Operator Algebra/The first K-group

Exercises

 * 1) Given a loop $$\gamma: [0,1] \to \operatorname{GL}_n(\mathbb C)$$, we associate to it its winding number $$\frac{1}{2\pi i} \int_0^1 \operatorname{tr} (\gamma(t)^{-1} \gamma'(t)) dt$$
 * 2) Prove that this number is an integer, appealing to the corresponding result in the one-dimensional case.
 * 3) Prove that if $$\gamma: [0,1] \to \operatorname{GL}_n(\mathbb C)$$ and $$\rho: [0,1] \to \operatorname{GL}_n(\mathbb C)$$ are loops and there exists a homotopy $$H: [0,1]^2 \to \operatorname{GL}_n(\mathbb C)$$ 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.
 * 4) Prove that if $$K_1$$ is regarded as a group, then the winding number induces a group homomorphism $$K_1(\mathcal C(S^1, \mathbb C)) \to \mathbb Z$$.
 * 5) Prove that this group homomorphism is surjective.
 * 6) Prove that the winding number of a matrix-valued path equals that of its point-wise determinant.