Operator Algebra/The first K-group

Definition (K1):

Exercises

1. Given a loop $\gamma :[0,1]\to \operatorname {GL} _{n}(\mathbb {C} )$, we associate to it its winding number ${\frac {1}{2\pi i}}\int _{0}^{1}\operatorname {tr} (\gamma (t)^{-1}\gamma '(t))dt$
   1. Prove that this number is an integer, appealing to the corresponding result in the one-dimensional case.
   2. Prove that if $\gamma :[0,1]\to \operatorname {GL} _{n}(\mathbb {C} )$ 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.
   3. Prove that if $K_{1}$ is regarded as a group, then the winding number induces a group homomorphism $K_{1}({\mathcal {C}}(S^{1},\mathbb {C} ))\to \mathbb {Z}$.
   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.