(1) For a continuous map f:S^n-->S^n, the induced map in top homology is, up to identification of H_n(S^n) with Z, just multiplication by an integer, and this integer is defined as the degree of f. (in Hatcher: http://www.math.cornell.edu/~hatcher/AT/ATpage.html)(adsbygoogle = window.adsbygoogle || []).push({});

Now, in the event that f is a homeomorphism, it follows from the elementary properties of the degree that deg(f)=±1. After this remark, Hatcher adds that in applications, it is usually not hard to determine which it is between +1 and -1.

Can someone give an example illustrating how one decides between +1 and -1?

(2) How to see why the above definition of degree coincide with the one in terms of preimage of regular value in the case of a differentiable f?

Thanks.

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# Two questions about degree

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