# Explicit action of the Dehn twist in the homology of punctured sphere with local coefficients

Let $$X=\mathbb{P}^1\setminus S$$, where $$S=\{a_1,\dots, a_k\}$$ is a finite subset of $$\mathbb{P}^1$$ and we may assume that $$|S|\geq 4$$. Let $$\mathbb{L}$$ be a local system on $$X$$ given by a monodromy representations $$\rho:\pi_1(X)\to GL_n(\mathbb{C})$$. We know that the First homology $$H_1(X,\mathbb{L})$$ is generated by the paths $$\delta_r, r=1,\dots, k-1$$, with values in $$\mathbb{L}$$ where $$\delta_r$$ is a path connecting $$a_r$$ to $$a_{r+1}$$ for $$r=1,\dots, k-1$$. We know moreover that for each path $$\delta_r$$, there is a homoeomorphism $$T_{\delta_r}$$ known as the Dehn twist. My question: How does each Dehn twist $$T_{\delta_r}$$ act on $$H_1(X,\mathbb{L})$$? I would like to have explicit formulas. I appreciate any relevant reference.

• You should know how it acts on the fundamental group, so its action on $H_1$ with local coefficients should be derivable from that, no? – Ryan Budney Mar 19 '20 at 0:31
• @ Ryan Budney. I don't know how the action on $\pi_1$ gives the action on $H_1$ with local coefficients. Of course if the coefficients are $\mathbb{Z}$ it is justbthe abelianization of the fundamental group. But how about the general local system? And by the way, I am not sure of this ist the best way, because the action on the fundamental group might ber vey complicated, even for integral coefficients, whereas the action on the homology should be more accessible. – Darius Math Mar 19 '20 at 1:23