In Hatcher's book, http://en.wikipedia.org/wiki/Talk:Coherent_topology, see the very end of page 135 and the beginning of page136 for the definition of the local degree of a map S^n-->S^n.(adsbygoogle = window.adsbygoogle || []).push({});

I don't get how the local degree degf|x_i could be anything else than ±degf: In the diagram of page 136 used to define the local degree, identify all the outer groups withZ. Start with 1 in the lower left [itex]H_n(S^n)[/itex] and follow the isomorphism that leads to [itex]H_n(U_i,U_i-x_i)[/itex].Since an isomorphism brings a generator to a generator, we end up with ±1. Then take this ±1 to [itex]H_n(V,V-y)[/itex] through f_*. We end up with ±degf|x_i by definition of the local degree. Now take that same initial 1 to [itex]H_n(V,V-y)[/itex] through the other outer route. We get [tex]1\stackrel{f}{\mapsto} \deg f\stackrel{\cong}{\mapsto}\pm \deg f[/tex], and so by commutativity of the diagram, \degf|x_i=±degf.

Where am I mistaken??

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# About the definition of local degree

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