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I am a bit confused with what I am suppose to prove here. I plan to go with prove isomorphism = homomorphism + bijection, but which function should I construct for the homomorphism?

Show that the isomorphism $\pi_1( X \times Y) \approx \pi_1(X) \times \pi_1(Y)$ in Proposition 1.12 is given by $[f] \mapsto (p_{1*}([f]), p_{2*}([f]))$ where $p_1$ and $p_2$ are the projections of $X \times Y$ onto its two factors.

I want to show that $\pi_1( X \times Y) \approx \pi_1(X) \times \pi_1(Y)$ is a homomorphism by considering $$f = (g, h), f^\prime = (g^\prime, h^\prime).$$

And I also know that $$[f \cdot f^\prime] = [f][f^\prime]$$ by the definition of group product.

1LiterTears
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1 Answers1

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For the sake of an answer,

If you look at the proof of Proposition 1.12 in the text, Hatcher proves that if $f=(g,h)\colon Z\to X\times Y$ is a map, then the isomorphism is given by $[f]\mapsto ([g],[h])$.

However, $p_{1*}([f])=[p_1f]=[g]$ since $p_1f=g$ as functions, and likewise $p_{2*}([f])=[h]$, so it's really just the same map.

Ben West
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