Operator Algebra/The first K-group

K1 Exercises Given a loop , we associate to it its winding number Prove that this number is an integer, appealing to the corresponding result in the one-dimensional case. Prove that if and are loops and there exists a homotopy through loops from to which is continuously differentiable in the component that varies when going along a fixed loop, then the winding numbers of and are equal. Prove that if is regarded as a group, then the winding number induces a group homomorphism . Prove that this group homomorphism is surjective. Prove that the winding number of a matrix-valued path equals that of its point-wise determinant.