I am trying to find if the winding number is invariant to some transformations. I already know that it is invariant respect to translation, rotation and scaling. Also, it is not invariant respect to every continuous / holomorphic function since $z^n$ changes it. However, it is true that it is invariant with respect to homotopy. But I am not able to see if an injective function represents an homotopy. I would like to know if adding the conditions of continuity or holomorphism the invariance is true. Or if there exists a counterexample.
2026-04-12 00:22:20.1775953340
Does injective continuous / holomorphic functions preserve winding number?
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