Singular chains, being by their nature "too" linearly connected, do not carry information about "continuous" cycles if the latter are not "sufficiently" linearly connected.
	eom.springer.de /s/s085560.htm (533 words)

	Singular homology - Wikipedia, the free encyclopedia
		In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms.Besides, many times instead of graded abelian groups we consider R-modules and R-linear homomorphisms.
		Singular homology is constructed by applying the general homology construction to the singular chain complex, the chain complex of formal sums of singular simplices.
		Since the number of homology theories has become large (see Category:Homology theory), the terms Betti homology and Betti cohomology are sometimes applied (particularly by authors writing on algebraic geometry), to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.
	en.wikipedia.org /wiki/Singular_homology (387 words)

	what is cohomology
		Homology is a concept that is used in many branches of algebra and topology.
		Singular homology groups of a space measure the extent to which there are finite (compact) boundaryless gadgets in that space, such that these gadgets are not the boundary of other finite (compact) gadgets in that space.
		With hindsight, general homology theory should probably have been given an inclusive meaning covering both homology and cohomology: the direction of the arrows in a chain complex is not much more than a sign convention.
	askville.amazon.com /cohomology/AnswerViewer.do?requestId=5266414 (2456 words)

	Homology (mathematics) - Wikipedia, the free encyclopedia
		In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space (singular homology) or a group.
		For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
		In abstract algebra, one uses homology to define derived functors, for example the Tor functors.
	en.wikipedia.org /wiki/Homology_(mathematics) (788 words)

	MA408 Algebraic Topology
		Singular homology is the generalisation of simplicial homology to arbitrary topological spaces.
		It is not that hard to prove that singular homology is a homotopy invariant but it is quite hard to compute singular homology from the definition.
		Aims: To introduce homology groups for simplicial complexes; to extend these to the singular homology groups of topological spaces; to prove the topological and homotopy invariance of homology; to give applications to some classical topological problems.
	www.maths.warwick.ac.uk /undergrad/pydc/mauve/mauve-MA408.html (401 words)

	[No title]
		The main idea of the singular theory is to examine all continuous maps from the standard simplices to a topological space to detect the "holes" of this space, i.e., to measure which "cycles" are no "boundaries".
		This norm induces a seminorm on singular homology.
		The fundamental homology class of an n-dimensional oriented closed con- nected manifold M is the generator of Hn (M),= Z which is compatible with the orientation of M [Massey, Theorem XIV 2.2].
	www.math.purdue.edu /research/atopology/Strohm/diploma_main.txt (17691 words)

	Math 215a Home Page
		Relative homology; long exact sequences of a pair and a triple; excision; isomorphism between relative homology and reduced homology of the quotient for "good pairs".
		Homology with coefficients, Tor, and the universal coefficient theorem.
		(11/1) Homology with coefficients, the universal coefficient theorem, and Tor.
	math.berkeley.edu /~hutching/teach/215a-2005/index.html (2991 words)

	Algebraic Topology: Homology
		A singular simplex in X is a map from a Euclidean (solid) simplex (with ordered set of vertices) into X, where two singular simplices are considered equal when they differ by the unique affine transformation of the domain simplices that preserves vertex order.
		It is a singular simplex and hence represents an element of H(S(1)).
		Thee are many homology theories (we have seen singular homology and.Cech homology), and it is possible to develop the theory axiomatically.
	www.win.tue.nl /~aeb/at/algtop-6.html (3450 words)

	Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by James W. Vick [ISBN: ...
		In particular, it is devoted to the foundations and applications of homology theory.
		The essentials of singular homology are given in the first chapter, along with some of the most important applications.
		The structure of the book is mostly solid, getting straight to the point with singular homology instead of wasting time with simplicial homology and its results (a rarity with algebraic topology books).
	www.gettextbooks.com /isbn_0387941266.html (443 words)

	Singular homology and cohomology (Site not responding. Last check: )
		Singular homology and cohomology is an algebraic and combinatorial method of constructing invariants for topological spaces.
		The singular version of homology is basic in most parts of algebraic topology.
		The axioms of homology theory and their verification for singular homology.
	www.imf.au.dk /da/uddannelse/beskrivelser/older/F2001/node18.html (140 words)

	Citations: Singular homology of abstract algebraic varieties - Suslin, Voevodsky (ResearchIndex)
		singular homology H sing (X; A) and cohomology groups H sing (X; A) for any scheme X of nite type over a eld k, and any abelian group A. For k = C, and A = Z=n, these groups generalize the usual singular homology groups.
		singular homology H sing (X; A) and cohomology groups H sing (X; A) for any scheme X of finite type over a field k, and any abelian group A. For k = C, and A = Z=n, these groups generalize the usual singular homology groups.
		The diculty in the application of the results of [SV1] to higher Chow groups lies in the fact that a priori higher Chow are not de ned as singular homology of a sheaf.
	citeseer.ist.psu.edu /context/391754/245290 (3242 words)

	55: Algebraic topology
		Cohomology theories are a slight change from homology theories in that the directions of some homomorphisms are reversed; they're roughly the dual groups of the homology groups.
		Homology groups are particularly well suited to computation via some inductive procedure: if a space is somehow pieced together from simpler spaces (as unions, say, or fibrations) then the homology theories of the large space reflect those of the smaller spaces, together with some algebraic information which indicates the nature of the piecing-together.
		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.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	Singular Homology of Arithmetic Schemes , by Alexander Schmidt
		Singular Homology of Arithmetic Schemes, by Alexander Schmidt
		We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties.
		For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.
	www.math.uiuc.edu /K-theory/0830 (74 words)

	Satya Deo, Rani Durgawati University, Jabalpur (Site not responding. Last check: )
		This is a basic text on algebraic topology designed for use in a one year course at the masters or beginning Ph.D. level.
		Basic concepts of the subject like the fundamental group, covering projections, simplicial complexes, and simplicial homology are discussed at length.
		Singular homology is introduced to give a glimpse of an abstract homology theory in the sense of Elienberg and Steenrod.
	www.hindbook.com /Home.asp?P=52 (100 words)

	Math 8306-07: Class Outlines
		The singular, cellular, and simplicial cohomology of spaces, CW complexes, and Delta-complexes, resp.
		Brush-up on using the algebraic homotopy to see that the singular homology of a convex set is trivial in all degrees but zero, in which the homology is G. [Hatcher: pp.
		Relative homology and reduced homology of the quotient.
	www.math.umn.edu /~voronov/8306/outline.html (1288 words)

	Tulane Math Graduate qualifying exam syllabi
		One is not expected to memorize proofs of major theorems such as the homotopy invariance of homology-the emphasis is on knowing the statements and being able to apply the results.
		Axioms (characterizing properties of singular homology for simplicial complexes): homotopy invariance (SA), exact homology sequence for pair (SA), excision (SA), dimension axiom (SA).
		Applications: students are expected to know basic facts such as homology of spheres and to do computations using axioms, Mayer-Vietoris sequence or techniques such as cellular homology computations, as in computation for projective spaces.
	www.math.tulane.edu /graduate/qualifying/topology.html (673 words)

	Singular Manual: homology
		If B induces a map M'-->N' (i.e BM=0) and if im(A) is contained in ker(B) (that is, BA=0) then ker(B)/im(A) is the homology of the complex
		homology returns a free module of rank m if ker(B)=im(A).
		User manual for Singular version 3-0-0, May 2005, generated by
	www.math.lsu.edu /singular/sing_714.htm (114 words)

	[math/0504103] Measure homology and singular homology are isometrically isomorphic
		Abstract: Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume.
		It is the aim of this paper to prove that this isomorphism is isometric with respect to the l^1-seminorm on singular homology and the seminorm on measure homology induced by the total variation.
		For example, measure homology can be used to prove the proportionality principle of simplicial volume.
	arxiv.org /abs/math.AT/0504103 (179 words)

	Peter Albers - Publications and Preprints
		In [PSS] an isomorphism between Hamiltonian Floer homology and the singular homology is established.
		In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general.
		They underly no degree restrictions and are provento be the natural analogs to the homomorphisms in singular homology induced by the inclusion map of the Lagrangian submanifold into the ambient symplectic manifold.
	www.cims.nyu.edu /~albers/Publications/lagrangian_PSS.html (160 words)

	CJM - The homology of singular polygon spaces
		For odd $n$, the rational homology of $M_n$ was determined by Kirwan and Klyachko [6], [9].
		The purpose of this paper is to determine the rational homology of $M_n$ for even $n$.
		For even $n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the resolution of the singularities.
	journals.cms.math.ca /cgi-bin/vault/view/kamiyama0695 (78 words)

	Homology Group -- from Wolfram MathWorld
		The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a
		In particular, singular homology groups form a measure of the
		In addition, there are "generalized homology groups" which are not singular homology groups.
	mathworld.wolfram.com /HomologyGroup.html (103 words)

	K-theory Preprint Archives
		524: November 8, 2001, The obstruction to excision in K-theory and in cyclic homology, by Guillermo Cortiñas.
		329: February 3, 1999, On the descent proplem for topological cyclic homology and algebraic K-theory, by Stavros Tsalidis.
		223: December 10, 1998, On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology, by Guillermo Cortiñas.
	www.mathematik.uni-osnabrueck.de /K-theory (13536 words)

	Simplicial Homology (Site not responding. Last check: )
		In fact, homology and cohomology were developed in concert with algebraic topology, to support the structures described in these pages.
		If homology theory is new to you, read the introduction, and I'll try to reference other theorems as they are needed.
		In fact, homology has been generalized from abelian groups to modules, where it serves as a foundation for many definitions and theorems.
	www.mathreference.com /at-sh,intro.html (486 words)

	285G, Lecture 5: Finite time extinction of the third homotopy group, I « What’s new
		As a consequence, every j-cycle, being the combination of singular simplices in a singular complex involving singular simplices of dimension at most j, can be expressed as the boundary of a (j+1)-chain, and the claim follows.
		Of course, being 3-dimensional, all higher homology groups vanish, and so M is a homology sphere.
		To be precise, a map between simply-connected spaces which is an isomorphism on integral homology groups is an isomorphism on homotopy groups, by the relative Hurewicz theorem; and a map between connected CW complexes which is an isomorphism on homotopy groups is a homotopy equivalence, by Whitehead’s theorem.
	terrytao.wordpress.com /2008/04/15/285g-lecture-5-finite-time-extinction-of-the-third-homotopy-group-i (4734 words)

	mat531 week 13
		supp(Y(s)) = supp (s) for any singular simplex s, where the support supp(s) of a singular simplex s : Delta_p --> X is the image s(Delta_p), and the support of a chain is the union of the supports of its simplexes.
		For any singular simplex s : Delta_p --> X there is an integer k such that the chain Y^k(s) lies in Delta^{U,V}_*(X).
		In fact, an inclusion map of this type is automatically a chain map, and so induces a homomorphism of homology groups.
	www.math.sunysb.edu /~tony/archive/top2/week13.html (528 words)

	Referativni Zhurnal Classification
		Axiomatics 271.19.17.17.17.17 Investigation of topological spaces and continuous mappings by homological methods 271.19.17.17.17.17.15 Homology theory of dimension 271.19.17.17.17.17.21 Spectral sequence of a continuous mapping 271.19.17.17.17.17.27 Homology theory of fixed points and coincidence points 271.19.17.17.17.17.33 Homology manifolds 271.19.17.17.17.19 Homology and cohomology with nonabelian coefficients 271.19.17.17.17.25 Homotopy and cohomotopy groups: definitions and basic properties.
		Singular points 271.27.15.31.27 Sequences and series of exponentials 271.27.15.31.27.17 Dirichlet series 271.27.15.33 Systems of functions 271.27.15.33.17 Problems of completeness.
		Singularities 271.27.19.27 Classes and boundary properties of functions of several variables 271.27.19.31 Meromorphic functions of several variables.
	www.ams.org /mathweb/Classif/RZhClassification.html (1545 words)

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