Timeline for Minimal Betti numbers of simply-connected threefolds with trivial canonical class
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8 events
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| Jan 11, 2023 at 18:48 | comment | added | Aleksandar Milivojević | @user69559 This follows from Theorem 1 in Wall's "Classification Problems in Differential Topology. V". | |
| Jan 11, 2023 at 18:27 | comment | added | YangMills | In their subsequent paper "The complex structures on connected sums of $S^3\times S^3$" Manifolds and geometry (Pisa, 1993), 284–293, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996, Lu and Tian construct complex structures with trivial canonical bundle on all $k(S^3\times S^3)$ for $k\geq 2$. | |
| Jan 11, 2023 at 17:49 | comment | added | Basics | @Aleksandar Milivojević: Could you give a reference for your statement "Topologically any .... homology sphere"? | |
| Jan 11, 2023 at 9:14 | comment | added | Aleksandar Milivojević | Topologically any simply connected threefold with $b_2= 0$ is a connected sum $k(S^3 \times S^3) \# Q$ where $Q$ is a rational homology sphere. Lu and Tian showed in "The complex structure on a connected sum of $S^3\times S^3$ with trivial canonical bundle" that $25(S^3 \times S^3)$ carries a complex structure with trivial canonical. I don't know if smaller examples have been found. | |
| Jan 11, 2023 at 9:07 | comment | added | abx | @Nick L: Hopf manifolds are diffeomorphic to $S^{1}\times S^{2n-1}$. I guess you are thinking of the Calabi-Eckmann manifolds, but their canonical class is not trivial. | |
| Jan 11, 2023 at 8:17 | comment | added | Nick L | How about the Hopf manifold $S^3 \times S^3$? | |
| Jan 11, 2023 at 6:47 | history | edited | Basics |
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| Jan 11, 2023 at 6:19 | history | asked | Basics | CC BY-SA 4.0 |