A local coefficient system $A\hookrightarrow E \to B$ is a fiber bundle $p:E\to B$ such that

  • The fiber is a discrete abelian group $A$
  • The structure group $G$ is a subset of Aut$(A)$

Is the action of $\pi_1(B)$ on $A$ necessarily a homomorphism?

  • $\begingroup$ I do not understand: Homomorphism to where? To $Aut(A)$? $\endgroup$ – Moishe Kohan Sep 18 '13 at 8:08
  • $\begingroup$ I mean I do not know whether or not $\pi_1(B) \subset Aut(A)$. $\endgroup$ – Hezudao Sep 19 '13 at 0:14
  • $\begingroup$ Now, it makes sense and is true, except you do not have an embedding, only a homomorphism, called the monodromy of the local system (the same will work even if A is not a group). You can find a detailed discussion for instance in Steenrod's book "Topology of fiber bundles". $\endgroup$ – Moishe Kohan Sep 19 '13 at 8:19

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