Recently I'm studying Serre spectral sequence in Hatcher's book. Let $\pi : X \to B$ is a fibration, it's an easy exercise to check that when B is path-connected then all the fibers are homotopy equivalent to a fixed fiber through a path $\gamma$ in $B$ that lifts to a homotopy equivalence, $L_{\gamma}: F_{\gamma_{0}}\to F_{\gamma_{1}}$. Particularly, when it is a loop then we get an action of fundamental group of $B$ on $\sum H_{n}(F)$. This action is interesting in case it's trivial as Hatcher wrote.

I'm so greatful if someone can give me a detailed explanations for this point and Hatcher's construction. Specifically in the seven-th line from the botton of the page 530.

  • $\begingroup$ Do you want to know how we get the action or why it's interesting when the action is trivial ? $\endgroup$ – Nicolas Hemelsoet Jun 30 '18 at 13:29
  • $\begingroup$ When the action is trivial. I can't understand the construction. $\endgroup$ – Bang Pham Khoa Jun 30 '18 at 22:25

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