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Operator Algebra/The first K-group

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Definition (K1):

Exercises

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  1. Given a loop , we associate to it its winding number
    1. Prove that this number is an integer, appealing to the corresponding result in the one-dimensional case.
    2. 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.
    3. Prove that if is regarded as a group, then the winding number induces a group homomorphism .
    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.