How can i see a CW-Structure of $S^n$ compatible with the antipodal map?

Question

I dont understand this notion of compatibility. I saw two possibles CW-structures to $S^n$. Please, anyone can explain me this notion?

Answer 1

You do not explain what it means that a "CW-structure is compatible with the antipodal map". So let us interpret it in the sense that the antipodal map $a : S^n \to S^n$ is cellular, which means that $a(C^{(i)}) \subset C^{(i)}$ for $i=0,\dots,n$ where $C^{(i)}$ denotes the $i$-skeleton of $S^n$ with respect to the given CW-structure.

There are many such CW-structures, but we are interested in the simplest possible one.

We have $S^m = \{ x \in \mathbb{R}^{m+1} \mid \lVert x \rVert = 1 \}$, where $\lVert - \rVert $ denotes the usual Euclidean norm. There are canonical embeddings $i_m : S^m \to S^{m+1}, i_m(x) = (x,0)$.

Let us construct the desired CW-structure inductively for $m = 0,1,\dots$ in such a way that the $m$-skeleton of $S^{m+1}$ is $i_m(S^m)$. There are two equivalent approaches to build CW-complexes, one based on "open cells" and one based on "closed cells". See my answer to cell complex structure of circle. Let us adopt the open cell approach.

For $m = 0$ it is obvious that $S^0 = \{-1, 1 \}$ has two $0$-cells $e^0_\pm$. Clearly $a(e^0_\pm) = e^0_\mp$.

Assuming that we have constructed our CW-structure on $S^m$, we already have the $m$-skeleton of $S^{m+1}$ which is $i_m(S^m)$. We have to find the open $(m+1)$-cells of $S^{m+1}$ whose union must be $S^{m+1} \setminus i_m(S^m)$. Clearly there are two such cells $e^{m+1}_\pm = \{ (x_1,\dots,x_{m+2}) \in S^{m+1} \mid (-1)^{\pm 1} x_{m+2} > 0 \}$. Obviuosly $a(e^{m+1}_\pm) = e^{m+1}_\mp$.