Let $G$ be a group and $M,N$ be normal subgroups of $G$ such that $G=MN$. Then there is a natural exact homology sequence

$Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} H_2(G) \xrightarrow{\alpha} H_2(G/M)\oplus H_2(G/N) \xrightarrow{\beta} \frac{M \cap N [G,G]}{[M,N]} \rightarrow 0 $.

The above sequence is obtained in VAN KAMPEN THEOREMS FOR DIAGRAMS OF SPACES by R Brown and J L Loday in Topology 26(3) using Eilenberg-Maclane spaces $K(G,1), K(G/M,1)$ and $K(G/N,1)$. I know little about Eilenberg-Maclane spaces (almost nothing). But, alternatively to visualize the above sequence, I can use free presentation and Hopf formula.

Let $F/R=G$ be a free presentation of $G$. Let $S$ and $T$ be two subgroups of $F$ such that $SR/R=M$ and $TR/R=N$. Then the above sequence takes the form

$Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} \frac{R \cap [F,F]}{R,F} \xrightarrow{\alpha} \frac{SR \cap [F,F]}{[SR,F]} \oplus \frac{TR \cap [F,F]}{[TR,F]} \xrightarrow{\beta} \frac{M \cap N [G,G]}{[M,N]} \rightarrow 0 $.

Now, I want to write and visualize the maps $\alpha$ and $\beta$ explicitly so that I can find the kernel and images and I can really see that the above sequence is exact. Some times I see the phrases that the map $G \rightarrow G/N$ induces a natural map from $H_2(G) \rightarrow H_2(G/N)$. But I am not able to visualize the natural map $\frac{R \cap [F,F]}{R,F} \xrightarrow{\alpha} \frac{SR \cap [F,F]}{[SR,F]} \oplus \frac{TR \cap [F,F]}{[TR,F]}$. I want an explicit maps $\alpha$ and $\beta$ in my case so that I can feel the situation.


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