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	Cohomology - Wikipedia, the free encyclopedia
		Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology.
		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)

	Eilenberg-MacLane space - Wikipedia, the free encyclopedia
		These spaces are important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations.
		Another version of this result, due to Peter J. Huber, establishes a bijection with the n-th Čech cohomology group when X is Hausdorff and paracompact and G is countable, or when X is Hausdorff, paracompact and compactly generated and G is arbitrary.
		A further result of Morita establishes a bijection with the n-th numerable Čech cohomology group for an arbitrary topological space X and G an arbitrary abelian group.
	en.wikipedia.org /wiki/Eilenberg-Mac_Lane_space (444 words)

	[No title]
		The bottom Cech cohomology CH0I(A; M) is sometimes known as the "ideal transfor* *m" of M.
		(ii) Although the usual Cech complex is concentrated in codegrees from 0 to n-1* , the given projective approximation is concentrated in codegrees from -n-1 to n; from Rema rk (ii) of Section 1, it is quasi-isomorphic to a complex concentrated in codegrees -1 to *n.
		For generalised cohomology t* heories the phenomenon takes on the present character; special cases of it show that the sp ectra con- sidered by Mahowald and others in relation to Lin's theorem are instances of Ta *te spectra (see [13, Sections 13 and 16] for further discussion and detailed references).
	www.math.purdue.edu /research/atopology/Greenlees/local_tate_cohomology.txt (7297 words)

	[No title]
		The De Rham cohomology of $M$ is precisely the cohomology of the differential complex $$\dots \to \Omega^n(M) \to \Omega^{n+1}(M) \to \dots$$ with differential operator $d$.
		More generally, for some sheaf $\Omega$, the cohomology of the complex $$\dots \to C^i(U,\Omega) \to C^{i+1}(U,\Omega) \to \dots$$ computes the Cech cohomology of $M$ with coefficients in $\Omega$, where $C^i(U,\Omega)$ is the sections of the sheaf $\Omega$ over the $i+1$-fold intersections.
		Then the Cech cohomology of a space with coefficients in a flasque sheaf are zero.
	math.berkeley.edu /~mgsa/qualquestions/syllab.txt (1625 words)

	ovaltrack.com Store \| Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)
		The de Rham cohomology, which is the main subject of the book, is explained in here in a way that gives the reader an intuitive and geometric understanding, which is sorely needed, especially for physicists who are interested in applications.
		The Mayer-Vietoris sequence is generalized to the case of countably many open sets in chapter 2, and shown to be isomorphic to the Cech cohomology for a "good" cover of a manifold.
		The authors show that the singular cohomology of a triangularizable space is isomorphic to its Cech cohomology with the constant presheaf the integers.
	www.ovaltrack.com /bookstore/AMStore/AsinInfo.asp?asin=0387906134 (1052 words)

	Seminar on Cohomology of Quasi-Coherent Sheaves
		The vanishing of Cech cohomology of a quasi-coherent sheaf on an affine scheme (lemma 2.17 and thm.
		Corollary: the vanishing of cohomology of a quasi-coherent sheaf on an affine scheme (using the result about the vanishing of Cech cohomology from the previous lecture, and a result in Godement's book, which we will use as a fact; see [EGA] III.1.3.1).
		II.6.3 in [G], esp. the footnote); and as application the cohomology of the sheaves associated to divisors of bidegree (a,b).
	www.math.leidenuniv.nl /~bogaart/seminarfall05.html (641 words)

	[No title]
		An important theorem says this class is necessarily an integral class, that is, it comes from an element of the 2nd cohomology with integer coefficients; moreover, isomorphism classes of line bundles over a manifold are in one-to-one correspondence with elements of its 2nd cohomology with integer coefficients.
		Note that importance of the SECOND cohomology group in the above story is twofold: 1) symplectic structures give elements of the second cohomology, 2) the curvature of a connection gives an element of the second cohomology, and in fact 2') line bundles are classified by elements of second cohomology.
		The beauty of 2nd cohomology is that integer classes in the 2nd cohomology of M correspond to line bundles on M; there is, in other words, a very nice geometrical picture of 2nd cohomology classes.
	math.ucr.edu /home/baez/twf_ascii/week25 (2476 words)

	PlanetMath: de Rham cohomology
		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.
	planetmath.org /encyclopedia/DeRhamCohomology.html (123 words)

	Painless intro to spectral sequences and algebraic topology \| Differential Forms in Algebraic Topology...
		The authors though explain that the spectral sequence is nothing other than a generalization of the double complex of differential forms that was considered in chapter 2.
		This text, developed from a first-year graduate course in algebraic topology, is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory.
		By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate.
	www.very-clever.com /information/dkoizdqake (1419 words)

	Koszul biography
		In 1950 Koszul published a major 62 page paper Homologie et cohomologie des algèbres de Lie in which he studied the connections between the homology and cohomology (with real coefficients) of a compact connected Lie group G and purely algebraic problems on the Lie algebra associated with G.
		The superb lecture notes were published in 1957 and covered: Cech cohomology with coefficients in a sheaf; resolutions; a theorem concerning the cohomology with coefficients in a sheaf for a paracompact space; isomorphism of ordinary Cech cohomology with de Rham-cohomology, Alexander-Spanier- cohomology, and singular cohomology.
		This was first introduced to define a cohomology theory for Lie algebras and turned out to be a useful general construction in homological algebra.
	www-groups.dcs.st-and.ac.uk /history/Biographies/Koszul.html (793 words)

	Sheaf Theory
		This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." The parts of sheaf theory covered here are those areas important to algebraic topology.
		The fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces.
		Among the items added are new sections on Cech cohomology, the Oliver transfer, intersection theory, generalized manifolds, locally homogeneous spaces, and homological fibrations.
	www.booksmatter.com /b0387949054.htm (223 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		Question: Is this statement true for sheaf cohomology with (trivial) coefficients in Z? Assume that X,Y are paracompact and hence H* is isomorphic to Alexander-Spanier and Cech cohomologies.
		If you are willing to use compact metric spaces then there is a very nice homology theory dual to Cech cohomology - it is called Steenrod homology and (remarkably enough) was developed by Steenrod in his 1940 paper in Mich Math.
		It has a wedge axiom - the homology of the strong wedge is the product of the homology of the individual pieces, and there is a UCT with Cech cohomology as well as a lim - lim^1 sequence for the inverse limit of finite complexes.
	www.lehigh.edu /~dmd1/mg123.txt (316 words)

	More algebraic geometry questions (Site not responding. Last check: 2007-10-30)
		There are many different theories of cohomology in algebraic geometry, usually with sheaf coefficients, but with various different topologies, such as Zariski topology, or etale topology, in which an "open set" is a covering map onto an actual open set in the space considered.
		There are several constructions of cohomology, the most intuitive being Cech cohomology used by Serre in his famous paper Faisceaux algebriques coherents, and more generally, derived functor cohomology, introduced by Grothendieck, and discussed in his famous Tohoku paper, "Sur quelques points d'algebre homologique".
		This quotient group is called the first cohomology group with coefficients in the "sheaf" of holomorphic, or regular algebraic, functions on the Riemann surface, or algebraic curve.
	www.physicsforums.com /showthread.php?p=311114 (1995 words)

	Amazon.ca: Sheaf Theory: Books: Glen E. Bredon (Site not responding. Last check: 2007-10-30)
		Primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems", the parts of sheaf theory covered here are those areas important to algebraic topology.
		The cohomology of sheaves is discussed in the next chapter, and many examples are given illustrating the main points, along with relative cohomology.
		Although short, the author's discussion is effective in that he clarifies the need for a paracompactifying family of supports, generalizing the paracompactness hypothesis needed in the usual cohomology theories.
	www.amazon.ca /exec/obidos/ASIN/0387949054 (753 words)

	Matches for: (Site not responding. Last check: 2007-10-30)
		Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local holomorphic functions on a complex manifold or the local sections of a vector bundle.
		To study schemes, it is useful to study the sheaves defined on them, especially the coherent and quasicoherent sheaves.
		For example, in studying ampleness, it is frequently useful to translate a property of sheaves into a statement about its cohomology.
	www.mathaware.org /bookstore?fn=20&arg1=mmonoseries&item=MMONO-197 (376 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		These geometric ideas will be introduced in Section (cohomologies for a topos) and Section (the general fundamental group).
		In Section the sheaf cohomology groups of an arbitrary topos are introduced.
		Homotopy and cohomology of topoi are discussed extensively in [] (for the fundamental group), [] (vol.
	www1.elsevier.com /homepage/saj/523281/h14.htm (667 words)

	The One-Dimensional Cech Cohomology of the Higson Compactification and its Corona by James Keesling (Site not responding. Last check: 2007-10-30)
		The One-Dimensional Cech Cohomology of the Higson Compactification and its Corona by James Keesling
		The One-Dimensional Cech Cohomology of the Higson Compactification and its Corona
		In this paper it is shown that under very general conditions the 1-dimensional Cech cohomology of the Higson compactification contains a subgroup isomorphic to the additive real numbers.
	at.yorku.ca /b/a/a/h/12.htm (108 words)

	Seminar On Cohomology
		Goal: Grothendieck topologies and etale and crystalline cohomologies; applications to zeta functions and the Weil conjectures; de Rham cohomology and its variation in families; calculation of cohomology groups.
		Review of homological algebra, sheaf cohomology via derived functors and via Cech cohomology, some key theorems, examples of cohomology groups (in particular, Zariski is no good for constant coefficients).
		Weil cohomology, zeta functions, cohomolgical interpretation, the zeta function of a curve and its Jacobian, variation of zeta functions, H0et and the zeta function of some zero dimensional schemes (including automorphicity), Grassmann varieties and the schubert calculus, the zeta function of Grassmann varieties.
	www.math.mcgill.ca /goren/SeminarOnCohomology.html (242 words)

	Theory and Applications of Categories, Vol. 9, No. 3, 2001, pp. 43--60. (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Abstract: The interpretation by Duskin and Glenn of abelian sheaf cohomology as connected components of a category of torsors is extended to homotopy classes.
		This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy.
		2 Eilenberg-MacLane toposes and cohomology (context) - Joyal, Wraith - 1984
	citeseer.ist.psu.edu /668643.html (322 words)

	Re: Cech cohomology and bundles
		More generally, if I'm not mixed up, we can think of a Cech >1-cocycle as defining a principal G-bundle, and a Cech 1-coboundary >as defining a bundle isomorphism, so that the 1st Cech cohomology >group consists of isomorphism classes of principal G-bundles.
		This is closely related >to the "stabilization" we see in the periodic table of n-categories.
		If we work with more general sorts of >n-bundles, we'll find that they're classified by some more general >kind of Cech cohomology, which is often alluded to under the name >of "nonabelian cohomology" - mainly by people saying they wish they >knew what this was.
	www.lns.cornell.edu /spr/2001-10/msg0036336.html (1151 words)

	Re: Cech cohomology and bundles
		I can discuss ONE-dimensional nonabelian cohomology, which is directly related to the interpretation John mentioned, but I don't know how to do nonabelian cohomology in other dimensions.
		I asked Klaus Schmidt (the dynamical systems world expert on cohomology) about this a few years ago, and he told me that he knows of no nonabelian cohomology other than the one-dimensional kind.
		the reference to sheaf cohomology: one rumour I have heard is that there is a nonabelian sheaf cohomology practiced in France.
	www.lns.cornell.edu /spr/2001-10/msg0036337.html (345 words)

	PlanetMath: Cech cohomology group
		, called the complex of Cech cochains relative to the covering
		Anyone with an account can edit this entry.
		This is version 1 of Cech cohomology group, born on 2004-10-10.
	planetmath.org /encyclopedia/CechCohomologyGroup2.html (81 words)

	cohomology - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "cohomology" is defined.
		Cohomology : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include cohomology: cech cohomology, cohomology class, cohomology group, dolbeault cohomology, galois cohomology, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=cohomology (101 words)

	The One-Dimensional Cech Cohomology of the Higson Compactification and its Corona (Site not responding. Last check: 2007-10-30)
		A detailed description of this compactification and its elementary properties is given in 1.
		In this paper we show that in very general circumstances the 1-dimensional Cech cohomology of the Higson compactification contains a subgroup isomorphic to the additive real numbers.
		The techniques of A. Calder and J. Siegel ([1] and [2]) are relevant to the study of the higher-dimensional Cech cohomology of
	at.yorku.ca /b/a/a/h/12.l2h (565 words)

	MAT 539 -- Algebraic Topology -- Spring 2003
		The guiding principle of the book is to use differential forms and in fact the de Rham theory of differential forms as a prototype of all cohomology thus enabling an easier access to the machineries of algebraic topology in the realm of smooth manifolds.
		The material is structured around four core sections: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes, and includes also some applications to homotopy theory.
		The Cech-de Rham complex: the generalized Mayer-Vietoris argument, sheaves and Cech cohomology, the de Rham theorem, sphere bundles, Euler class, the Hopf index theorem, the Thom isomorphism in general, monodromy
	www.math.sunysb.edu /~sorin/539 (708 words)

	Categorified Gauge Theory
		Just as the electromagnetic vector potential should really be regarded as a connection on a U(1) bundle, the Kalb-Ramond field should really be thought of as a connection on a "U(1) gerbe".
		Moreover, just as U(1) bundles are classified by the 1st Cech cohomology with coefficients in the sheaf of smooth U(1)-valued functions, U(1) gerbes are classified by the 2nd Cech cohomology with coefficients in this sheaf.
		We conclude by sketching how nontrivial C-2-bundles can be classified by the 2nd nonabelian Cech cohomology.
	math.ucr.edu /home/baez/gauge (604 words)

	Math 262
		Universal Coefficients Theorem (for both cohomology and homology)
		Equivalence of Cech, singular, and De Rham cohomology
		Kenneth S. Brown, Cohomology of Groups, Springer-Verlag 1982.
	www.cgtp.duke.edu /~psa/cls/262 (117 words)

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