Restriction of "$\pi_{1}$" to topological groups Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is  a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$? 
 A: No. Take $G=SO(5)$ and $H=SO(3)$, both of which have fundamental group $\mathbb{Z}/2$. I claim that there is no continuous map $f: G\to H$ which induces the identity homomorphism.
If there were, then $f$ would induce  a nontrivial homomorphism $f_\ast: H_1(G;\mathbb{Z}/2)\to H_1(H;\mathbb{Z}/2)$, and a graded ring homomorphism $f^\ast: H^\ast(H;\mathbb{Z}/2)\to H^\ast(G;\mathbb{Z}/2)$ which is nontrivial on $H^1$. To see that no such ring homomorphism exists, recall (see Hatcher's Algebraic Topology, section 3.D) that $$H^\ast(H;\mathbb{Z}/2)\cong \mathbb{Z}/2[\alpha_1]/(\alpha_1^4)$$ and $$H^\ast(G;\mathbb{Z}/2)\cong \mathbb{Z}/2[\beta_1]/(\beta_1^8)\otimes\mathbb{Z}/2[\beta_3]/(\beta_3^2),$$ where each $\alpha_i,\beta_i$ is in degree $i$.
A: Not always. Take $G=H=\mathrm{PGL}(n,\mathbb{R})$, with $n\geq 7$. Every nontrivial continuous homomorphism $G\rightarrow G$ is a Lie automorphism; the group of such automorphisms is generated by inner automorphisms and $A\mapsto {}^tA^{-1} $, so it acts  on $\pi _1(G)$ through {$\pm 1$}. On the other hand $\pi _1(G)=\mathbb{Z}/n$ has  automorphisms $\neq \pm 1$.
Edit: Sorry, I misread the question - this answers a different question.
