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Topic: Cohomology theories

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	Cohomology - Wikipedia, the free encyclopedia
		That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries.
		Cohomology arises from the algebraic dualization of the construction of homology.
		A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms.
	en.wikipedia.org /wiki/Cohomology (705 words)

	Group cohomology - Wikipedia, the free encyclopedia
		Some general theory was supplied by Mac Lane and Lyndon; from a module-theoretic point of view this was integrated into the Cartan-Eilenberg theory, and topologically into an aspect of the construction of the classifying space BG for G-bundles.
		Galois cohomology is a large field, and now basic in the theories of algebraic groups and étale cohomology (which builds on it).
		The analogous theory for Lie algebras, called Lie algebra cohomology and largely developed after early papers in the late 1940s, by Jean-Louis Koszul, is formally similar, starting with the corresponding definition of invariant.
	en.wikipedia.org /wiki/Group_cohomology (809 words)

	List of cohomology theories - Wikipedia, the free encyclopedia
		This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra.
		These are the theories satisfying the "dimension axiom" of the Eilenberg-Steenrod axioms that the homology of a point vanishes in dimension other than 0.
		The cohomology functors of ordinary cohomology theories are represented by Eilenberg-Mac Lane spaces.
	en.wikipedia.org /wiki/List_of_cohomology_theories (1101 words)

	Cohomology (Site not responding. Last check: 2007-11-03)
		In mathematics, cohomology is a general termfrom homological algebra, for a sequence of abelian groups defined from a cochain complex.
		Although cohomology is fundamental to modern algebraictopology, its importance was not seen for some 40 years after the development of homology.
		A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereofsuch as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms
	www.therfcc.org /cohomology-35908.html (620 words)

	Cohomology (Site not responding. Last check: 2007-11-03)
		In mathematics cohomology is a general term from homological algebra for a sequence of abelian groups defined from a cochain complex.
		Although cohomology is fundamental to modern algebraic topology its importance was not seen for 40 years after the development of homology.
		De Rham cohomology and the theory of characteristic classes are not only two of the most important topics in mathematics, but also in theoretical physics.
	www.freeglossary.com /Cohomology (821 words)

	PlanetMath: Lie algebra cohomology (Site not responding. Last check: 2007-11-03)
		There are analogous cohomology theories for groups, associative algebras, and commutative rings.
		The aim was to calculate the cohomology, in the topological sense, of a compact Lie group by using the finite-dimensional data of the corresponding Lie algebra.
		This is version 9 of Lie algebra cohomology, born on 2003-08-14, modified 2006-02-03.
	www.planetmath.org /encyclopedia/Cohomology2.html (647 words)

	Motivic cohomology
		Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by Grothendieck during the 1960s.
		At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry.
		It had a basis in category theory for drawing consequences from those conjectures; Grothendieck and Bombieri showed the depth of this approach by deriving a conditional proof of the Weil conjectures by this route.
	www.libraryoflibrary.com /E_n_c_p_d_Motivic_cohomology.html (251 words)

	Lecture Notes on Motivic Cohomology
		The theory of (étale, Nisnevich and Zariski) sheaves with transfers is developed in parts two, three and six, respectively.
		Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry.
		We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory.
	www.claymath.org /publications/Motivic_Cohomology (280 words)

	Institute for Advanced Study: Press Releases: INSTITUTE FOR ADVANCED STUDY FACULTY MEMBER VLADIMIR VOEVODSKY WINS 2002 ...
		The notion of cohomology first arose in topology, which can be loosely described as "the science of shapes." Examples of shapes studied are the sphere, the surface of a doughnut, and their higher-dimensional analogues.
		Generalized cohomology theories extract data about properties of topological objects and encode that information in the language of groups.
		One of the most important of the generalized cohomology theories, topological K-theory, was developed chiefly by another 1966 Fields Medalist, Michael Atiyah.
	www.ias.edu /midcom-permalink-e91c75d0f32fb8e65d6bf132d4fddde6 (909 words)

	Springer Online Reference Works
		The notion of cohomology is dual to that of homology (see Homology theory; Homology group; Aleksandrov–Čech homology and cohomology).
		There are two cohomology theories with values (or coefficients) in sheaves of Abelian groups: Čech cohomology and Grothendieck cohomology.
		A disadvantage of Čech cohomology is that (for non-paracompact spaces) it does not form a cohomology functor (see Homology functor).
	eom.springer.de /c/c023060.htm (852 words)

	Interests
		One of the definition is given in the joint preprint of I.Panin and A.Smirnov " Push-forward in oriented cohomology theories of algebraic varieties" (http://www.math.uiuc.edu/K-theory/0459).
		Among the oriented cohomology theories in the since of the very first mentioned concept are usual singular cohomology (if the groun field is the field of complex numbers), etale cohomology, K-theory, motivic cohomology, algebraic cobordism and many others.
		An oriented cohomology theory $A$ is a ring cohomology theory (in the since of the preprint "Push-forwards in oriented cohomology theories of algebraic varieties" (http://www.math.uiuc.edu/K-theory/0459) equipped with an Euler structure (another term is Chern structure and aposteriory with an integration).
	www.pdmi.ras.ru /~panin/research.html (495 words)

	Nick Kuhn, University of Virginia (Site not responding. Last check: 2007-11-03)
		My work in topology has included work on the development of a character theory for complex oriented cohomology theories, iterated loopspace theory, Goodwillie functor calculus, periodic homotopy, topological realization questions, stable homotopy groups, and model categories.
		One novelty was the use of modular representation theory of finite semigroups.
		These take the form: the mod p cohomology of a topological space must be either 'very small' or 'very large,' where cohomology is organized by using both the nilpotent and Krull filtrations of the category of unstable modules over the Steenrod algebra.
	www.math.virginia.edu /Faculty/Kuhn (1090 words)

	学術専門洋書販売(株)ニュートリノ: Generalized Cohomology (Site not responding. Last check: 2007-11-03)
		The purpose of the book is to give an exposition of generalized (co)homology theories that can be read by a wide group of mathematicians who are not experts in algebraic topology.
		Then the authors discuss various types of generalized cohomology theories, such as complex-oriented cohomology theories and Chern classes, $K$-theory, complex cobordisms, and formal group laws.
		A separate chapter is devoted to spectral sequences and their use in generalized cohomology theories.
	www.neutrino.co.jp /publication/0821835149 (213 words)

	Conference in Goemetric Topology
		Cohomology theories of quandles, that are analogues of cohomology theories of groups and other algebraic systems, have been developed.
		State-sum invariants called quandle cocycle invariants were defined for knots and knotted surfaces, using quandle colorings and cocycles of a quandle cohomology theory, and applications to non-invertibility and triple point numbers of knotted surfaces were given.
		A cohomology theory for set-theoretic Yang-Baxter equations is defined from this context, and corresponding cocycle state-sum invariants will be discussed.
	www.cs.uiowa.edu /~wu/gtc/abs/MasaSait.htm (133 words)

	Springer Online Reference Works
		A generalization of the various cohomology theories of algebraic varieties.
		in the classical theory of correspondences, and the use of this theory in the study of the zeta-function of a curve
		The theory of motives is universal in the sense that every geometric cohomology theory, of the type of the classical singular cohomology for algebraic varieties over
	eom.springer.de /M/m065040.htm (476 words)

	Amazon.com: Elliptic Cohomology (University Series in Mathematics): Books: Charles B. Thomas (Site not responding. Last check: 2007-11-03)
		The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms.
		A map from the subgroups of G to the (complex-oriented) cohomology of their classifying space is a Mackey functor, and the author shows that in fact it is a 'Green functor' in that the family of modules has a multiplicative structure and the restriction and induction maps satisfy Frobenius reciprocity.
		This gives rise to a group cohomology that generalizes that of the ordinary cohomology of the classifying space BG of the group G. The author shows how to compute this cohomology for BG, the first result being that for cyclic prime subgroups of G, the (complex oriented) cohomology is generated by the Chern classes.
	www.amazon.com /Elliptic-Cohomology-University-Mathematics-Charles/dp/0306460971 (2040 words)

	New Contexts for Stable Homotopy Theory
		The cohomology theories themselves are represented by geometric objects; only very recently have really satisfactory models been found, opening a wide range of new possibilities for exploiting algebraic ideas in topology, geometry, number theory and physics.
		The development of cohomology theories which behaved in positive characteristic like the classical cohomology used by Lefschetz was a major achievement of the 20th century.
		It is natural to ask whether there is a functorial way to associate a complex orientable cohomology theory to a formal group, lifting parts of the category of formal groups to the category of spectra itself, not merely to the homotopy category.
	www.newton.cam.ac.uk /reports/0203/nst.html (2154 words)

	Etale Cohomology. (PMS-33)
		Milnes account of \etale cohomology is sometimes useful---in that it proves various little lemmata and criteria that come handy in practice---but disorganised the rest of the time.
		One example of this are the integer cohomology groups, which for an irreducible variety are all zero except in dimension zero.
		Cech cohomology is introduced mostly as a device to compute the cohomology groups, which is difficult to do directly when expressed in terms of derived functors.
	www.wkonline.com /a/Etale_Cohomology_PMS33_0691082383.htm (1098 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.
		While the general theory of their development is more properly a topic in homological algebra, certain spectral sequences are very commonly used in algebraic topology.
		Homotopy theory focuses on the most intrinsically invariable features of a space -- for example, all Euclidean spaces are in homotopy theory considered the same, since all are contractible; the circle is decidedly distinct, since it has a hole.
	www.math.niu.edu /~rusin/known-math/index/55-XX.html (2581 words)

	[No title]
		After all, he became famous for his work on homotopy theory, he invented the axioms of conformal field theory - borrowing lots of ideas from string theory, of course - and I'm sure he mastered the theory of elliptic curves one weekend when he was a kid.
		To be a generalized cohomology theory, this gadget must satisfy a bunch of axioms called the Eilenberg-Steenrod axioms.
		Eventually, you learn that underlying any generalized cohomology theory there is a list of spaces E(n) such that h^n(X) = [X, E(n)] where the right-hand side is the set of homotopy classes of maps from X to E(n).
	www.math.niu.edu /~rusin/known-math/00_incoming/gen_coho (3543 words)

	John Greenlees's Preprint Archive
		The cohomology ring of a finite group is shown to enjoy certain global duality properties in the sense that the local cohomology theorem holds: in particular this gives a new proof of the result of Benson and Carlson that if the cohomology ring of a finite group is Cohen-Macaulay it is also Gorenstein.
		For each elliptic curve A over the rational numbers we construct a 2-periodic S^1-equivariant cohomology theory E whose cohomology ring is the sheaf cohomology of A; the homology of the sphere of the representation z^n is the cohomology of the divisor A(n) of points with order dividing n.
		This is a thumbnail sketch introduction to equivariant cohomology theories and a a very brief summary of the cases where an algebraic model of rational G-spectra is known.
	www.greenlees.staff.shef.ac.uk /preprints.html (2954 words)

	Cornell Math - Thesis Abstracts (Lie Groups)
		Abstract: This work concerns the computation of various L_2 cohomology theories, where L_2 cohomology is an analogue of de Rham cohomology on complete Riemannian manifolds which demands the forms under consideration be square integrable.
		Finally, the manifolds under study are sufficiently close to arithmetic quotients of rank one symmetric spaces that the cohomology of the links of the cusp points at infinity admits a weight space decomposition, allowing the definition of an analogue of the weighted cohomology of [GHM94] on them.
		Warped cohomology allows results on such spaces which are similar to the weighted L_2 construction of weighted cohomology (theorem A) derived on arithmetic quotients of symmetric spaces (of any rank) in [Nai99].
	www.math.cornell.edu /Research/Abstracts/lie_groups.html (1159 words)

	[No title]
		Araki determined the mod p cohomologies of E_k for the rest of the all cases, namely, for (p,k)=(2,6),(2,7),(2,8),(3,7),(3,8), by a deep considerations on results of Bott-Samelson making use of a theorem of R.Bott which claims that the integral cohomology of the loop space of a simply connected compact Lie group is torsion free (J. Math.
		In this lecture note, he made use of the theory of P.Cartier on typical curves over formal groups to do a unified treatment in showing the Quillen decomposition of localized complex cobordism theory and the Adams decomposition of localized complex K-theory.
		He determined the ring structure of the ground ring of such formal group and showed that this ring is isomorphic to the image of the canonical map from the coefficient ring of the complex cobrodism theory to that of the oriented cobordism theory if p=2.
	www.lehigh.edu /~dmd1/ay1012 (1091 words)

	PlanetMath: de Rham cohomology (Site not responding. Last check: 2007-11-03)
		This action on differentiable maps makes the de Rham cohomology into a contravariant functor from the category of paracompact
		It turns out to be homotopy invariant; this implies that homotopy equivalent manifolds have isomorphic de Rham cohomology.
		This is version 5 of de Rham cohomology, born on 2004-06-12, modified 2004-10-12.
	www.planetmath.org /encyclopedia/DeRhamCohomologyGroup.html (123 words)

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