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Topic: Hurewicz theorem

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	user.websites.theSite.pubs
		Hurewicz fiber maps with ANR fibers (with T.A. Chapman), Topology, 16 (1977), 131-143.
		Strongly regular mappings with compact ANR fibers are Hurewicz fiberings, Pacific Journal of Mathematics, 75 (1978), 373-382.
		A stable converse to the Vietoris-Smale theorem with applications to shape theory, Trans.
	math.rutgers.edu /~sferry/pubs.html (502 words)

	AMCA: Hurewicz Theorem for Nagata-Assouad dimension by Nikolay Brodskiy (Site not responding. Last check: 2007-09-17)
		Theorem B: asdim(G) ≤ asdim(K)+asdim(H) for any short exact sequence 1→ K→ G→ H→ 1 of countable groups.
		Both Theorems A and B generalize results of Bell and Dranishnikov in which f is Lipschitz and X is geodesic and G, K are finitely generated, respectively.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/q/u/63.htm (145 words)

	[No title]
		\end{enumerate} \item{Margulis' lemma} This theorem states that there is a universal constant $\epsilon$ such that for $\Gamma$ a discrete group of isometries of ${\Bbb H}^3$, the subgroup $\Gamma_\epsilon$ which is generated by elements which translate points a distance $\le \epsilon$ is virtually nilpotent.
		However, it is proved by using the Weierstrass theorem, which states that if $f_n(z)$ is analytic in a region $\Omega$ and the sequence $f_n$ converges uniformly on compact sets to $f$, then $f$ is analytic in $\Omega$, and moreover $f_n'$ converges uniformly to $f'$ on every compact subset of $\Omega$.
		The log of the ratio of the inner and outer circles of $N$ is the modulus of $A$.
	abel.math.harvard.edu /~tomc/qualfolder/syllab.txt (1625 words)

	Grad course descriptions
		Fundamentals of smooth manifolds, Sard's theorem, Whitney's embedding theorem, transversality theorem, piecewise linear and topological manifolds, knot theory.
		Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.
		Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.
	www.math.upenn.edu /grad/courses.html (2365 words)

	[No title]
		The Hurewicz theorem (see 3.35 and 3.57), and some of its natural con- sequences, will easily follow from Theorem 7, at least once the notion of A1- 1 chain complex and the corresponding notion of A1-homology sheaves HA*(X) of a space X are introduced; see Section 3.3.
		An important consequence of the Hurewicz theorem is the unstable A1-connectivity theorem (see Theorem 3.38 in Section 3.3): 6 Theorem 15 Let X be a pointed space and n 0 be an integer.
		The Hurewicz Theorem implies furthermore that for n 2, the first non- trivial A1-homotopy sheaf of the (n - 2)-A1-connected sphere An - {0} ~=A1 1 n-1(Gm ^n) is its ssAn-1and is the free strongly (or strictly by Theorem 14) A1-invariant sheaf of abelian groups generated by the pointed 0-dimensional sphere (Gm)^n.
	hopf.math.purdue.edu /Morel/A1homotopy.txt (19047 words)

	University of Chicago Department of Mathematics
		Hartogs' Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces.
		Inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem.
		Geodesics and the associated variational formalism (formulas for the 1st and 2nd variation of length), the exponential map, completeness, and the influence of curvature on the structure of a manifold (positive versus negative curvature).
	www.math.uchicago.edu /firstyear.html (506 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-09-17)
		In this theorem, the condition that the sum be finite or countable may be replaced by the condition of local finiteness.
		The statement for the large and small inductive dimensions analogous to this sum theorem already fails in the class of Hausdorff compacta.
		Hurewicz, which he originally obtained for the class of spaces with a countable base.
	eom.springer.de /D/d032450.htm (2461 words)

	[No title]
		The proof of Theorem 1.3 proceeds by first showing in Theorem 1.12 that (i) and (ii) are each equivalent to a statement about one noninvariant cellular con* struc- tion and then showing in Theorem* 1.18 that (ii) and (iii) are each equivalent t* *o a statement about another noninvariant cellular construction.
		The first noninvari* ant construction, codified in Theorems* 1.9 and 1.11, is based on the notions of nuc* *lear complexes and cores introduced in [10].
		Proof.Since a minimal atomic complex is equivalent to a nuclear complex, the second statement of Theorem 1.17 implies that (iii) is equivalent to the condit* *ion specified in the statement and that (iii) implies (ii).
	www.math.purdue.edu /research/atopology/Baker-May/CoresMay30.txt (6865 words)

	Math 8306-07: Class Outlines
		4/18/05: Finishing the proof of the Dold-Thom Theorem.
		The fundamental theorem of algebra: a topological proof.
		29-31 (The Fundamental Group of the Circle through Theorem 1.8), and pp.
	www.math.umn.edu /~voronov/8306/outline.html (1288 words)

	2006-2007 Course Register
		Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus.
		Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.
		Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.
	www.upenn.edu /registrar/register/math.html (4476 words)

	Luitzen Egbertus Jan Brouwer (Stanford Encyclopedia of Philosophy)
		The basic theorems of intuitionistic analysis — the bar theorem, fan theorem, and continuity theorem — are in ‘Über Definitionsbereiche von Funktionen’ (‘On the Domains of Definition of Functions’) of 1927.
		The fan theorem is, in fact, a corollary of the bar theorem; combined with the continuity principle, which is not classically valid, it yields the continuity theorem.
		The bar and fan theorems on the other hand are classically valid, although the classical and intuitionistic proofs for them are not exchangeable.
	plato.stanford.edu /entries/brouwer (4205 words)

	[No title]
		After this I was asked for a statement and proof of Liouville's theorem.
		When I blanked on that, she settled for rough discussion of how I would construct an analytic function with a given set of zeroes, and a statement of the condition under which an infinite product will converge.
		Next: what could I say about the homotopy groups of a manifold of negative sectional curvature (by Hadamard's theorem the universal cover is R^n, so the higher homotopy groups vanish and pi_1 must be infinite).
	www.math.princeton.edu /graduate/generals/milley_peter (776 words)

	Atlas: Hurewicz Theorem for Nagata-Assouad dimension by J. Dydak (Site not responding. Last check: 2007-09-17)
		Theorem B. asdim(G) ≤ asdim(K) + asdim(H) for any short exact sequence 1 → K → G → H → 1 of countable groups.
		Both Theorems A and B generalize results of Bell and Dranishnikov in which f is Lipschitz and X is geodesic and G, K are finitely generated, respectively.
		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 # caqs-99.
	atlas-conferences.com /cgi-bin/abstract/caqs-99 (158 words)

	Chronology for 1920 to 1930
		He considers the growth, success, and impact of "scientific materialism" which is the notion that nature is merely matter and energy.
		Krull proves the "Krull-Schmidt theorem" for decomposing abelian groups of operators.
		Hurewicz proves his embedding theorem for separable metric spaces into compact spaces.
	www-gap.dcs.st-and.ac.uk /~history/Chronology/1920_1930.html (383 words)

	Matches for:
		Hurewicz first studied set theory and dimension, and his papers on this topic are especially clear and precise, making them accessible to beginning mathematicians.
		His work in algebraic topology is marked by five fundamental papers which provide an introduction to homotopy groups and the Hurewicz Theorem concerning the relation between homotopy and singular homology.
		Hurewiczs papers should also be consulted as prototypes for mathematical writing.
	www.mathaware.org /bookstore?fn=20&arg1=cworksseries&item=CWORKS-4 (755 words)

	Collected Works of Witold Hurewicz - (American Mathematical Society Bookstore) (Site not responding. Last check: 2007-09-17)
		Hurewiczs papers should also be consulted as prototypes for mathematical writing.
		His work in algebraic topology is marked by five fundamental papers which provide an introduction to homotopy groups and the Hurewicz Theorem concerning the relation between homotopy and singular homology.
		Hurewicz first studied set theory and dimension, and his papers on this topic are especially clear and precise, making them accessible to beginning mathematicians.
	mirror.math.nankai.edu.cn /mirror/www.ams.org/CWORKS-4.html (1273 words)

	Hurwitz biography
		It was also Schubert who persuaded Hurwitz's father to allow him to go to university and who sent him with warm recommendations to Klein at Munich.
		We note that this first paper by Hurwitz, written jointly with Schubert, was on Chasles's theorem.
		We mention a generalization of Eneström's theorem and give an application to a similar result by Hurwitz.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Hurwitz.html (1636 words)

	Whitehead manifold - Wikipedia, the free encyclopedia
		For dimensions 1 and 2, the answer is classical and it is "yes".
		In dimension 2, it follows, for example, from the Riemann mapping theorem.
		It follows from our previous observation, the Hurewicz theorem, and Whitehead's theorem on homotopy equivalence, that X is contractible.
	en.wikipedia.org /wiki/Whitehead_manifold (423 words)

	[No title]
		In most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology.
		A unique feature of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find.
		Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and a full exposition of Steenrod squares and powers.
	www.lycos.com /info/algebraic-topology--spaces.html (468 words)

	Luitzen Egbertus Jan Brouwer
		In classical mathematics, he founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem.
		Brouwer's classes were also attended by Max Euwe, the later world chess champion, who published a game-theoretical paper on chess from the intuitionistic point of view (Euwe, 1929), and who would much later deliver Brouwer's funeral speech.
		Among Brouwer's assistants were Heyting, Hans Freudenthal, Karl Menger, and Witold Hurewicz, the latter two of whom were not intuitionistically inclined.
	www.economyprofessor.com /theorists/luitzenegbertusjanbrouwer.php (604 words)

	UIC Graduate College -- Courses: Mathematics (Site not responding. Last check: 2007-09-17)
		Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics.
		Prerequisite: Math 411 and 417 and 481, or consent of the instructor.
		Prerequisites: Math 320 and 417 and 481, or consent of the instructor.
	www.uic.edu /depts/grad/courses/math.shtml (2065 words)

	HERS Output
		Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.
		Elements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, Geodesics, Riemann curvature, Darboux’s theorem, moment maps and symplectic quotients, complex and Kaehler manifolds, Dolbeault and de Rham cohomology.
		Riemann zeta function and the Prime Number Theorem; Dirichlet’s theorem on primes in arithmetic progressions; lower bounds on discriminants etc. from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.
	webdocs.registrar.fas.harvard.edu /courses/Mathematics.html (5117 words)

	Mathematics Graduate Course Descriptions
		The ultimate goal of the course will be to use these techniques to prove the sphere theorem of Micallef and Moore (1988) and other geometric results.
		This sphere theorem extended the earlier sphere theorems of Berger, Klingenberg and Toponogov, and showed that a new type of curvature, isotropic curvature, is relevant to the study of minimal surfaces.
		Our sphere theorem states that a simply connected Riemannian manifold of positive isotropic curvature (of dimension at least four) is homeomorphic to a sphere.
	www.math.ucsb.edu /grad/courses.php (1354 words)

	V. V. Uspenskij (Site not responding. Last check: 2007-09-17)
		According to Alexandrov's theorem, dimension can be characterized in terms of extension of maps to spheres: for a normal space X the condition \dim X\le n is equivalent to X\tau S
		We consider generalizations of Hurewicz' theorems on dimension-lowering and dimension-raising maps to the case of the extensional dimension.
		We have already cited Hurewicz' theorem stating that in this case \dim X \le \dim Y +k.
	www.utm.edu /staff/jschomme/topology/c/a/a/i/114.htm (410 words)

	Amazon.com: "Morera's Theorem": Key Phrase page (Site not responding. Last check: 2007-09-17)
		Let G be a region and let f : G- C be a continuous function such that I if =...
		In a subsequent theorem (Theorem 2.6.5: Morera's Theorem) we show that if f (z) is continuous and Eq.
		Morera's theorem The conventions D(a, r) def { z : lz - al < r} H(1l) def { f : f is...
	www.amazon.com /phrase/Morera's-Theorem (504 words)

	3.3 Fractal Dimension
		On a line segment like the Koch coastline, we arbitrarily chose the length of one side of the first iterate as a unit length.
		On the Euclidean coordinate plane the distance between any two points is given by the Pythagorean theorem
		A study of twelve definitions of dimension appeared in 1981 (see Harrison) and an entire book on Dimension Theory was written in 1941 (Hurewicz and Wallman).
	hypertextbook.com /chaos/33.shtml (1028 words)

	[No title]
		The courese starts with a study of singular homology, homological algebra (exact sequences, axioms), Mayer-Vietoris sequence, CW-complexes and cellular homology, calculation of homology of cellular spaces, homology with coefficients.
		It then moves on to cohomology theory, Universal Coefficients theorems, Bockstein homomorphism, Künneth formula, cup and cap products, Hopf invariant, Borsuk-Ulam Theorem, Brouwer and Lefschetz-Hopf Fixed Point Theorems.
		The course ends with an introduction to higher homotopy groups, Hurewicz and Whitehead theorems, Eilenberg-MacLane spaces, orientability, and duality theorems (Poincaré, Lefschetz, Alexander).
	www.math.neu.edu:16080 /grad/semconv/Topology3.doc (276 words)

	Atlas: Hurewicz Theorem for Nagata-Assouad dimension by Nikolay Brodskiy (Site not responding. Last check: 2007-09-17)
		Theorem A: asdim(X) ≤ asdim(f)+asdim(Y) for any coarse function f: X→ Y. As an application we prove
		Theorem B: asdim(G) ≤ asdim(K)+asdim(H) for any short exact sequence 1→ K→ G→ H→ 1 of countable groups.
		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 # caqu-63.
	atlas-conferences.com /cgi-bin/abstract/caqu-63 (147 words)

	Chronology for 1930 to 1940
		G D Birkhoff proves the general ergodic theorem.
		This will transform the Maxwell-Boltzmann kinetic theory of gases into a rigorous principle through the use of Lebesgue measure.
		Vinogradov publishes Some theorems concerning the theory of prime numbers in which he proves that every sufficiently large odd integer can be expressed as the sum of three primes.
	www-gap.dcs.st-and.ac.uk /~history/Chronology/1930_1940.html (364 words)

	Algebraic Topology - Cambridge University Press (Site not responding. Last check: 2007-09-17)
		The author emphasizes the geometric aspects of the subject, which helps students gain intuition.
		A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.
		Additional topics the universal coefficient theorem for homology; 49.
	www.cambridge.org /us/catalogue/catalogue.asp?isbn=0521795400 (285 words)

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