Question about Milnor's topology fromthe diffable viewpoint

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The discussion centers on Lemma 1 from a topology text, which asserts that the index of an isolated zero of a vector field in R^m remains invariant under a diffeomorphism. The lemma is proven by first simplifying the scenario to a small open ball around the zero, then addressing cases where the diffeomorphism preserves and reverses orientation. The key insight is that analyzing reflections suffices for understanding orientation-reversing diffeomorphisms, as they encapsulate the necessary properties for the proof.

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quasar987
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Lemma 1 of page 35 says that the index at an isolated zero z of vector field v on an open set U of R^m is the same as the index at f(z) of the pushfoward f_*v = df o v o f^-1 of v by a diffeomorphism f:U-->U'. For the proof, he first reduces the problem to the case where z=f(z)=0 and is U a small open ball. Then he proves the lemma in the case where f preserves the orientation. Then he arrives to the case when f reverses it and says, "To consider diffeomorphisms which reverse orientation, it is sufficient to consider the special case of a reflection."

Why is this sufficient?!?
 
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Ok, got it.
 

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