To test Morse theory, one day I tried to compute the homology of Lens spaces. You can build a lens space out of $S^3 = \{ |z|^2 + |w|^2 = 1: z, w \in \mathbb{C} \}$ using the quotient by $z \mapsto e^{2\pi i /p}z, w \mapsto e^{2\pi i q/p}w$. The quotient by this action L(p,q) is a lens space.

With Morse theory estimates the Betti numbers of a manifold by counting the critical points of a specified index. For any real-valued function $f: L(p,q) \to \mathbb{R}$, we look for critical points where $\nabla f(p_0) = 0$. Expanding around p₀ $f(p) = f(p_0) + (p - p_0)^T A (p - p_0) + O(|p - p_0|^2)$ for some matrix $A$, the Hessian. If the eigenvalues of A are all real and nonzero and the critical values of f are all different then $f$ is called "Morse". The Morse theory says L(p,q) is homotopic to a CW-complex with a cell complex of dimension k for each critical point of index k.

The Morse function I chose is $h = r \cos p\theta$ where $r,\theta$ come from $z = r e^{i \theta}$. Also let $w = \rho e^{i \phi}$. This function is well-defined on the lens space L(p,q) and its critical points are (+/-1,0,0,0) and its images under the deck transformation. To see this, notice the gradient in this coordinate system is $$\nabla = \left( \frac{\partial }{\partial r}, \frac{1}{r}\frac{\partial }{\partial \theta},, \frac{\partial }{\partial \rho}, \frac{1}{\rho} \frac{\partial }{\partial \phi}\right) $$ and you look for points where the gradient is normal to S³. However, $H_1[L(p,q)]= \mathbb{Z}/p$ suggesting we should have a saddle point of index 1.

I am told that Morse theory only gives you the singular $\mathbb{R}$ homology. Did I find all the critical points correctly? Is there a way to get the $\mathbb{Z}$ homology using Morse homology or other piece of differential topology?