Timeline for Stein fillable tight contact structures on the 3-torus
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2021 at 21:20 | comment | added | KSackel | To clarify, the quotient is therefore just in bijection with primitive elements in $\mathbb{Z}^3$ (given by the image of $\gamma$). | |
| Nov 1, 2021 at 20:53 | comment | added | KSackel | Because contact isotopy generates contactomorphisms, we have that up to contact isotopy, tight contact structures up to isotopy are in bijection with $MCG(T^3)/\mathrm{Stab}(\xi_0)$. On the one hand, $MCG(T^3) = GL_3(\mathbb{Z})$. On the other hand, given $\xi_0$, $T^3$ comes with a canonical generator $\gamma \in H_1(T^3)$ coming from representing the unique class nulhomologous with respect to the Stein filling, and $\mathrm{Stab}(\xi_0)$ consists precisely of those elements fixing this class, i.e. $\mathrm{Stab}(\xi_0) = \mathbb{Z}^2 \rtimes \mathrm{GL}_2(\mathbb{Z})$. | |
| Nov 1, 2021 at 19:34 | comment | added | magicker72 | These are all stated up to contactomorphism. Up to contact isotopy, there are infinitely distinct many tight contact structures that are all contactomorphic to the standard Stein fillable one. | |
| Oct 30, 2021 at 5:10 | history | answered | KSackel | CC BY-SA 4.0 |