CW-Structure of Spin(n)

Question

I cannot find any information about the CW-Structure of $Spin(n)$ groups. Clearly $\pi_1=\pi_2=0$ and I think $H_3(Spin(n))=\mathbb Z$ $(n\geq 5)$, so the $3$-skeleton is $S^3$. What is the $4$-skeleton?

Answer 1

For each $n$ there is a fibration sequence

$$Spin_n\rightarrow Spin_{n+1}\rightarrow S^n$$ covering the corresponding fibration sequence $SO_n\rightarrow SO_{n+1}\rightarrow S^n$. The point is that the inclusion $Spin_n\hookrightarrow Spin_{n+1}$ is $(n-1)$-connected, so the $4$-skeleton of $Spin_n$ is the same as the $4$-skeleton of $Spin_6$ for each $n\geq 6$.

Now $Spin_5\cong Sp_2$ and it is know that $Sp_2\simeq S^3\cup_{\nu'} e^7\cup e^{10}$, so the $4$-skeleton of $Spin_5$. Of course $Spin_3\cong S^3$ and $Spin_4\cong S^3\times S^3$ which have $4$-skeletons $S^3$, $S^3\vee S^3$ and $S^3$, respectively.

Also $Spin_6\cong SU_4$ and the $5$-skeleton of $SU_4$ is $\Sigma \mathbb{C}P^2\simeq S^3\cup_{\eta_3}e^5$, so by the previous comments, in fact the $5$-skeleton of $Spin_n$ for $n\geq 6$ is $S^3\cup_{\eta_3}e^5$.

We also have a fibration $G_2\rightarrow Spin_7\rightarrow S^7$, and it's not that difficult to see that $Spin_7\simeq S^3\cup_{\eta_3}\cup e^6\cup e^6\cup\dots$, which gives the $6$-sekelton of $Spin_n$ for $n\geq 7$. In fact, the previous fibration splits when localised away from $2$ so there is a $\frac{1}{2}$-local homotopy equivalance $Spin_7\simeq G_2\times S^7$. Finally we mention that there is a homeomorphism $Spin_8\cong Spin_7\times S^7$, so you get the $7$-skeleton of $Spin_8$ from that of $Spin_7$ wedged with an extra $7$-cell. I'm not aware of any further isomorphisms for the higher rank spinor groups.

A full CW decomposition was given by Araki in his paper On the homology of spinor groups, which was predated by some related results of Borel in Sur I'homologie et la cohomologie des groupes de Lie compacts connexes. As you might expect, the results tend to get a bit complicated as the rank grows, and whilst Araki calculates explicit attaching maps, he does not explicitly calculate their homotopy classes, so depending on what information you are looking for, you may be more interested in his homological calculations. And to this end I should refer you to the paper The integral homology and cohomology rings of $SO(n)$ and $Spin(n)$ of Pittie, which does exactly what it says.