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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.

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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 $$\omega_*([\sigma_1] \oplus [\sigma_2]) = [\sigma_1] + [\sigma_2].$$

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