# Problem in Proof of Theorem $5.3$ Chapter $2$ of "Cohomology of Groups by Kenneth S.Brown"

I am reading proof of Theorem $$5.3$$ in Chapter $$2$$ of "Cohomology of groups by Brown" on page number $$42$$. I have a problem with understanding the following consequence given in the proof:

Let $$G= F/R$$ where $$F$$ is a free group over the set $$S$$. Let $$Y$$ be a wedge sum of $$|S|$$ many circles and $$\tilde{Y}$$ to be the connected regular covering of $$Y$$ corresponding to the normal subgroup $$R$$ of $$F$$. Consider the exact sequence $$C_1{\tilde{Y}} \to C_0{\tilde{Y}} \to \mathbb{Z} \to 0$$ of free $$\mathbb{Z}[G]$$ modules. Then $$H_2(G) \cong \ker\{ (H_1(\tilde{Y})_G \to H_1(Y)\}.$$

I don't understand why $$H_2(G) \cong \ker\{ (H_1(\tilde{Y})_G \to H_1(Y)\}$$?

The author has used the following result: Let $$F_n \to \cdots \to F_0 \to \mathbb{Z} \to 0$$ be an exact sequence of $$\mathbb{Z}[G]$$-modules where each $$F_i$$ is projective. Then $$0 \to H_{n+1}(G) \to (H_nF)_G \to H_n(F_G) \to H_n(G) \to 0$$ is exact.

So using the above result I can only see that $$H_2(G) \cong \ker\{ H_1(\tilde{Y})_G \to \ker\big({C_1\tilde{Y}\otimes_{\mathbb{Z}[G]} \mathbb{Z} \to C_0\tilde{Y}\otimes_{\mathbb{Z}[G]} \mathbb{Z}\big)}\}.$$

Can someone tell what I am missing?

• I'm pretty sure that the thing you wrote on the right is $H_1(Y)$, so I don't think you are wrong Commented Feb 26, 2020 at 18:25
• @AndresMejia: I don't know how to prove it.
– eyp
Commented Feb 27, 2020 at 6:05