I'm trying to solve an exercise from chapter 1 in Using the Borsuk-Ulam Theorem.
The exercise goes as follows. Consider an $m$ by $n$ chessboard. Let each square of the board be a vertex in an abstract simplicial complex and let each simplex be a combination of squares where rooks could be placed without threatening each other. Call this complex the chess complex.
What is the geometric shape of the polyhedron of the geometric simplicial complex derived from the abstract simplicial complex in the $3$ by $4$ case? According to the book the answer is the torus but I do not see it.
So I labeled the vertices of the simplicial complex like this:
Then I found all of the triangles that include $A$. These are $AFK, AFL, AGJ, AGL, AHJ, AHK$. Since this is a simplicial complex we also have all the sides of the triangles as simplices. Similarly for $B$ we find $BEK, BEL, BGI, BGL, BHK, BHI$. For $C$ we have $CEJ, CEL, CFI, CFL, CHJ, CHI$ and finally for $D$ we have $DEJ, DEK, DFI, DFK, DGJ, DGI$.
So after finding all of the triangles in the complex I drew it like so:
But I am having a difficult time visualizing how this is the torus. Can anyone help me see it? Maybe I need to draw the triangles in a smarter way?
As you see the six faces round a vertex fit together in a nice hexagonal fashion. Each edge is in two faces. So we get a nice manifold. The complex has $12$ vertices, $36$ edges and $24$ faces. So Euler characteristic is $12-36+24=0$ the same as the torus. If it's orientable, it must be a torus.
I can't see a quick way to prove it's not non-orientable though.