Number of cells in a minimal cell structure for a non-simply connected manifold?

Question

I have obtained a cell structure of a connected (but not simply connected) manifold using Morse theory. Is there any way for me to know whether this cell structure is minimal?

Answer 1

While it's far from a complete answer, the Morse inequalities give you bounds. Namely if $C_i$ is the number of cells of dimension $i$ and if $\beta_i$ is the $i^{th}$ Betti number then for any $i$ you have the Morse inequality $$C_i - C_{i-1} + C_{i-2} - \cdots \pm C_0 \ge \beta_i - \beta_{i-1} + \beta_{i-2} - \cdots \pm \beta_0 $$