# Kodaira-Thurston example

I am trying to understand the Kodaira-Thurston example as it's done in the book "Lectures on Symplectic Geometry" by Ana Cannas.

Let's take $$\mathbb{R}^{4}$$, with local coordinates $$\{x_1,x_2,y_1,y_2\}$$, and following sympletic form $$\omega = dx_{1}\wedge dy_{1} + dx_{2}\wedge dy_{2}$$. Now one can consider the following discrete group $$\Gamma$$ which is generated by the following symplectomorphisms:

\begin{align*} &\gamma_{1}: (x_1,x_2,y_1,y_2) \longrightarrow (x_1,x_2+1,y_1,y_2)\\ &\gamma_{2}: (x_1,x_2,y_1,y_2)\longrightarrow (x_1,x_2,y_1,y_2+1)\\ &\gamma_{3}: (x_1,x_2,y_1,y_2)\longrightarrow (x_1+1,x_2,y_1,y_2)\\ &\gamma_{4}: (x_1,x_2,y_1,y_2)\longrightarrow (x_1,x_2,y_1+1,y_2) \end{align*} and take $$M=\mathbb{R}^4/\Gamma$$. Now we know that $$\pi_1(M)\cong \Gamma$$, but I am having some trouble understanding why this will have rank $$3$$. Also I am not sure how one can define a symplectic structure on this manifold $$M$$.

Any help is appreciated, thanks in advance.

• Not sure if you are using an earlier version, but the one I found online, $\gamma_4$ is different. Jul 4 at 3:29
• What you described here is the usual torus $\mathbb{T}^4 = \mathbb{R}^4 / \mathbb{Z}^4$ which has $\mathbb{Z}^4$ as fundamental group. There must be a typo somewhere Jul 4 at 12:27

The purpose of the example of Thurston is to describe a symplectic manifold which is not Kahlerian by showing that the first Betti number of the example is odd. Obviously there is mistake in Ana Cannas, since the example that you describe is the 4-dimensional torus which is Kahlerian.

the example is:

$$(x,y,z,t)\rightarrow (x,y,z+1,t)$$

$$(x,y,z,t)\rightarrow (x,y,z,t+1)$$

$$(x,y,z,t)\rightarrow (x,y+1,z,t)$$

$$(x,y,z,t)\rightarrow (x+1,y,z+t,t)$$

https://www.jstor.org/stable/2041749?origin=crossref

• In the book that I found here, $\gamma_{4}: (x_1,x_2,y_1,y_2)\longrightarrow (x_1,x_2+y_2,y_1+1,y_2)$ so it is the OP that makes a typo (or that they are using an earlier version which had this mistake...). Jul 4 at 3:25