The induced homomorphism on the cohomology under the pinching map $S^n \rightarrow S^n \vee S^n$ There is a continuous map from $S^n$ to the wedge sum of spheres. I imagine that the induced pull back map on cohomology groups $H^n( S^n \vee S^n) \cong \mathbb{Z} \oplus \mathbb{Z} \rightarrow H^n( S^n ) \cong \mathbb{Z}$ is just addition though I'm not sure. Is there an easy way to show this?

A map is given explicitly by composing the diagonal map $x \rightarrow (x,x)$ with quotienting by a point. Let $\alpha_1 \in H^n(S^n \vee S^n)$ send the generator of the homology of the the "first" sphere $\sigma_1$ to $1$ and $\alpha_2$ send the generator of the second sphere $\sigma_2$ to $1$. Under the diagonal map the generator of $H_n(S^n)$ gets sent to the sum of generators $\sigma_1+\sigma_2$ so that the pull back of $\alpha_1+\alpha_2$ on the generator of $H_n(S^n)$ gives $f^*(\alpha_1+\alpha_2)(\sigma)=(\alpha_1+\alpha_2)(\sigma_1+\sigma_2)=\alpha_1(\sigma_1)+\alpha_2(\sigma_2)$.