**Hatcher Exercise 2.1.9**

Consider the space obtained as the quotient space of when all faces of the same dimension are identified. Fix an ordering on the vertices of , say . We compute the homology groups:

We have, for each , where is the equivalence class containing all -dimensional faces. WLOG, we can use as a representative for .

First, consider . We have:

And (note, is not the zero map, because ):

So, when is even, we have and . When is odd, we have , and . Then, in both cases, .

Now consider . If is even, then , by the same argument as above. Since is the zero map, and we have . However, if is odd, we have , by the same argument as above. But, since is still the zero map, we have and . To summarize,