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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!

Liz
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    You don't seem to be using the simply-connectedness of $X_2$. – Angina Seng Apr 25 '18 at 16:19
  • Use the fact that $\pi_1(X\times Y)=\pi_1(X)\times \pi_1(Y): $ https://math.stackexchange.com/questions/291311/the-fundamental-group-of-a-product-is-the-product-of-the-fundamental-groups-of-t – Mike Earnest Apr 25 '18 at 16:34
  • @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. – Liz Apr 25 '18 at 16:58
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    $Z$ is simply connected iff $\pi_1(Z)$ is trivial. The product of two trivial groups is trivial. – Mike Earnest Apr 25 '18 at 17:00
  • @MikeEarnest Oh, got it! Thanks for the help! – Liz Apr 25 '18 at 17:01
  • As for where you went wrong: there are many, many maps from $S^1 \times I \to X_1\times X_2$ that project to $h_1$. Which one of them are you calling $h$? There is no natural lift of $h_1$ to $X_1 \times X_2$ – Paul Sinclair Apr 26 '18 at 03:29

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