# Effect of the multiplication map $G\times G\rightarrow G$ on homology groups

Let $$G$$ be a topological group with multiplication $$m:G\times G\rightarrow G$$. Let $$\omega$$ denote the composition $$G\vee G \subset G\times G\xrightarrow{\text{m}}G$$

I have to calculate the effect of $$\omega$$ on homology groups using the isomorphism $$\widetilde{H_*}(G\vee G)\cong\widetilde{H_*}(G)\oplus\widetilde{H_*}(G)$$. My guess is it should be $$[\sigma_1]\oplus[\sigma_2]\mapsto [\sigma_1+\sigma_2]$$ but can't prove it rigorously. Any help is welcome.

Since $$\omega_*$$ is linear, we have $$\omega_*([\sigma_1] \oplus [\sigma_2]) = \omega_*([\sigma_1] \oplus 0) + \omega_*(0 \oplus [\sigma_2])$$.
Let $$i_1 : G \to G \vee G$$ be the inclusion of the first factor in the wedge sum. Note that $$[\sigma] \oplus 0 = (i_1)_*([\sigma])$$, and that $$\omega \circ i_1$$ is the identity of $$G$$ (assuming that the base point of $$G$$ is the unit of $$m$$). It follows that $$\omega_*([\sigma] \oplus 0) = (\omega \circ i_1)_* ([\sigma]) = [\sigma]$$. Of course, the same holds for $$\omega_*(0 \oplus [\sigma])$$, so in the end, you get $$\omega_*([\sigma_1] \oplus [\sigma_2]) = [\sigma_1] + [\sigma_2].$$