Timeline for Homotopy type of an oriented, closed, simply connected manifold
Current License: CC BY-SA 3.0
8 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Jul 7, 2016 at 19:37 | comment | added | Danny Ruberman | The vanishing of Betti numbers doesn't mean that the homology groups vanish! So you'd need cells to account for the torsion in the homology, which is not ruled out by the Kaehler condition. If there's a lot of torsion, you need a lot of cells. So you need to know the integral homology, not just what you can deduce from Hodge theory. | |
| Jul 7, 2016 at 18:55 | comment | added | Bilateral | Thanks a lot for the info. Since a compact Kahler manifold has vanishing odd betti numbers, is there a simpler cell decomposition of these manifolds with a controlled number of odd-dimensional cells? | |
| Jul 7, 2016 at 15:30 | comment | added | Danny Ruberman | It's essentially an algebraic problem. Look at the homology groups, and figure out a chain complex with the fewest of generators in each dimension that could give those groups. Then Smale says you can find a cell decomposition with exactly the number of cells in each dimension as there are generators of the chain groups. For instance, if the homology groups are free (as in your example) then you'd have one cell per homology generator and boundary operators 0. If there's torsion in the homology, you'd need some additional cells because there must be some non-trivial boundary operators. | |
| Jul 6, 2016 at 21:37 | comment | added | Bilateral | For example, from the fact that $M$ is closed and oriented we deduce that it is homotopy equivalent to a complex having a unique 0-cell and a unique 8-cell. Is there a way to deduce that other k-cells should be absent, or there should be a given number of them? | |
| Jul 5, 2016 at 21:13 | comment | added | Danny Ruberman | What do you mean by simplest? Can you give a non-trivial example of what this might mean? | |
| Jul 5, 2016 at 19:49 | comment | added | Bilateral | Thanks for the reference. However, the same topological space, in this case manifold, can admit many different cell decomposition. What would be the simplest one in this case? | |
| Jul 5, 2016 at 12:21 | history | answered | Danny Ruberman | CC BY-SA 3.0 |