TITLE: Homology of lens spaces using Morse theory?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: I tried a computation, I hope it's correct. I considered the map h = Re $z^p$ + Re $w^p$ on the sphere, and I found four sets of critical points: (1) ($\mu_{1}^{+} / \sqrt{2}$, $\mu_{2}^{+} / \sqrt{2}$), (2) ($\mu_{1}^{-}/ \sqrt{2}$, $\mu_{2}^{-} / \sqrt{2}$), (3) ($\mu_{1}^{+}$, 0) and (0, $\mu_{2}^{+}$), (4) ($\mu_{1}^{-}$, 0) and (0, $\mu_{2}^{-}$), where $\mu_{j}^+$ denotes a p-root of 1 and $\mu_{j}^-$ denotes a p-root of -1. 
The first set gives maxima (where h = 2), the second set gives minima (h=-2) and the other two give index 1 and 2 points (I think). The first and second set contribute to p points in the lens space, the other two contribute 2 points each. Perhaps doing Morse homology as Pietro suggested counting connecting flow lines gives you the correct result.... but I have not tried.