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this is a problem in my topology class:

True or false: the product of two simply connected spaces also simply connected.


What I have right now:

Let X1 and X2 be two simply connected spaces. Let f: $S^1$ --> X1 $\times$ X2. Define f1: $S^1$ --> X1, and use $\pi_1$(X1) = 0 to get a homotopy h1: $S^1 \times$ I --> X1 from f1 to a constant. Then because of the universal property of the product, h: $S^1 \times I$ --> X1 $\times$ X2, which projects to h1. And we can prove that h is a homotopy from f to a constant, so the product X1 $\times$ X2 is simply connected.

Is this correct? If not, please point out where I got wrong.

Thank you so much!

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    $\begingroup$ You don't seem to be using the simply-connectedness of $X_2$. $\endgroup$ – Lord Shark the Unknown Apr 25 '18 at 16:19
  • $\begingroup$ Use the fact that $\pi_1(X\times Y)=\pi_1(X)\times \pi_1(Y): $ math.stackexchange.com/questions/291311/… $\endgroup$ – Mike Earnest Apr 25 '18 at 16:34
  • $\begingroup$ @MikeEarnest I see what you mean, and I can directly get the statement that π1(X×Y) is simply connected? I'm a bit confused. $\endgroup$ – Liz Apr 25 '18 at 16:58
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    $\begingroup$ $Z$ is simply connected iff $\pi_1(Z)$ is trivial. The product of two trivial groups is trivial. $\endgroup$ – Mike Earnest Apr 25 '18 at 17:00
  • $\begingroup$ @MikeEarnest Oh, got it! Thanks for the help! $\endgroup$ – Liz Apr 25 '18 at 17:01

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