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I solved other problems, except (d): if $\beta$ is injective, $\alpha$ and $\gamma$ are surjective, then $\gamma$ is injective.

Unlike others, I don't know where to start.

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This is a typo in the book: the first $\gamma$ should be a $\varphi$. See the errata on Foote's webpage. –  Dylan Moreland Mar 7 '12 at 3:01
Yep, a typo, you can easily find a counterexample to the statement as it is written. –  Álvaro Lozano-Robledo Mar 7 '12 at 3:16
Thanks, guys! I lost a few hours thinking over this. –  Gobi Mar 7 '12 at 4:42
@Dylan Making your comment an answer, perhaps with a counter example, will resolve this question. Please consider this. Regards, –  user21436 Mar 14 '12 at 4:24

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