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Answer by unknown (google) for How Many 4-Manifolds are Symplectic?

2012-10-30T08:45:23Z

By Donaldson, symplectic manifolds are the same as topological Lefschetz pencils. So you may ask how many 4-manifolds are topological Lefschetz pencils.

Answer by unknown (google) for Dilogarithm, tetrahedrons, and hyperbolic space

2012-11-28T06:12:30Z

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 $$\sum_{i=0}^4\left(-1\right)^i vol(\partial_i\sigma)=\sum_{i=0}^4\left(-1\right)^i \tilde{D}\left(z_0,\ldots,\hat{z}_i,\ldots,z_4\right).$$

Answer by unknown (google) for Can bilipschitz models of hyperbolic 3-manifolds be made effective?

2012-10-28T10:12:26Z

See Bowditch: link text Systems of bands in hyperbolic 3-manifolds

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.

Answer by unknown (google) for Busemann function on Hyperbolic space

2012-10-15T15:30:08Z

Heintze-ImHof "Geometry of Horospheres" contains a proof that the Busemann functions are C^2.

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.

The level sets of Busemann functions are horospheres, in some sense the opposite from totally geodesic submanifolds.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jdg/1214434219

Codimension zero immersions

2011-06-16T15:43:49Z

Given an immersion of the n-1-sphere into a (closed) n-manifold, when does it extend to an immersion of the n-disk?

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?

Comment by

2013-05-14T08:35:38Z

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.

Comment by

2013-05-10T05:25:18Z

Because the Suspension of the Hopf fibration is a Map from S^4 to S^3.

Comment by

2013-05-10T05:23:53Z

The references are maths.ed.ac.uk/~aar/papers/koschork.pdf and maths.ed.ac.uk/~aar/papers/koschsan.pdf

Comment by

2013-05-08T12:32:03Z

At least homotopy classes from X to K(G,1) correspond to homomorphisms of fundamental groups mod inner autos.

Comment by

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 math.uiuc.edu/K-theory/0691/KZsurvey.pdf

Comment by

2013-05-05T12:00:42Z

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/…

Comment by

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.

Comment by

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.

Comment by

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 "Quasifaserungen und Symmetrische Produkte".

Comment by

2013-04-04T06:53:35Z

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.

Comment by

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.)

Comment by

2013-02-04T12:38:40Z

Perhaps a more interesting question: is every measurable cocycle cphomplogous to a smooth one?

Comment by

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.)

Comment by

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.

Comment by

2013-01-03T22:51:32Z

Charter 5 in cis.upenn.edu/~jean/surfclass-n.pdf