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Topic: Universal coefficient theorem

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	Universal coefficient theorem - Wikipedia, the free encyclopedia
		In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A.
		The form of the result is that other coefficients A may be used, at the cost of using a Tor functor.
		The theorem describes the cokernel of ι as
	en.wikipedia.org /wiki/Universal_coefficient_theorem (340 words)

	PlanetMath: universal coefficient theorem (Site not responding. Last check: 2007-11-03)
		The Universal Coefficient Theorem for homology expresses the homology groups with coefficients in an arbitrary abelian group
		The universal coefficient theorem for cohomology expresses the cohomology groups of a complex in terms of its homology groups.
		This is version 17 of universal coefficient theorem, born on 2003-02-17, modified 2004-04-21.
	planetmath.org /encyclopedia/UniversalCoefficentTheorem.html (278 words)

	CMB - A universal coefficient decomposition for subgroups induced by submodules of group algebras
		Dimension subgroups and Lie dimension subgroups are known to satisfy a `universal coefficient decomposition' and {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the `universal' coefficient rings given by the cyclic groups of infinite and prime power order.
		Here this fact is generalized to much more general types of induced subgroups and notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series and as well as certain common generalisations of these which occur in the study of the former.
		This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi and Parmenter and Seghal) and to all additive subgroups of group rings.
	journals.cms.math.ca /cgi-bin/vault/view/hartl6842 (190 words)

	[No title]
		In particular, a finite spectrum satisfies the hypotheses on Y and so a special case of Theorem 2.6 is that all maps from an Eilenberg-Mac Lane spectrum to a finite spectrum are phantom.
		Theorem 6.4 of [3] identifies this subgroup as PExt (H-1 X, G), where we write PExt (A, B) for the 8 J. subgroup of Ext (A, B) consisting of the divisible elements.
		A corollary of this theorem is that Wn-1 defines a contravariant self-equivalence of the category of F (n)-local spectra of n-finite type.
	jdc.math.uwo.ca /papers/all-or-nothing.txt (8072 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		This is the approach to an equivariant freeness theorem taken by the second author in [11].
		Unfortunately, the Universal Coeffici* ent Theorem* for equivariant ordinary cohomology seems inherently less powerful that the corresponding result for nonequivariant ordinary cohomology in that it appl* *ies only to finite, rather than finite-type, complexes.
		One of the particularly attractive aspects of the main freeness theorem in [* 5] is that, since it applies when the cell-attaching boundary maps are nonzero, it * is reasonable to hope that this result could be extended to groups other than Z=p.
	hopf.math.purdue.edu /Ferland-Lewis/FerlandLewis.txt (12652 words)

	Courses
		Vector analysis: algebra and geometry of vectors, vector differential and integral calculus, theorems of Green, Gauss, and Stokes; complex analysis: analytic functions, complex integrals and residues, Taylor and Laurent series.
		Fundamental theorem of arithmetic, quadratic residues and quadratic reciprocity, number-theoretic functions, certain diophantine equations, Farey fractions, continued fractions.
		Differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stoke's theorem, De Rham theory, Riemannian metrics.
	www.wisc.edu /grad/catalog/letsci/mathemC.html (3682 words)

	[No title]
		Rosenberg and C. Schochet, The Kunneth Theorem and the Universal Coefficient Theorem for Equivariant K- and KK- Theory, Mem.
		Rosenberg and C. Schochet, The Kunneth Theorem and the Universal Coefficient Theorem for Kasparov's Generalized K-functor, Duke Math J., 55 (1987), 431-474.
		Schochet, The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups, K-theory, to appear.
	www.math.wayne.edu /~claude/vita.html (611 words)

	CEU, Department of Mathematics and its Applications
		Parametrix for elliptic differential operators, fundamental decomposition theorem for self-adjoint elliptic operators and complexes.
		The Peter-Weyl Theorem on the matrix elements of irreducible representations of compact groups, decomposition of the regular representation.
		Ergodic theorems of von Neumann and of Birkhoff and Khinchin.
	www.ceu.hu /math/Courses/fall0405.html (2845 words)

	[No title]
		J. Rosenberg and C. Schochet, The Kunneth Theorem and the Universal Coefficient Theorem for Equivariant K- and KK-Theory, Mem.
		J. Rosenberg and C. Schochet, The Kunneth Theorem and the Universal coefficient Theorem for Kasparov's Generalized K-functor, Duke Math J., 55 (1987), 431-474.
		C. Schochet, The UCT, the Milnor Sequence, and a canonical decomposition of the Kasparov Groups, K-theory 10 (1996), 49-72.
	www.math.wayne.edu /~claude/vita810.doc (820 words)

	MATH 734. Algebraic Topology
		One can't quite use the Acyclic Models Theorem, since representability of singular homology requires having all continuous maps in the category, and simplicial chains are not functorial for non-simplicial continuous maps.
		Prove that T is a chain homotopy equivalence for finite polyhedra, using the fact that a finite polyhedron has a finite CW structure for which the simplicial chains are just the cellular chains, and the simplicial boundary map coincides with the cellular boundary map.
		Bredon Theorem VI.7.15 shows orientability is equivalent to existence of an atlas of charts for which all the transition functions have positive Jacobian.
	www.math.umd.edu /~jmr/734 (1448 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The universal coefficient theorem says there's an extra term: H^2(G,A) = Hom(H_2(G,Z),A) + Ext(H_1(G,Z),A) I forget if we ever came up with a snappy description of the universal property of H_2(G,Z) - i.e., a snappy description of what a homomorphism from H_2(G,Z) to A amounts to.
		Having finally convinced myself that H^2(G,A) classifies extensions of G by A, I find it terrifying that H_2(G,Z) is sometimes the universal central extension of G. My last shred of sanity relies on the fact that this only happens when G is perfect, and that in general, H_2(G,Z) has some other meaning.
		John McKay -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.
	www.math.niu.edu /~rusin/known-math/00_incoming/schur_mult (535 words)

	Math 215a Home Page
		Homology with coefficients, Tor, and the universal coefficient theorem.
		As applications, outlined proofs of the Brouwer fixed point theorem in two dimensions, the Borsuk-Ulam theorem in two dimensions, and the "fundamental theorem of algebra".
		(11/1) Homology with coefficients, the universal coefficient theorem, and Tor.
	math.berkeley.edu /~hutching/teach/215a-2005/index.html (2991 words)

	The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized -functor, Jonathan ...
		The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized -functor, Jonathan Rosenberg, Claude Schochet
		The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor
		[5] L. Brown, The universal coefficient theorem for ${\mathrm Ext}$ and quasi-diagonality, Operator Algebras and Group Representations, I (Neptun, 1980), Monographs and Studies in Math., vol.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1077306030 (844 words)

	Recent Publications of Jonathan M. Rosenberg (Site not responding. Last check: 2007-11-03)
		(with Claude Schochet) The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor, Duke Math.
		Slides for a talk on "A selective history of the Stone-von Neumann theorem" at the AMS meeting in Baltimore, January, 2003 (special session in honor of the 100th birthdays of Stone and von Neumann).
		Slides for a talk on "Another look at the Universal Coefficient Theorem for Ext" at the AMS meeting in Baltimore, January, 2003 (special session in honor of Larry Brown's 60th birthday).
	www.math.umd.edu /~jmr/jmr_pub.html (1232 words)

	Department of Mathematics - Graduate Course Descriptions
		Equicontinuous families of functions, Weierstrass’ theorem, inverse function theorem and implicit function theorem, integration of differential forms, and Stokes’ Theorem.
		Topics include the fundamental group, covering spaces, covering transformations, the universal covering spaces, graphs and subgroups of free groups, and the fundamental groups of surfaces.
		The course is devoted to cohomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula.
	www.math.virginia.edu /grad_desc_coursespage.htm (1458 words)

	Graduate Courses
		A study of the underlying mathematical principles, and the use of sophisticated software for numerical problems such as spline interpolation, ordinary differential equations, nonlinear equations, optimization, and singular-value decomposition of a matrix.
		Differential and integral calculus in Euclidean spaces, implicit and inverse function theorems, differential forms, and Stokes' Theorem.
		Study of the fundamental theorems of analytic function theory.
	www.math.virginia.edu /grad/grad6.htm (1548 words)

	Topology (Ph.D.)
		Computations of homology groups of finite graphs and manifolds, the Jordan-Brower separation theorem, relation between the fundamental group and the first homology group.
		Homology with arbitrary coefficients, the universal coefficient theorem.
		Definition of cohomology groups, the universal coefficient theorem, excision property, Eilenberg- Steenrod axioms, the Mayer-Vietoris sequence.
	www.math.okstate.edu /~graddir/long-hbk/Topology_Ph_D.html (247 words)

	php-deluxe.net - description Universal coefficient theorem (via CobWeb/3.1 planetlab1.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		Quite generally, the result indicates the relationship that holds between the Betti numbers b i of X and the Betti numbers b i, F with coefficients in a field (mathematics) F.
		These can differ, but only when the characteristic of F is a prime number p for which there is some p -torsion in the homology.
		There is also a universal coefficient theorem for cohomology, involving the Ext functor.
	www.php-deluxe.net.cob-web.org:8888 /wiwimod,index.page,Universal-coefficient-theorem.htm (239 words)

	NCGOA Seminar, Spring 2003
		Abstract: The validity of the universal coefficient theorem in KK-theory is shown to be equivalent to an approximation property for residually finite dimensional C*-algebras.
		Abstract: A well-known theorem of T. Wolff asserts that for every $f \in L^\infty $ on the unit circle $T$, there is a non-trivial $q \in $ QA $=$ VMO$\cap H^\infty $ such that $fq \in $ QC.
		The particular $g$ that we construct also serves to show that a famous factorization theorem of S. Axler for $L^\infty $-functions on the unit circle $T$ cannot be generalized to $S^{2n-1}$ when $n \geq 2$.
	www.math.vanderbilt.edu /~bisch/NCGOA_seminar_spring03.html (1605 words)

	[No title]
		The main result is a Fubini-type theorem, based on the Riemann integral defined as the limit of Riemann sums, with no mention of continuity, A short sketch proof is given.
		This note presents an alternate proof of van Kampen's Theorem from the pushout point of view, for the case where the space is covered by two open sets.
		The easy case of the Universal Coefficient Theorem when the ground ring is a field is discussed, starting from integer coefficients as in Munkres' book.
	www.math.jhu.edu /~jmb/course.html (1740 words)

	COURSE MOD
		Differentiable structures on manifolds, vector fields and flows, tensor bundles, distributions and Frobenius theorem, metric geometry, differential forms, Stokes theorem, Lie groups, connections on manifolds, geodesics, geometry of tangent bundle, curvature, torsion, exponential map, Rienammian geometry, geometry of submanifolds and submersion, relative Gauss-Bonnet theorem, homogeneous and symmetric spaces, topics in differential geometry.
		Simplicial and cellular complexes, simplicial and cellular homology, universal coefficient theorem, Kunneth theorem, cohomology theories, cohomology operations, duality on manifolds, general homotopy theory, fibration nd cofibration, higher homotopy groups, weak homotopy equivalence, Hurewicz theorem, Eilenberg-Maclane spaces, classifying spaces, spectral sequences.
		Simplicial and cellular complexes, simplicial and cellular homology, universal coefficient theorem, Kunneth theorem, cohomology theories, cohomology operations, duality on manifolds, general homotopy theory, fibration and cofibration, higher homotopy groups, weak homotopy equivalence, Hurewicz theorem, Eilenberg-Maclane spaces, classifying spaces, spectral sequences.
	astro1.panet.utoledo.edu /~ascur/0511/CmodMATH.htm (2811 words)

	PhD Requirements (Site not responding. Last check: 2007-11-03)
		The regulations are interpreted by the Graduate Studies Committee which, on written petition from a student, may permit deviations from the rules, provided there are exceptional circumstances.
		In addition to the departmental regulations, there are university requirements which must also be satisfied.
		A student is considered to have a deficiency if in his first semester as a graduate student at SUNYAB he officially enrolls in and completes Math 519 (introductory algebra) or Math 531 (introductory real variables).
	www.math.buffalo.edu /gr_reqts_phd.html (1504 words)

	Mathematics MATH Courses - Graduate Catalog Fall 2006 - University of Maryland (Site not responding. Last check: 2007-11-03)
		Metric spaces, topological spaces, connectedness, compactness (including Heine-Borel and Bolzano-Weierstrass theorems), Cantor sets, continuous maps and homeomorphisms, fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem), surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem), elements of combinatorial topology (graphs and trees, planarity, coloring problems).
		Introduction to differential forms and their applications, and unites the fundamental theorems of multivariable calculus in a general Stokes Theorem that is valid in great generality.
		Inverse and implicit function theorems, Sard's theorem, orientability, degrees, smooth vector bundles, imbeddings and immersions, transversality approximation theorems and applications, isotopy extension theorem, tubular neighborhoods.
	www.gradschool.umd.edu /catalog/courses/MATH.html (2932 words)

	[No title]
		Cohomology of spaces and the universal coefficient theorem.
		The universal coefficient theorem for homology, the cross product in homology and cohomology.
		Matousek: Using the Borsuk-Ulam Theorem; Lectures on Topological Methods in Combinatorics and Geometry (Springer 2002).
	www.math.chalmers.se /~janalve/AlgTopV05/text.html (382 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Read a very detailed proof of the Universal Coefficient Theorem(s), as well as a survey of the Künneth formulas - the final essay for my algebraic topology course.
		Attn: I just noticed some stupid english mistakes and the references didn't work properly in the.pdf version - the only difference being that it was created on a different computer.
		Basically an introduction to complex line bundles and sheaf cohomology, along with a proof of the Riemann-Roch theorem.
	www.math.ubc.ca /people/faculty/aclay/work.html (175 words)

	PlanetMath 2004-01-12 Snapshot: Index of Contributors
		theorem for the direct sum of finite dimensional vector spaces
		$n$th root of $2$ is irrational for $n\ge 3$ (proof using Fermat's last theorem)
		proof of calculus theorem used in the Lagrange method
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/people.html (391 words)

	http://www.math.wisc.edu/graduate/guide-qe.htm (Site not responding. Last check: 2007-11-03)
		Galois extensions and the fundamental theorem of Galois theory.
		Infinite series, theorems of Bolzano-Weierstrass and Heine-Borel, uniform continuity, uniform convergence, Weierstrass theorem (density of polynomials in C[a,b]), Ascoli's theorem, the Riemann integral, differentiation of series and integrals, the contraction principle (in R
		Cauchy-Riemann equations (both homogeneous and inhomogeneous), Cauchy's theorem, Cauchy's formula, the residue theorem, singularities, local behavior, the principle of maximum, Schwarz's lemma, analytic continuation (including the Schwarz reflection principle), Runge's theorem, theorems of Weierstrass and Mittag Leffler, normal families, conformal mapping, harmonic functions.
	www.math.wisc.edu /graduate/guide-qe.htm (1829 words)

	Graduate Courses (Site not responding. Last check: 2007-11-03)
		Three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals are among the subjects studied in detail.
		The Riemann integral, the Lebesgue integral and the convergence theorems.
		Riemann surfaces, theorems of Liouville, Weierstrass and Mittag-Leffler.
	www.math.buffalo.edu /gr_course_list.html (2399 words)

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