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This concerns Hatcher's exercise 1.3.26.

It says almost this: Given a universal cover there are two actions of $\pi_1$ on the fibers, one given by lifting loops and one given by restricting deck transformation to fibers. Are these equivalent ? When are they equivalent? and he gives two examples, lets say $S^1 \times S^1$.

I am new to this subject and I am not exactly sure how to start.. Can someone please explain this better.. I would appreciate a solution to the problem as well.

sifsa
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For starters, you should use Proposition 1.39. It says that

$$ \pi_1 \to Deck$$

given by these actions is an isomorphism. Look into the details of the proof and also into what he writes prior to the proof, namely that Deck transformations are uniquely determined by what they do on one point. That should give you enough insights.

Daniel Valenzuela
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