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?



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