Want to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$, using Hurewicz theorem. This is one of the questions on the previous topology qualifying exams. Any help will be appreciated!
I am thinking in stead to calculate $\pi_3$ of the universal cover and use Hurewicz; But I got something that doesn't make sense. What do I know about the homomorphism: $h:\pi_3(\mathbb{R}P^4 \vee S^3) \rightarrow H_3(\mathbb{R}P^4 \vee S^3)$?
Yes, use the universal cover which is $S^4\vee S^3\vee S^3$. The first nonzero homology group for this guy is $H_3$ (and very easy to compute), so Hurewicz says that the homomorphism is actually an isomorphism.