Hatcher example 3H.3. Local coefficients via Modules I'm trying to understand the following example made by Hatcher at page 329 in the section "Local Coefficients via Modules"

The problem arises when I start to prove that a basis for $C_n^+(X')$ is given by the chains $\sigma + \tau (\sigma)$ where $\tau$ is the non-trivial deck transformation. Now I don't know how to prove that it is a generating set. Take a $\tau$-invariant simplex $\alpha$. I want to write $\alpha = c + \tau(c)$ for some chain $c$. I don't know where to start, because from what I know there could be only $\alpha$ as a simplex.
I almost sure there is some kind of trick here. 
I'd prefer hints rather than solution. Thanks in advance!

Hint: There are no $\tau$-invariant simplices, because $\tau$ acts freely on $X'$.  To show that a general $\tau$-invariant chain $\sum a_i\sigma_i$ can be written in the form $c+\tau(c)$, try grouping the $\sigma_i$ into their orbits under the action of $\tau$.
