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I came across an argument for the fact that the degree of the map [itex]R_n[/itex] which reflects the n-sphere through a plane is -1. It goes as follows:

Describe [itex]S^n[/itex] as two disks whose overlap is [itex]S^{n-1}[/itex] (in such a way that [itex]R_n[/itex] restricted to this overlap is [itex]R_{n-1}[/itex])

Then due to naturality of the Mayer-Vietoris sequence, the following commutes:

[itex]\begin{array}{ccc}

H_n(S^n) &\to^\cong &H_{n-1}(S^{n-1}) \\

\downarrow R & & \downarrow R \\

H_n(S^n) &\to^\cong &H_{n-1}(S^{n-1})

\end{array}[/itex]

Hence deg(R) is independent of the dimension n (and then we calculate deg(R) = -1 for [itex]S^1[/itex]).

However:what if instead of the reflection [itex]R[/itex] we had used the inversion [itex]\pi[/itex]? Every piece of the argument goes through (what would change?) but the conclusion would bewrong! (The degree of reflection depends on n, i.e. [itex]\deg \pi = (-1)^{n+1}[/itex].)

What is going wrong? Thanks! (andmerry christmas)

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# Naturality/commutativity of Mayer-Vietoris giving wrong answer?

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