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My question relates to Hatcher $p.143$, Example $2.40$.

Suppose $n=1$ and $G=\mathbb{Z}_2$, i.e. $\mathbb{Z}$ modulo $2$. Then (using the construction in the second paragraph) he defines $X$ to be a shape that is homeomorphic to the closed disk. But $X$ has different homology groups than a closed disk.

This is not possible. (Help!)

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    This is not a disk because the attaching map for $\mathbb S^1$ is of degree $2$. In fact you can easily see this is $\mathbb RP^2$ thus it has $H_1(X) = \mathbb Z_2$ as claimed. – 2017-01-09
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    I see that (I think). What if degree = 3? – 2017-01-10
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    The space is the quotient by a hexagon when you glue approprietly the edges. It has fundamental group $\mathbb Z/3 \mathbb Z$. I don't think there is a simple description of this space other that description of its construction. – 2017-01-10
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    Thanks. Please rewrite your comment(s) as an answer so I can give credit. – 2017-01-10

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This is not a disk since the attaching map has degree $2$ : what you get is $\mathbb RP^2$ so there is no contradiction.

The same construction with a map $f : S^1 \to S^1$ with $\deg(f) = n$ gives you a space with fundamental group $\mathbb Z/n \mathbb Z$.