Let $X$ be a simply connected space. Prove that any map from $X \to {(S^1 )}^k$ induces a 0 map on reduced homology.
My attempt:
Let $f : X \to {(S^1 )}^k$ be a map. $X$ simply connected implies that $\pi_1 (X)=0$ and thus consider the Universal cover of ${(S^1 )}^k$ , $p : \Bbb R^k \to {(S^1 )}^k$ then , $0=f_*(\pi_1 (X)) \subseteq p_*(\pi_1(\Bbb R^k))=0$ . So by General lifting lemma, $f$ admits a unique lift $\tilde{f}:X \to \Bbb R^k$ , since $\Bbb R^k$ is contractible we get that $\tilde{f}$ is null-homotpoic and thus $f=p \circ \tilde{f}$ is homotopic to constant, thus the induced map on homology is $0$ and thus the induced map is $0$ also on the reduced homology.
Please point out mistakes, if any!