So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a map as the projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})\cong S^{5}$, but I have no idea how to write out such a map in coordinates (to prove it is smooth) and I have some trouble proving that last isomorphism.
Many thanks for the help!
First of all, $S^2\times S^3$ has a CW structure consisting of one $0$-cell, one $2$-cell, one $3$-cell, and one $5$-cell. The subspace $S^2\vee S^3$ has a CW structure consisting of one $0$-cell, one $2$-cell, and one $3$-cell; these are just the corresponding cells of $S^2\times S^3$. Therefore, the space $S^2\times S^3/(S^2\vee S^3)$ has a CW structure consisting of one $0$-cell and one $5$-cell, so $S^2\times S^3/(S^2\vee S^3) \cong S^5$.
I assume you just need to find a smooth degree one map $S^2\times S^3 \to S^5$, you don't actually have to show that the given map is such a map. In that case, note that the given map $S^2\times S^3 \to S^5$ is a continuous degree one map, and is homotopic to a smooth map by the Whitney Approximation Theorem. As degree is preserved by homotopy, we're done.