# Computing morphisms between stalks of proper pushforward sheaf

Consider the map $$\pi : (0,n]\to S^1$$ sending $$t\mapsto e^{2\pi it}$$ and let's look at the proper pushforward sheaf $$F\cong \pi_!\mathbb{C}_{(0,n]}$$. The stalk of this sheaf is $$\mathbb{C}^n$$ at every point on $$S^1$$. Consider the paths, $$\gamma_1(t)=t\pi$$ and $$\gamma_2(t)=(2-t)\pi$$ connecting $$0$$ and $$\pi$$. Now these paths induce maps on stalks, i.e. we have maps from $$F_{0}\to F_{\pi}$$, namely $$f_1$$ and $$f_2$$ induced by $$\gamma_1$$ and $$\gamma_2$$ respectively. Now I have computed $$f_2$$ and it turns out to be the identity matrix. I am having trouble with computing $$f_1$$. I think this map should be given by the Jordan block matrix of size $$n\times n$$ with eigenvalue $$1$$.(As far as I understand, these pairs of maps $$(f_1, f_2)$$ should correspond to indecomposable representations of the $$2-$$Kronecker quiver, and when the vector spaces have the same dimension, then those representations are given by pairs $$(\text{Id}, J_\lambda)$$ for some Jordan block with eigenvalue $$\lambda$$.)

To compute these maps, what I have been doing is picking neighbourhoods $$U$$ of $$0$$ and $$V$$ of $$\pi$$. Then I choose a neighbourhood $$W_i$$ containing the image of $$\gamma_i$$ and use the following composition of arrows : $$F_0\cong F(U)\cong F(W_i)\stackrel{res_{U, W_i}}{\longrightarrow}F(V)\cong F_\pi$$ But I am stuck with the case when it is $$\gamma_1$$. Any help?