Let $z_j=x_j+iy_j $ $(j=1, 2, ... n)$ be the coordinates in $ℂ^n$.
We can identify $ℂ^n$ with $ℝ^ {2n}$ by writing $(z_1, ... z_n) =(x_1,y_1, ... x_n,y_n) $ Let $D⊂ℂ$ be the square of the points $z=x+iy $ such that $ |x|≤1, |y|≤1$
Define $:D → ℂ^2 $ by $(z) = (e^z, e^{-z})$. Compute
$$\int_^{ }dx_1\wedge dy_1 + dy1\wedge dx_2 +dx_2 \wedge dy_2 \,\mathrm .$$
My attempt was to consider $:D → ℂ^2 $ as $:D → ℝ^ {4} $. Then $(x,y)=(e^xcosy, e^xsiny, e^{-x}cosy, -e^{-x}siny)$ and $$\int_^{ }dx_1\wedge dy_1 + dy_1\wedge dx_2 +dx_2 \wedge dy_2 \, = \int_^{ }(e^{2x}+e^{-2x}+cos^2y-sin^2y)dx \wedge\ dy \, \mathrm .$$ So now I tried to compute the integral on the right hand side by using a pullback:
$$\int_^{ }(e^{2x}+e^{-2x}+cos^2y-sing^2y)dx \wedge\ dy \, = \int_D^{ }*((e^{2x}+e^{-2x}+cos^2y-sing^2y)dx \wedge\ dy)\, \mathrm .$$
But now, I am stuck, because the dimensions don't match when trying to calculate the pullback.
Can anyone give me some hints, please?