How to construct Morse function

Question

I am studying cellular homology theory and there is a theorem saying that critical points of morse function corresponds to cells. For example, there should be a morse function of $\mathbb{R}P^n$ with $n+1$ critical points, how to construct such funtion?

Answer 1

Let $Q$ be the quadratic form on $\mathbb{R}^{n+1}$ defined by $$ Q(x_0,\dots,x_n)=\sum_{i=1}^{n+1}\alpha_ix_i^2,$$ where $0<\alpha_1<\dots<\alpha_{n+1}$ is a sequence of $n$ positive real numbers. Now take the standard embedding $S^n\subset\mathbb{R}^{n+1}$ of the $n$-sphere and let $\widetilde f=Q|_{S^n}:S^n\rightarrow\mathbb{R}$ the restriction of $Q$. Then $\widetilde f$ is a Morse function on $S^n$ with $2(n+1)$ critical points $Cr(\widetilde f)=\{e_1,-e_1,\dots,e_{n+1},-e_{n+1}\}$.

Now, $\widetilde f$ is invariant under the antipodal $\mathbb{Z}_2$ action on $S^n$, so factors to over the quotient as a map $f=\widetilde f/\mathbb{Z}_2:\mathbb{R}^n\rightarrow\mathbb{R}$. The critical points of $f$ are the $\mathbb{Z}_2$ equivalence classes of the critical points of $\widetilde f$. In particular the critical set consists of the $n+1$ points $$Cr(f)=\{[1,0,\dots,0],[0,1,0,\dots,0],\dots,[0,\dots,0,1]\}.$$