# Čech cocycles and monodromy

It is well known that over a topological space $$X$$ (and choosing an open cover $$\mathfrak{U}$$) every locally constant Cech cocycle $$g$$ on $$\mathfrak{U}$$ with coefficients in a group $$G$$ yields a $$G$$-covering space $$X_g \rightarrow X$$. As such, the monodromy action of this covering space gives a homomorphism from the fundamental group $$\pi_1(X,x)$$ on a point $$x\in X$$ to $$G$$.

I am trying to write this homomorphism explicitly in terms of the cocycle $$g$$. In his Bachelor Thesis, Lemma 5.5, M.P. Noordman claims that this can be done in the following way. You consider a loop $$\sigma:[0,1]\rightarrow X$$ and apply the Lebesgue number lemma to get a finite subcover $$\{U_1,...,U_n\}$$ of $$\mathfrak{U}$$ and a partition $$t_0 of the interval $$[0,1]$$ in such a way that $$\sigma([t_{i-1},t_i])\subset U_i$$. Now, you can define the homomorphism $$f:\pi_1(X,x)\rightarrow G$$ as $$f([\sigma])=g_{12} g_{23} g_{34} \cdots g_{(n-1) n}.$$

However, it is not clear to me why this does not depend on the choice of the "Lebesgue subcover" $$\{U_1,...,U_n\}$$ or on the choice of the representative of the class $$[\sigma]$$.

For example, consider the case where $$\mathfrak{U}=\{U,V,W\}$$ consists on three open sets with $$U\cap V \neq \varnothing$$ and $$U,V \subset W$$. If we choose a path contained in $$U\cup V$$, we could choose the Lebesgue covering to be $$\{U,V\}$$, which would yield $$f([\sigma])=g_{UV}$$ or we could choose the covering to be simply $$\{W\}$$, which would yield $$f([\sigma])=1$$, and I do not see why these should coincide.

• A Cech cocycle always produces a bundle which is trivializable on each open set in the cover, so its monodromy on any loop contained in the open set should be trivial. May 6 '20 at 14:37
• Also, you need to choose the first and last open sets to be the same for this formula to be correct. May 6 '20 at 14:37

As already pointed out in the comments, the open cover must be cyclic: $$U_0=U_n$$.
By induction, it suffices to show invariance under adding a single new point $$t_{i-1} with $$[t_{i-1},t_{i-1/2}]⊂σ^*U_{i-1/2}$$ and $$[t_{i-1/2},t_i]⊂σ^*U_i$$.
But this invariance is precisely what the Čech cocycle condition says for the open sets $$U_{i-1}$$, $$U_i$$, and $$U_{i-1/2}$$: $$g_{i-1,i}=g_{i-1/2,i}g_{i-1,i-1/2}.$$