Operator Algebra/The first K-group

Exercises

1. Given a loop $\gamma :[0,1]\to {\mathrm{GL}}_n(ℂ)$, we associate to it its winding number $\frac{1}{2\pi i}\underset{0}{\overset{1}{\int }}\mathrm{tr}(\gamma (t{)}^{-1}{\gamma }^{\prime }(t))dt$
   1. Prove that this number is an integer, appealing to the corresponding result in the one-dimensional case.
   2. Prove that if $\gamma :[0,1]\to {\mathrm{GL}}_n(ℂ)$ and $\rho :[0,1]\to {\mathrm{GL}}_n(ℂ)$ are loops and there exists a homotopy $H:[0,1{]}^2\to {\mathrm{GL}}_n(ℂ)$ through loops from $\gamma$ to $\rho$ which is continuously differentiable in the component that varies when going along a fixed loop, then the winding numbers of $\gamma$ and $\rho$ are equal.
   3. Prove that if $K_1$ is regarded as a group, then the winding number induces a group homomorphism $K_1(𝒞(S^1,ℂ))\to \mathbb{Z}$.
   4. Prove that this group homomorphism is surjective.
   5. Prove that the winding number of a matrix-valued path equals that of its point-wise determinant.