User thku - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:03:39Z http://mathoverflow.net/feeds/user/23650 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105878/how-should-one-understand-orbifold-fundamental-groups/105912#105912 Answer by thku for How should one understand orbifold fundamental groups? thku 2012-08-30T07:55:43Z 2012-08-30T07:55:43Z <p>A reference for the basics in the topology of orbifolds is <a href="http://kaist.academia.edu/SuhyoungChoi/Papers/236402/Geometric_Structures_on_Orbifolds_and_Holonomy_Representations" rel="nofollow">http://kaist.academia.edu/SuhyoungChoi/Papers/236402/Geometric_Structures_on_Orbifolds_and_Holonomy_Representations</a></p> <p>It is a Theorem of Thurston (Theorem 8 on Page 18 of the above notes) that every connected orbifold has a universal covering and that the orbifold fundamental group is the same as the group of deck transformations of the universal covering.</p> <p>Of course this helps only for the fundamental group, not the higher homotopy groups....</p> http://mathoverflow.net/questions/105803/berger-sphere-theorem/105805#105805 Answer by thku for Berger sphere theorem thku 2012-08-29T08:56:46Z 2012-08-29T08:56:46Z <p>By Hurewicz (n-1)-connected implies vanishing of the first n-1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have H_n=Z. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by Hurewicz' converse a homotopy sphere.</p> http://mathoverflow.net/questions/100451/lagrangian-kleinian-bottles Lagrangian Kleinian bottles thku 2012-06-23T12:52:26Z 2012-06-24T17:25:43Z <p>I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for which the problem were open. Question 1: Where to find references for the other surfaces? Question 2: What is the current status for Kleinian bottles? Do there exist written proofs now?</p> http://mathoverflow.net/questions/5277/simplicial-volume/97649#97649 Answer by thku for Simplicial volume thku 2012-05-22T10:06:39Z 2012-05-22T10:06:39Z <p>Just Take an irreduzible, but not atorial, 3-manifold with the property that all (or at least One) of the pieces in the JSJ-decomposition are hyperbolic.</p> <p>This is nonpositively curved by Leeb's thesis, but the fundamental Group is Not hyperbolic because of the abelian subgroups coming from incompressible tori. The simplicial Volume is the sum of the simplicial Volumina of the hyperbolic pieces.</p> http://mathoverflow.net/questions/96780/manifolds-are-paracompact Manifolds are paracompact thku 2012-05-12T15:57:08Z 2012-05-12T16:35:33Z <p>By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom. I have heard (but never seen written) that these assumptions imply paracompactness (and thus the existence of a Riemannian metric by the well-known construction using Partition of unity). Does anybody know a reference or Proof for paracompactness?</p> http://mathoverflow.net/questions/107458/manifold-whose-universal-covering-is-a-sphere-but-which-is-not-a-space-form/107461#107461 Comment by thku thku 2012-09-22T13:24:32Z 2012-09-22T13:24:32Z This Page should contain much of what is known about classification of Fake Lens spaces: <a href="http://www.map.mpim-bonn.mpg.de/Fake_lens_spaces" rel="nofollow">map.mpim-bonn.mpg.de/Fake_lens_spaces</a>