Timeline for When are Brieskorn Manifolds Homeomorphic?
Current License: CC BY-SA 3.0
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| Apr 9, 2013 at 17:34 | history | edited | user02138 | CC BY-SA 3.0 |
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| Apr 9, 2013 at 2:10 | history | edited | user02138 | CC BY-SA 3.0 |
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| Apr 9, 2013 at 2:02 | history | edited | user02138 | CC BY-SA 3.0 |
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| Apr 8, 2013 at 20:09 | history | edited | user02138 | CC BY-SA 3.0 |
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| Apr 8, 2013 at 19:56 | comment | added | Bruno Martelli | Mmmhhh... I realise after reading Milnor's paper that what I said is (hopefully...) true only for coprime triples, not in general. More complicate triples yield Seifert manifolds which may fiber on more complicate orientable surfaces with fibers that are not necessarily "three fibers of order $a_1$, $a_2$, $a_3$" as I thought. Anyway, I don't understand the formula for the genera given by user02138: what kind of genus is that? | |
| Apr 8, 2013 at 18:31 | answer | added | Misha | timeline score: 8 | |
| Apr 8, 2013 at 16:42 | comment | added | Bruno Martelli | When $n=2$ you should get two Seifert 3-manifolds fibering over the 2-sphere with three singular fibers of order $a_i$, (resp. $b_i$). In that case, If $a_i, b_i >1$ then by standard theorems of Seifert different triples (up to reordering) give rise to non-diffeomorphic manifolds (see any book on Seifert manifolds). If some parameter is 1 you get lens spaces and the situation may be more complicate. As far as I see, your conditions are not enough even in the homology sphere case: two distinct unordered triples of coprime integers always give rise to non-homeomorphic manifolds. | |
| Apr 8, 2013 at 10:23 | history | edited | Serge Lvovski |
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| Apr 8, 2013 at 1:34 | comment | added | Alex Suciu | The 3-dimensional Brieskorn manifolds are discussed at length by John Milnor in his paper On the 3-dimensional Brieskorn manifolds M(p,q,r), in: Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 175–225. Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N. J., 1975, MR0418127. | |
| Apr 7, 2013 at 23:46 | comment | added | user02138 | Hi Liviu, yes, I just thumbed through "Singularities and Topology of Hypersurfaces". | |
| Apr 7, 2013 at 23:42 | comment | added | Liviu Nicolaescu | Have checked Dimca's book on singularities of hypersurfaces? The exact title excapes me at this moment. | |
| Apr 7, 2013 at 23:00 | comment | added | user02138 | Hi Misha, yes, I've read the paper. As far as I can tell it doesn't answer my questions. | |
| Apr 7, 2013 at 22:49 | comment | added | Misha | For n=2 you get Seifert manifolds and the exponents $a_i$ are topological invariants of such manifolds. Did you read Hirzebruch's paper "Singularities and exotic spheres"? He has a thorrough discussion of topology of Brieskorn manifolds there. | |
| Apr 7, 2013 at 22:20 | history | asked | user02138 | CC BY-SA 3.0 |