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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.


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HINT: Consider the induced map on the fundamental groups and the fact that there exists a universal covering space for Y. – mixedmath May 6 '12 at 20:29
up vote 6 down vote accepted

(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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