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Topic: Homology groups

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Homology (mathematics)

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	Algebraic topology - Wikipedia, the free encyclopedia
		The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic.
		As another example, the top-dimensional integral cohomology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not.
		Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
	en.wikipedia.org /wiki/Algebraic_topology (627 words)

	PlanetMath: homology (Site not responding. Last check: 2007-11-07)
		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).
		This is version 10 of homology, born on 2002-12-10, modified 2004-06-08.
	planetmath.org /encyclopedia/HomologyTopologicalSpace.html (519 words)

	Abstracts of my papers
		We also exhibit an automorphism of a subgroup of finite index in the mapping class group of a sphere with four punctures (or a torus) such that it is not the restriction of an endomorphism of the whole group.
		Our main result is that the group of automorphisms of the complex of curves of a surface is isomorphic to the extended mapping class group of the surface, if the surface is a sphere with at least five punctures or is a tori with at least three punctures.
		We conclude that the outer automorphism group of a finite index subgroup of the extended mapping class group is finite.
	www.math.metu.edu.tr /~korkmaz/abstracts.html (1346 words)

	Algebraic topology - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		If an n-th homology group of a simplicial complex has torsion, then the complex is nonorientable (query this).
		Beyond simplicial homology, one can use the differential structure of smooth manifolds via de Rham cohomology, or Cech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
		De Rham showed that all of these approaches were interrelated and that the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through De Rham cohomology.
	encyclopedia.learnthis.info /a/al/algebraic_topology.html (438 words)

	[No title]
		Although homology can be defined in terms of huge (infinitely-generated) chain complexes, one may in fact show that the this complex is chain-homotopic to the finite chain complex freelyl generated by the cells of the simplicial complex.
		In the case of the cube, for example, we simply observe that the space is homeomorphic to the CW complex formed by adjoining a 2-cell (a closed disk in R^2) to a 0-cell (a point) by sending the whole boundary of the disk to the point.
		I want to give this project to my M Sc student and have her find the homology groups of various examples of this kind - to be completely explicit these complexes will be neighbourhood complexes of Kneser graphs, and will be glad to define all that if you wish.
	www.math.niu.edu /~rusin/known-math/96/homology_calc (1030 words)

	Kids.net.au - Encyclopedia Algebraic topology - (Site not responding. Last check: 2007-11-07)
		Fundamental groups give us crucial information about the structure of a topological space, but they are often nonabelian and can be difficult to work with.
		Beyond simplicial homology, one can use DeRham cohomology[?] to investigate the differential structure of manifolds, or Cech or Sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
		In particular, fundamental groups, homology and cohomology groups are invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups.
	www.kids.net.au /encyclopedia-wiki/al/Algebraic_topology (367 words)

	Glossary: Homology (Site not responding. Last check: 2007-11-07)
		The k-th homology group is made up of classes of k-cycles, where two k-cycles are in the same class if together the form the boundary of some (k+1)-dimensional object.
		The identity in the k-th homology group is the class made up of the k-cycles that bound regions all by themselves.
		We have seen that for the sphere, the 1st homology group is trivial.
	www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/Homology.html (349 words)

	PlanetMath: relative homology groups (Site not responding. Last check: 2007-11-07)
		Relative homology groups are important for a number of reasons, principally for computational ones, since they fit into long exact sequences, which are powerful computational tools in homology.
		Cross-references: exact sequences, reduced, quotient topology, relation, equivalence classes, groups, homology, complex, quotient, map, boundary, subgroup, inclusion map, subspace, topological space
		This is version 2 of relative homology groups, born on 2002-12-10, modified 2002-12-12.
	planetmath.org /encyclopedia/RelativeHomologyGroups.html (107 words)

	[No title]
		For every group is shown the group id, and a table describing the correspondence between S.cerevisiae genes in the homology group (whose names are shown on the top), and the S.paradoxus pORFs in the homology group (whose ids are shown on the side).
		The corresponding table entries show: amino acid identity, percentage query coverage, percentage subject coverage Also, for those matches that belong to a synteny block, a star is shown next to the three numbers.
		Group 86: 1 pORFs match 3 S.cer genes in 3 correspondences.
	www.broad.mit.edu /personal/manoli/yeasts/S2c.Homology_Groups/Sbay_homology_groups_small.txt (17793 words)

	[No title]
		In this paper, we construct homology groups Hi(X, G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG x X, an idea first suggested by Andrei Suslin.
		Hi(X, G; A), whose source is the usual group homology of the discrete group G(R) of R-points of the algebraic group G. Moreover, this map is an isomorphism if R = k and k is algebraically closed, so that these groups capture the homology of the discrete group G(k).
		HOMOLOGY OF LINEAR GROUPS 7 Proof.The abelian group H1(Spec (R), Gm ; Z) is generated by classes of non- zero complex numbers, [z] for z 2 Cx, modulo the relations [z] = [__z] and [zw] = [z] + [w], for all z, w 2 Cx.
	hopf.math.purdue.edu /Knudson-Walker/hom11-19.txt (4410 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		To make sure it's what you are counting with homology you pretty much have to define "hole" as a cycle which is not a boundary, in which case, sure, cohomology counts co-holes, which are co-cycles which are not coboundaries.
		For example, the first homology group of a space (using integer coefficients) is the abelianization (G/[G,G]) of the fundamental group.
		You can do it with homotopy groups too (in which case you ought even to ask about non-abelian groups for \pi_1) and the answer is still "yes"; the groups are certain CW complexes called Eilenberg-MacLane spaces.
	www.math.niu.edu /~rusin/papers/known-math/94/detect (797 words)

	Singular homology (Site not responding. Last check: 2007-11-07)
		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.
		Singular homology is constructed by applying the general homology construction to the singular chain complex, the chain complex of formal sums of singular simplices.
		If we consider the free abelian groups generated by all singular n-simplices and extend the boundary operator d to formal sums of singular n-simplices, we obtain a chain complex of abelian groups.
	www.sciencedaily.com /encyclopedia/singular_homology (305 words)

	Homology Groups
		Homology group 18: Avian histone H5 + EP11067 Gg histone H5 - EP11066 Cm histone H5 Homology group 19: Mammalian, avian beta-actin.
		Homology group 157: Mammalian elastase II + EP29006 Rn elastase II - EP16055 Mm elastase II Homology group 158: Mammalian alpha cardiac actin.
		Homology group 217: Mammalian histone H3 genes group 1.
	www.epd.isb-sib.ch /current/HG.html (1451 words)

	Algebraic Topology: Homology
		A morphism of graded Abelian groups is a homomorphism of degree 0.
		Similarly, one usually defines homology H using E'(V), notices that there are formal complications in degree 0, and then looks at the augmented chain complex E(V) with reduced homology H-tilde, which equals H everywhere, except in degree 0, in case V is nonempty, where H_0 has an additional summand Z.
		The second result here is that H_1 equals pi_1 made Abelian, that is, the first homology group of a space (for singular homology) equals the quotient of the fundamental group by its commutator subgroup.
	www.win.tue.nl /~aeb/at/algtop-6.html (3450 words)

	Homology Of Gaussian Groups (ResearchIndex)
		Abstract: We describe new combinatorial methods for constructing explicit free resolutions of Z by ZG-modules when G is a group of fractions of a monoid where enough least common multiples exist ("locally Gaussian monoid"), and, therefore, for computing the homology of G. Our constructions apply in particular to all Artin--Tits groups of finite Coxeter type.
		3 The homological algebra of Artin groups (context) - Squier - 1995
		3 Gaussian groups are torsion free (context) - Dehornoy - 1998
	citeseer.ist.psu.edu /685447.html (732 words)

	Topology (Site not responding. Last check: 2007-11-07)
		Be able to compute directly the homology groups of a low- dimensional simplicial complex, and understand how the ranks of the homology groups are related to information about higher-dimensional `holes' in the space.
		Be able to compute homology groups of low- dimensional spaces using appropriate triangulations, and to be able to state the usual invariance and well-definedness properties of homology groups.
		To understand how higher homology groups may be used (assuming the standard results on simplicial approximation) to prove Brouwer fixed point theorem.
	www.mth.uea.ac.uk /~h720/sheets/topology (246 words)

	Homology ring mod 2 of free loop groups of exceptional Lie groups - Hamanaka (ResearchIndex)
		H.Hamanaka, Homology ring mod 2 of free loop groups of exceptional Lie groups, J. Math.
		2 The mod 2 homology of the space of loops on the exceptional..
		Adjoint Actions on the Modulo 5 Homology Groups of E8 and..
	citeseer.ist.psu.edu /hamanaka95homology.html (500 words)

	Homotopy and homology groups in general topology, by Katsuya Eda
		Infinite groups, commutative or non-commutative, have been studied mostly in their own regions and not so many applications are known.
		Eda, Free $\sigma$-products and fundamental groups of subspaces of the plane, preprint.
		Schlitt, Sheaves of abelian groups and the quotients $A^{**}/A$, J. Algebra 158 (1993), 50--60.
	at.yorku.ca /t/a/i/c/04.htm (528 words)

	[No title]
		The main novelty is defining the homology groups of a (pointed path-connected) CW complex as the homotopy groups of its infinite symmetric product SP $X$.
		Nowhere does this book mention that the first homology group is the abelianization of the fundamental group.
		After the chapters on homology and cohomology are four thorough chapters on vector bundles, K-theory, Adams operations, and characteristic classes.
	www.lehigh.edu /~dmd1/gitrev.txt (797 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Topological Hochschild Homology of Extensions of Z=pZ by Polynomials, and of Z[x]=(xn) and Z[x]=(xn 1) Ayelet Lindenstrauss Princeton University Princeton, NJ 08544 Introduction The Dennis trace is a map from the algebraic K-theory of a ring into Hochsc* hild homology*.
		The Hochschild homology groups are easier to calculate,but they are r* *elatively simple.
		The Hochschild hom* ol- ogy groups* are homotopy groups of aspace jC H:(R)j,the realization as a simplic* ial space of the standard (bar) complex used to calculate Hochschild homology*.
	hopf.math.purdue.edu /Lindenstrauss/thh6.abstract (492 words)

	Bivariant Periodic Cyclic Homology (Site not responding. Last check: 2007-11-07)
		In this self-contained exposition, the author's purpose is to understand the functorial properties of the Cuntz-Quillen theory, which motivaties his explorations of what he calls cyclic pro-homology.Simply stated, the cyclic pro-homology of an (associative) algebra A is short for the Z/2 Z-graded inverse system of cyclic homology groups of A, considered as a pro-vector space.
		He explains the relation to the Cuntz-Quillen groups in a Universal Coefficient Theorem and in a Milnor lim1-sequence.
		It is interesting to note that for the excision result, this lifting procedure goes through without constraints.For those new to cyclic homology, Dr. Grønbaek takes care to provide an introduction to the subject, including the motivation for the theory, definitions, and fundamental results, and establishes the homological machinery needed for application to the Cuntz-Quillen theory.
	www.ramex.com /title.asp?id=6233 (553 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		TITLE: "Cellular Homology and applications" ABSTACT: Cellular Homology is a very powerful and easy way to compute the homology groups of certain topological spaces known as CW Complexes.
		] Cellular Homology is built "on top of" singular homology, leading some people to refer to it as "Homology Squared." I will assume you know something about singular homology -- specifically you need to know about relative homology groups and the long exact sequence.
		I should have enough time to prove that the cellular homology groups are equivalent to "normal" homology groups, and to compute the homology groups of several spaces.
	www.math.umn.edu /atac/rogness.txt (175 words)

	UCL/AGEL - (Site not responding. Last check: 2007-11-07)
		The more delicate treatment of the Brauer-Taylor group has also been achieved: it opens new problems involving "categories without units"; applications in functional analysis, in particular the theory of compact operators, are presently investigated.
		We have exhibited a characterization of those categories in which the groups of automorphisms are representable (like the category of all groups).
		The recent discovery of an internal notion of crossed module opens a new perspective for the developments of this theory in the context of semi-abelian categories and offers a unified approach to the study of the homological properties of crossed modules, crossed rings and other structures internal to varieties of algebras.
	www.math.ucl.ac.be /AGEL/AGELrech1.html (1085 words)

	Differential Topology and Homology
		of the sphere is trivial, while those of the other surfaces are profound; and this homology is linked with the connectedness or disconnectedness of the surface after one or more cuts.
		Analogous topological structures arise in quantum gravity, but inasmuch as the manifolds involved are multidimensional rather than two-dimensional, higher homology groups play a role as well.
		Nevertheless, the higher homology groups can be perceived, at least approximately, via a suitable multidimensional (nonlinear) logic.
	www.physics.nyu.edu /faculty/sokal/transgress_v2/node4.html (482 words)

	DED: Database of Evolutionary Distances -- Veeramachaneni and Makalowski 33 (Supplement 1): D442 -- Nucleic Acids ...
		Note that while the proteins are 100% identical, the alignment picture shows that the gene structure is not—there appears to be an intron gain in the rat lineage.
		Group structure and phylogenetic tree for a homology group.
		Pairwise comparison analysis suggests that the homology relationship between fugu and zebrafish genes can be ignored and the group split into two smaller groups.
	nar.oxfordjournals.org /cgi/content/full/33/suppl_1/D442 (1744 words)

	Floer Homology Groups in Yang-Mills Theory - Cambridge University Press (Site not responding. Last check: 2007-11-07)
		The concept of Floer homology has been one of the most striking developments in differential geometry over the past 20 years.
		It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles.
		The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory.
	www.cambridge.org /catalogue/print.asp?isbn=0521808030&print=y (248 words)

	Dwyer: Vanishing homology over nilpotent groups (Site not responding. Last check: 2007-11-07)
		Suppose that G is a finitely generated nilpotent group, and that M is a finitely generated module over Z[G].
		The main theorem of the paper states that if the zero dimensional group homology of G with coefficients in M vanishes, then all of the higher homology groups of G with coefficients in M also vanish.
		In the special case in which G is an infinite cyclic group, this is closely related to the statement that if f is a surjective endomorphism of a finitely generated module over a noetherian ring, then f is actually an automorphism (think of the increasing sequence of kernels of powers of f).
	www.nd.edu /~wgd/Html/Vanishing.Homology.Nilpotent.Groups.html (176 words)

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