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May 14 |
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fundamental class and simplicial volume For any k, therearemaps f_k:S^2--->S^2 of degree k, e.g. The Suspension of z--->z^k. Then f_k[\partial\Delta^3] represents k times the fundamentalclass, but it is just a sum of 4 simplices. So the simplicial volume is bounded abhöbe by 4/k, which goesto 0. |
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May 10 |
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Visualize Fourth Homotopy Group of $S^2$ Because the Suspension of the Hopf fibration is a Map from S^4 to S^3. |
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May 10 |
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Visualize Fourth Homotopy Group of $S^2$ The references are maths.ed.ac.uk/~aar/papers/koschork.pdf and maths.ed.ac.uk/~aar/papers/koschsan.pdf |
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May 8 |
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Homotopy classes of maps At least homotopy classes from X to K(G,1) correspond to homomorphisms of fundamental groups mod inner autos. |
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May 6 |
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What arithmetic information is contained in the algebraic K-theory of the integers By the way, we DO know the K theory of Z, except in degree 8,12,16,20,24.... (where it is conjectured to be 0), see math.uiuc.edu/K-theory/0691/KZsurvey.pdf |
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May 5 |
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What is the oldest known evidence of application of mathematics? Just a hint to literature concerning the Ishango Bone (or rather the difficulty to say something about it): reunion.iufm.fr/recherche/irem/IMG/pdf/… |
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Apr 8 |
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Manifold with nonzero pontryagin number? To compute p_1 you can use the Hirzebruch Signature Theorem which relates the signature to the Pontrjagin numbers. For CP^2, the second cohomology is 1-dimensional and the generator has self-intersection 1, so the signature is 1. |
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Apr 8 |
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Manifold with nonzero pontryagin number? The sphere is the boundary of the 5-ball, hence 0-bordant. Since Pontrjagin numbers are bordism invariant, this implies p_1=0. |
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Apr 7 |
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Homotopy equvalence from contractibility of fiber. In many cases f happens to be a quasifibration and then contractibility of the fiber implies f is a weak homotopy equivalence. Conditions under which f is a quasifibration are to be found in Dold-Thom "Quasifaserungen und Symmetrische Produkte". |
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Apr 4 |
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Characteristic classes for general $G$ bundles, not just $G=SO(n)$ or $G=U(n)$ Have a Look at Mimura-Toda "Topology of Lie Groups" or Borel "Topology of Lie groups and characteristic classes." Bull. Amer. Math. Soc. 61 (1955), 397–432. |
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Mar 9 |
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Reference request for translating from Top to C*-alg In Point 14 the right hand side should not be algebraic K theory, but topological K theory of C^* algebras. (Which agrees with algebraic K theory only in degree 0.) |
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Feb 4 |
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Is a measurable homomorphism on a Lie group smooth? Perhaps a more interesting question: is every measurable cocycle cphomplogous to a smooth one? |
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Feb 4 |
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Trichotomies in mathematics The trichotomy elliptic-parabolic-hyperbolic holds for Riemann surfaces, but in higher dimensions there are far more possibilities. In dimension 3, there are Thurston's 8 geometries ( and usually 3-manifolds have to be decomposed into pieces to be geometric). In dimension 4, Wallach had a list of 17 geometries, but one item in his list actually contains an infinite number of geometries. (And certainly not all 4-manifolds are locally homogeneous.) |
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Jan 21 |
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Chern-Simons for 2n-dimensional manifolds Of course you can consider Chern-Simons forms on any manifold. But integrating them over the manifold - to get a numerical invariant - only works if the dimension of the manifold equals the degree of the form. |
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Dec 28 |
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Explicit homeomorphism between Thurston’s compactification of Teichmuller space and the closed disc By the way the boundary is S^{6g-7}. |
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Dec 22 |
awarded | ● Commentator |
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Dec 4 |
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Journals and other sources with “easy reading” papers ? deleted 5 characters in body |
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Nov 28 |
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How Many 4-Manifolds are Symplectic? added 1 characters in body |
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Nov 28 |
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What are some Applications of Teichmüller Theory? @ E.Vargas: The same comment would apply to Bieberbach groups, the Kneser finiteness theorem, Tietze's extension theorem, Witt rings, the Hasse principle and the Blaschke conjecture, not to forget Heegard splittings and Pontryagin numbers. Arguably most of these people played a more influential role in the expulsion of jews from the universities as did then 20 years old fanatic students like Teichmüller. (Which is of course not an exculpation of the latter.) |
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Nov 28 |
answered | Dilogarithm, tetrahedrons, and hyperbolic space |
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Nov 28 |
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What are some Applications of Teichmüller Theory? @Qiaochu: I guess you meant $g>1$? |
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Nov 27 |
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Can a sphere be a phase space? One may still ask whether a sphere can arise as the unit tangent bundle of some manifold. But then of course ist should have odd dimension. |
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Nov 27 |
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Can a sphere be a phase space? $T^*M$ is not compact. |
