# Morse Homology of some function on torus

Consider the torus $$\Bbb T^2=\Bbb R^2/\Bbb Z^2$$. And define a Morse function $$f:\Bbb T^2\to \Bbb R$$ by $$f([(x,y)])=\cos(2\pi x)+\cos(2\pi y)$$. Now the critical points are minimum $$\big[\big(\frac{1}{2},\frac{1}{2}\big)\big]$$, maximum $$[(0,0)]$$ and two critical points of index 1, namely $$\big[\big(\frac{1}{2},0\big)\big]$$ and $$\big[\big(0,\frac{1}{2}\big)\big]$$. Now, I want to compute Morse Homology w.r.t. coefficient in $$\Bbb Z_2$$. Note that, $$C_0=\Bbb Z_2$$ and $$C_1=\Bbb Z_2\oplus\Bbb Z_2$$ and $$C_2=\Bbb Z_2$$. Now I have to compute differential i.e. for $$a\in C_k$$ we have to find $$\partial_X(a)=\sum_{b\text{ is a critical point of index } k-1}n(a,b)b$$ where $$n(a,b)\in \Bbb Z_2$$ is the number of trajectory of the pseudo-gradient field $$X$$ corresponding to $$f$$ modulo $$2$$ going from $$a$$ to $$b$$.

My question is how to find $$n(a,b)$$. I saw that, this can be done by looking pseudo-gradient flow in square and counting. Is there any other way doing this without drawing? Alos I have not understand the way of drawing this pseudo-gradient flow in square. Also I want to know the stable and unstable manifolds, how do they look like. Any help will be appreciated.

• Are you able to draw the level sets of $f$ in the plane? If you can, then the flow is easy to draw, using orthogonal curve to the level sets. – Lee Mosher Sep 24 '19 at 20:54