User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:19:09Z http://mathoverflow.net/feeds/user/15817 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110808/how-many-4-manifolds-are-symplectic/111061#111061 Answer by unknown (google) for How Many 4-Manifolds are Symplectic? unknown (google) 2012-10-30T08:45:23Z 2012-11-28T07:25:21Z <p>By Donaldson, symplectic manifolds are the same as topological Lefschetz pencils. So you may ask how many 4-manifolds are topological Lefschetz pencils.</p> http://mathoverflow.net/questions/109015/dilogarithm-tetrahedrons-and-hyperbolic-space/114734#114734 Answer by unknown (google) for Dilogarithm, tetrahedrons, and hyperbolic space unknown (google) 2012-11-28T06:12:30Z 2012-11-28T07:24:18Z <p>In fact this follows from Stokes' theorem. Consider the 4-simplex $\sigma$ with vertices ABCDE. Since the volume form $\omega$ is closed we have $$\int_{\partial\sigma}\omega=\int_{\sigma} d\omega=0.$$ But the integral of the volume form over $\partial\sigma$ is exactly the alternating sum $$\sum_{i=0}^4\left(-1\right)^i vol(\partial_i\sigma)=\sum_{i=0}^4\left(-1\right)^i \tilde{D}\left(z_0,\ldots,\hat{z}_i,\ldots,z_4\right).$$</p> http://mathoverflow.net/questions/110875/can-bilipschitz-models-of-hyperbolic-3-manifolds-be-made-effective/110892#110892 Answer by unknown (google) for Can bilipschitz models of hyperbolic 3-manifolds be made effective? unknown (google) 2012-10-28T10:12:26Z 2012-10-29T04:39:58Z <p>See Bowditch: <a href="http://msp.org/pjm/2007/232-1/pjm-v232-n1-p01-s.pdf" rel="nofollow">link text</a> Systems of bands in hyperbolic 3-manifolds</p> <p>with an approach to the Brock-Canary-Minsky Theorem (though not through their model manifold) that is, in principle, effective. Though I am not aware of an explicit algorithmic realization.</p> http://mathoverflow.net/questions/109716/busemann-function-on-hyperbolic-space/109728#109728 Answer by unknown (google) for Busemann function on Hyperbolic space unknown (google) 2012-10-15T15:30:08Z 2012-10-15T15:30:08Z <p>Heintze-ImHof "Geometry of Horospheres" contains a proof that the Busemann functions are C^2. </p> <p>When F is a Busemann function associated to a point z in the ideal boundary, then Z:=-grad(F) is the vectorfield showing towards z and the derivative of Z in direction v is Y'(0), where Y is the Jacobi field with Y(0)=v. </p> <p>The level sets of Busemann functions are horospheres, in some sense the opposite from totally geodesic submanifolds.</p> <p><a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.jdg/1214434219" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.jdg/1214434219</a></p> http://mathoverflow.net/questions/67961/codimension-zero-immersions Codimension zero immersions unknown (google) 2011-06-16T15:43:49Z 2011-06-28T08:03:30Z <p>Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?</p> <p>Remark: If the sphere had dimension k smaller than n-1, then such an immersion would exist if and only if the corresponding map from the k-sphere to the Stiefel manifold is 0-homotopic. This is the Hirsch-Smale Theorem and in fact an example of an h-principle. However the case k=n-1 is exactly the exceptional case which does NOT obey an h-principle. Easy examples (Figure 8.1. in the book by Eliashberg-Mishachev) show that there exist immersions of the circle in the plane which have a formal extension but not a genuine extension to the 2-disk. So, is there anything known about sufficient conditions for extendability?</p> http://mathoverflow.net/questions/130493/fundamental-class-and-simplicial-volume Comment by 2013-05-14T08:35:38Z 2013-05-14T08:35:38Z For any k, therearemaps f_k:S^2---&gt;S^2 of degree k, e.g. The Suspension of z---&gt;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 abh&#246;be by 4/k, which goesto 0. http://mathoverflow.net/questions/128821/visualize-fourth-homotopy-group-of-s2/130192#130192 Comment by 2013-05-10T05:25:18Z 2013-05-10T05:25:18Z Because the Suspension of the Hopf fibration is a Map from S^4 to S^3. http://mathoverflow.net/questions/128821/visualize-fourth-homotopy-group-of-s2/128827#128827 Comment by 2013-05-10T05:23:53Z 2013-05-10T05:23:53Z The references are <a href="http://www.maths.ed.ac.uk/~aar/papers/koschork.pdf" rel="nofollow">maths.ed.ac.uk/~aar/papers/koschork.pdf</a> and <a href="http://www.maths.ed.ac.uk/~aar/papers/koschsan.pdf" rel="nofollow">maths.ed.ac.uk/~aar/papers/koschsan.pdf</a> http://mathoverflow.net/questions/130065/homotopy-classes-of-maps Comment by 2013-05-08T12:32:03Z 2013-05-08T12:32:03Z At least homotopy classes from X to K(G,1) correspond to homomorphisms of fundamental groups mod inner autos. http://mathoverflow.net/questions/129762/what-arithmetic-information-is-contained-in-the-algebraic-k-theory-of-the-integer Comment by 2013-05-06T12:44:33Z 2013-05-06T12:44:33Z 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 <a href="http://www.math.uiuc.edu/K-theory/0691/KZsurvey.pdf" rel="nofollow">math.uiuc.edu/K-theory/0691/KZsurvey.pdf</a> http://mathoverflow.net/questions/129705/what-is-the-oldest-known-evidence-of-application-of-mathematics Comment by 2013-05-05T12:00:42Z 2013-05-05T12:00:42Z Just a hint to literature concerning the Ishango Bone (or rather the difficulty to say something about it): <a href="http://www.reunion.iufm.fr/recherche/irem/IMG/pdf/Keller_prehistoire_geometrie.pdf" rel="nofollow">reunion.iufm.fr/recherche/irem/IMG/pdf/&hellip;</a> http://mathoverflow.net/questions/126826/manifold-with-nonzero-pontryagin-number Comment by 2013-04-08T08:12:34Z 2013-04-08T08:12:34Z 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. http://mathoverflow.net/questions/126826/manifold-with-nonzero-pontryagin-number Comment by 2013-04-08T08:06:05Z 2013-04-08T08:06:05Z The sphere is the boundary of the 5-ball, hence 0-bordant. Since Pontrjagin numbers are bordism invariant, this implies p_1=0. http://mathoverflow.net/questions/126425/homotopy-equvalence-from-contractibility-of-fiber Comment by 2013-04-07T15:20:39Z 2013-04-07T15:20:39Z 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 &quot;Quasifaserungen und Symmetrische Produkte&quot;. http://mathoverflow.net/questions/126476/characteristic-classes-for-general-g-bundles-not-just-gson-or-gun Comment by 2013-04-04T06:53:35Z 2013-04-04T06:53:35Z Have a Look at Mimura-Toda &quot;Topology of Lie Groups&quot; or Borel &quot;Topology of Lie groups and characteristic classes.&quot; Bull. Amer. Math. Soc. 61 (1955), 397–432. http://mathoverflow.net/questions/82871/reference-request-for-translating-from-top-to-c-alg/82960#82960 Comment by 2013-03-09T19:58:06Z 2013-03-09T19:58:06Z 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.) http://mathoverflow.net/questions/120738/is-a-measurable-homomorphism-on-a-lie-group-smooth Comment by 2013-02-04T12:38:40Z 2013-02-04T12:38:40Z Perhaps a more interesting question: is every measurable cocycle cphomplogous to a smooth one? http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120629#120629 Comment by 2013-02-04T12:31:32Z 2013-02-04T12:31:32Z 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.) http://mathoverflow.net/questions/119353/chern-simons-for-2n-dimensional-manifolds/119359#119359 Comment by 2013-01-21T07:29:57Z 2013-01-21T07:29:57Z 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. http://mathoverflow.net/questions/118005/homology-cohomology Comment by 2013-01-03T22:51:32Z 2013-01-03T22:51:32Z Charter 5 in <a href="http://www.cis.upenn.edu/~jean/surfclass-n.pdf" rel="nofollow">cis.upenn.edu/~jean/surfclass-n.pdf</a>