I want to solve this question, but I have no idea where should I start. I'm not sure even if I understand the question correctly or not. The question is:
Let $X$ be a path connected and locally path connected space, and let $Y=S^1 \times S^1 \times \ldots \times S^1$, a product of $n$ copies of the circle with $n\ge 1$. Show that if the fundamental group of $X$ at $x_0$ is finite, then every map $f:X\to Y$ is null-homotopic.
Thanks,
$\endgroup$ 11 Answer
$\begingroup$(1) What's $\pi_1(Y)$?
(2) What's the universal covering space, $\tilde{Y}$, of $Y$?
(3) What possibilities are there for the group homomorphism $f_* : \pi_1(X) \rightarrow \pi_1(Y)$?
(4) Show there exists a lift $\tilde{f} : X \rightarrow \tilde{Y}$ with $f = p \circ \tilde{f}$, where $p : \tilde{Y} \rightarrow Y$ is the covering map.
(5) Argue $\tilde{f}$ is nullhomotopic.
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