Integration: a group homomorphism from fundamental group to (C,+)

I got an idea today in my complex analysis course.

Math Essays: ... Fundamental group and Homotopy as Natural Transformation (wuyulanliulongblog.blogspot.com)

When we deal with integration for exact form, it looks like a group homomorphism from the fundamental group of a multi-connected domain ${\pi }_1(\mathrm{\Omega })$$\pi_1(\Omega)$ to $(\mathbb{C},+)$$(\mathbb C,+)$.

Firstly, it is the integration of exact form, thus it is homotopy invariance. Thus it is well-defined for the element in the fundamental group. i.e. we could talk about the integration of the path homotopic equivalence class $⟨\gamma ⟩$$\langle\gamma\rangle$.

Then, by the property of integration, we have

$$\begin{array}{}\text{(1)}& {\int }_{⟨-⟩}f(z)dz:{\pi }_1(\mathrm{\Omega })⟶(\mathbb{C},+),{\int }_{⟨{\gamma }_1⟩\circ ⟨{\gamma }_2⟩}f(z)dz={\int }_{⟨{\gamma }_1\circ {\gamma }_2⟩}f(z)dz={\int }_{⟨{\gamma }_1⟩}f(z)dz+{\int }_{⟨{\gamma }_2⟩}f(z)dz\end{array}$$

In particular, the group homomorphism preserve the identity, i.e.

$$\begin{array}{}\text{(2)}& {\int }_{⟨e⟩}f(z)dz=0\end{array}$$

My friend told me that will related to Riemann-Hillbert correspondence, I will back to it after I learn them.