I'm reading Hatcher's Algebraic topology, where is said:
Proposition 3D.3 (Page 299.): $H_∗(SO(n); Z)$ is a direct sum of $Z$’s and $Z_2$’s.
The claim is that there are chain complex isomorphisms $C_∗(SO(2k + 1))≈ C^2⊗C^4⊗··· ⊗C^{2k}$ and $C_∗(SO(2k + 2))≈C^2⊗C^4⊗ ··· ⊗C^{2k}⊗C^{2k+1}$, where $C^{2k+1}$ has basis $e^0$ and $e^{2k+1}$ and $C^{2i}$ is subcomplex of $C_∗(SO(n))$ with basis the cells $e^0$, $e^{2i}$, $e^{2i−1}$ and $e^{2i}e^{2i−1}$.
My question is: why this isomorphisms hold and how to prove that boundary maps agree?
I would be welcome for detailed answers in order to understand this better. Thanks in advance.