In Border's book about fixed point theory, the following proof of Sperner's Lemma is given:
Let $\bar{T}$ simplicially subdivided and properly labeled by the function $\lambda$. Then there are an odd number of completely labeled n-subsimplexes in the subdivision.
My difficulty is with the definition of incidence and the consequent computation of deegree, in particular for incidence between n-simplexes. In fact, we are basically connecting n-simplexes which share a face 'rainbow coloured', in the sense that it presents n-1 different labels.
- Shouldn't the deegree depend on n, the deimension of the simplex? I
would expect n+1 links for an element of C and n for one in A, since an n simplex has n+1 faces, which in the first case are all 'rainbow coloured', while in the second, as in the proof itself is noticed, one label is repeated, yielding only n+1-1=n 'rainbow coloured faces'. This fact seem correct in the bidimensional case:
where we have 2 and 3 deegrees, respectevely. - Moreover, I understand that an element of B is self linked, but why is it linked to the simplex of which is a face?
Thanks in advance
There is a disparity between the definitions in the proof and the image. Let us say that red = $0$, green = $1$ and blue = $2$. According to the proof, two triangles should only be connected by an edge if the $1$-simplex joining them is labeled with $\{0,1\}$, that is, red and green. In the picture, however, we see edges between every pair of triangles whose shared border has two different colors, not just red and green. Therefore, the picture should be ignored.
If a simplex $c$ is in $C$, so is labeled with $\{0,1,\dots,n\}$, then it has exactly one face labeled $\{0,1,\dots,n-1\}$, so it is incident on exactly one edge.
If a simplex $a$ is in $A$, then by definition its set of labels is $\{0,1,\dots,n-1\}$, so one of the labels between $0$ and $n-1$ is repeated. Either of the faces which are to the opposite of a repeated label will be an edge, so the degree of $a$ is $2$.