- #1
nonequilibrium
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- 2
Hello,
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 be wrong! (The degree of reflection depends on n, i.e. [itex]\deg \pi = (-1)^{n+1}[/itex].)
What is going wrong? Thanks! (and merry christmas)
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 be wrong! (The degree of reflection depends on n, i.e. [itex]\deg \pi = (-1)^{n+1}[/itex].)
What is going wrong? Thanks! (and merry christmas)