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Topic: Betti cohomology

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	PlanetMath: Lie algebra cohomology (Site not responding. Last check: 2007-10-07)
		Generalizing a bit, Lie algebra cohomology is just the cohomology of a particular kind of algebraic theory.
		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.
	planetmath.org /encyclopedia/Cohomology2.html (647 words)

	De Rham cohomology - ExampleProblems.com
		In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.
		It is a cohomology theory based on the existence of differential forms with prescribed properties.
		The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.
	www.exampleproblems.com /wiki/index.php/De_Rham_cohomology (1086 words)

	Étale cohomology Information
		This theory is an example of a Weil cohomology theory in algebraic geometry, and as such it continues to play an important role in the more general theory of motives.
		The formal definition of étale cohomology is as the derived functor of the functor of sections,
		In general the l-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the l-adic integers (or numbers) rather than the integers (or rationals).
	www.bookrags.com /L-adic_cohomology (1173 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-07)
		One of the constructions of cohomology of abstract algebraic varieties and schemes.
		Cohomology with coefficients in rings of characteristic zero is used for various purposes, mainly in the proof of the Lefschetz formula and its application to zeta-functions.
		It is obtained from étale cohomology by passing to the projective limit.
	eom.springer.de /L/l057020.htm (221 words)

	The Hodge Filtration And Cyclic Homology - Weibel (ResearchIndex)
		We relate the "Hodge filtration" of the cohomology of a complex algebraic variety X to the "Hodge decomposition" of its cyclic homology.
		If X is smooth and projective, HC (i) n (X) is the quotient of the Betti cohomology H 2i\Gamman (X(C); C) by the (i + 1) st piece of the Hodge filtration.
		1 Cyclic homology and the de Rham cohomology of commutative al..
	citeseer.ist.psu.edu /256060.html (627 words)

	Weil cohomology theory - Weilcohomologytheory
		Mathematically, the theory of motives is then the conjectural "universal" cohomology theory for such objects.
		While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 1960-1970, that hope for it remains unfulfilled.
		The general idea is that one motive has the same structure in any reasonable cohomology theory with good formal properties; in particular, any Weil cohomology theory will have such properties.
	www.kopete.org /Weil-cohomology-theory.html (844 words)

	Algebraic topology
		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[?].
		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.ebroadcast.com.au /lookup/encyclopedia/al/Algebraic_topology.html (335 words)

	Remarks on absolute de Rham and absolute Hodge cycles
		by the comparison between the Leray spectral sequences for the Betti and the de Rham cohomologiesi, and the regularity of Gauss-Manin.
		Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ.
		cohomology in the Poincaré metric, Annals of Math.
	www.imsc.res.in /~kapil/papers/ln/index.html (889 words)

	Algebraic topology - ExampleProblems.com
		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.
		De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology.
	www.exampleproblems.com /wiki/index.php/Algebraic_topology (621 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)

	Citations: An incremental algorithm for Betti numbers of simplicial complexes - Delfinado, Edelsbrunner (ResearchIndex)
		Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it.
		For example # 0 = # n 0 = 1, # 1 = # n 1, and # 2 = # n 2 = 1 are the Betti numbers of K = K n.
		For example fi 0 = fi n 0 = 1, fi 1 = fi n 1, and fi 2 = fi n 2 = 1 are the Betti numbers of K = K n.
	citeseer.ist.psu.edu /context/30254/0 (3410 words)

	Ugo Betti - UgoBetti
		Ugo Betti (Camerino, February 4, 1892 - Rome, June 9, 1953) was an Italian judge, better known as an author, who is considered by many the greatest Italian playwright next to Pirandello.
		Betti studied law in Parma at the time when World War I broke out, and he volunteered as a soldier.
		After the war he finished his studies and became a judge.
	www.kopete.org /Ugo-Betti.html (314 words)

	LMS Proceedings Abstract, paper PLMS 1470 (Site not responding. Last check: 2007-10-07)
		Let $(L, M)$ be an admissible pair of simplices whose faces are defined as hyperplanes in ${\mathbb P}_F^n$ such that they do not have common faces of the same dimension.
		When $L$ and $M$ are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology $H^{\ast}({\mathbb P}_F^n\setminus L, M\setminus L; {\mathbb Q})$.
		The motivic cohomology provides a simple explanation of why the generic part of $A_\bullet$ forms a Hopf algebra with well-defined coproduct.
	www.lms.ac.uk /publications/proceedings/abstracts/p1470a.html (178 words)

	Wikinfo \| Betti number (Site not responding. Last check: 2007-10-07)
		In the case that X is a simplicial complex, assumed built up from a finite number of simplices, the sequence of Betti numbers is 0 from some points onwards, and consists of natural numbers.
		The Betti numbers do not take into account any torsion in the homology groups, but they are very useful basic topological invariants.
		If M is compact and oriented, the dimension of its kernel acting upon the space of p-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree p: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms.
	www.wikinfo.org /wiki.php?title=Betti_number (527 words)

	Abstracts
		Given a variety defined over the complex numbers (even singular), the problem of defining a suitable De Rham cohomology and its comparison with other cohomologies was posed several years ago by many authors (see for example Grothendieck, Hartshorne, Deligne...).
		This has led (among other reasons) to the definition of a infinitesimal sites and finally to the comparison between the Algebraic De Rham Cohomology of a singular scheme, its Infinitesimal (or Crystalline) Cohomology, and the Betti Cohomology of its associated analytic space (see works of Grothendieck, Hartshorne, Deligne, Herrera-Lieberman, Illusie, Berthelot, Ogus, and others).
		We define a cohomology theory for locally analytic representations of p-adic Lie groups in the sense of Schneider/Teitelbaum.
	www.uam.es /otros/aag2006/abstracts.html (1598 words)

	UC Berkeley Combinatorics Seminar (Site not responding. Last check: 2007-10-07)
		Hypertoric varieties are algebraic varieties which play roughly the same role for hyperplane arrangements that toric varieties play for convex polyhedra.
		whose dimensions are given by important combinatorial invariants of the matroid of the arrangement: for instance, the cohomology betti numbers of a rationally smooth hypertoric variety are the h-numbers of the independence complex of the matroid, while the intersection cohomology betti numbers of an affine hypertoric variety are the h-numbers of the "broken-circuit" complex.
		We use this construction to produce a canonical dual pairing between the top-degree cohomology of a smooth hypertoric variety and the "Morse" intersection cohomology of the affine hypertoric variety coming from the Gale dual arrangement.
	math.berkeley.edu /~kwoods/Seminars/Mar20.html (232 words)

	Proceedings of the American Mathematical Society
		Abstract: Betti numbers for the Heisenberg Lie algebras were calculated by Santharoubane in his 1983 paper.
		K.Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Annals of Math.
		Cairns, Grant; Jessup, Barry; Pitkethly, Jane, On the Betti numbers of nilpotent Lie algebras of small dimension, Integrable systemsand foliations/Feuilletages et systèmes intégrables (Montpellier, 1995), Birkhäuser, Boston, 1997, pp.
	www.mathaware.org /proc/1997-125-02/S0002-9939-97-03607-1/home.html (248 words)

	The Crime Package
		The computation of group cohomology involves several calculations, the results of which are reused in later calculations, and are thus collected in an object of type
		The cohomology object is used to store, in addition to the items mentioned above, the boundary maps, the Betti numbers, the multiplication table, etc.
		The following example calculates the homomorphism on cohomology induced by the inclusion of the cyclic group of size 4 into the dihedral group of size 8.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/crime/doc/crime.xml (1260 words)

	NSDL Metadata Record -- 1-skeleta, Betti numbers, and equivariant cohomology
		Project Euclid: A partnership of independent publishers of mathematics and statistics journals.
		Goresky, R. Kottwitz, and R. MacPherson show that for such a manifold this 1-skeleton has the structure of a "labeled" graph, (?,?), and that the equivariant cohomology ring of M is isomorphic to the "cohomology ring" of this graph.
		In this article we show that this "topological" result is, in fact, a combinatorial result about graphs.
	nsdl.org /mr/469267 (141 words)

	Singular homology - Wikipedia, the free encyclopedia
		The cohomology groups of X are defined as the cohomology groups of this complex.
		They form a graded R-module, which can be given the structure of a graded R-algebra using the cup product.
		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 (393 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-07)
		It has many nice properties of a cohomology theory; in particular there is a motivic Chern character mapping (a sum of projections after tensoring with
		Thus, one obtains a good cohomology theory, with supports, Poincaré duality, even a homological counterpart, satisfying the axioms of a Poincaré duality theory in the sense of S.
		Already the simplest explicit examples suggest one should restrict to "integral motivic cohomology" and one is led to Beilinson's regulator mappings
	eom.springer.de /B/b110220.htm (1070 words)

	Homology and Cohomology
		The n-th Betti number (see also ChapterÂ 14, Algebra) is the rank of the n-th homology group.
		For a closed, orientable surface of genus g, the Betti numbers are p
		The Euler characteristic can be expressed in terms of homology (EquationÂ 9.2, âEuler Characteristic and Betti Numbersâ).
	xistrat.sourceforge.net /docbook/ch09s02.xhtml (152 words)

	Wikinfo \| Algebraic topology (Site not responding. Last check: 2007-10-07)
		Finitely generated abelian groups can be completely classified and are particularly easy to work with.
		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.
	www.wikinfo.org /wiki.php?title=Algebraic_topology (493 words)

	Atlas: Weighted L^2 cohomology of Coxeter groups by Michael Davis (Site not responding. Last check: 2007-10-07)
		-cohomology of of a CW complex lies somewhere between its ordinary cohomology and its cohomology with compact support.
		-cohomology spaces interpolate between the ordinary cohomology of X and its cohomology with compact support.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cami-13.
	atlas-conferences.com /cgi-bin/abstract/cami-13 (243 words)

	Amazon.com: "betti sequences": Key Phrase page (Site not responding. Last check: 2007-10-07)
		As a potential source of bounded Betti sequences, consider periodic modules- if m is a positive integer, then a non-zero R-module M is said to be periodic of...
		On the other hand, there exist rings over which all infinite Betti sequences are unbounded, cf.
		The simplest pattern of bounded Betti numbers is highlighted in...
	www.amazon.com /phrase/betti-sequences (501 words)

	Projects (Site not responding. Last check: 2007-10-07)
		Most of my current research centres on p-adic Hodge theory and its applications to the arithmetic of modular forms and modularity of Galois representations.
		p-adic Hodge theory is a fairly young branch of number theory which seeks to study the p-adic analogue of the relationship between Betti cohomology and de Rham cohomology over the complex numbers.
		The p-adic analogue of Betti cohomology is p-adic etale cohomology, and it is related to other cohomology theories using the p-adic period rings introduced by J.M Fontaine.
	www.math.uchicago.edu /~kisin/projects.html (260 words)

	1-skeleta, Betti numbers, and equivariant cohomology, V. Guillemin, C. Zara
		The 1-skeleton of a G-manifold M is the set of points p \in M, where dim G
		Formule de localisation en cohomologie équivariante, C. Acad.
		[Ki] F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Math.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.dmj/1091736759 (706 words)

	Amazon.com: "local cohomology": Key Phrase page (Site not responding. Last check: 2007-10-07)
		The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry (Graduate Texts in Mathematics) by David Eisenbud
		Finally, I have included two appendices to help the reader: Appendix 1 explains local cohomology and its relation to sheaf cohomology, and Appendix 2 surveys, without proofs, the relevant commutative algebra.
		Finally, I have included two appendices to help the reader: Appendix 1 ex- plains local cohomology and its relation to sheaf cohomology, and Appendix 2 surveys, without proofs, the relevant commutative algebra.
	www.amazon.com /phrase/local-cohomology (531 words)

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