Timeline for How many simplicial complexes on n vertices up to homotopy equivalence?
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| Feb 3, 2016 at 23:37 | comment | added | Gil Kalai | Dear Vidit, This is my expectation. But maybe I am pushing it to assume it will be doubly exponential. Since you dont specify the dimension it is reasonable to think it will typically be n/2. Which also mean that the complex will typically be simply connected. The possibilities for sequences of Betti numbers are well known. I think there are roughly $2^{n^2}$ possible sequences of Betti numbers. The point is that these sequences can be very different for different characteristics. But I admit that the guess that the number is doubly exponential is pretty wild. | |
| Feb 3, 2016 at 20:17 | comment | added | Vidit Nanda | I understand why the number of labelled and unlabelled complexes on $n$ vertices differs by the $n!$ factor of course, but it is not clear why counting up to homotopy shouldn't make a bigger difference than that factor. | |
| Feb 3, 2016 at 19:55 | history | answered | Gil Kalai | CC BY-SA 3.0 |