Does the intersection of two n-simplexes have to be a sub-simplex or can it be any n-simplex?
I'm trying to create the simplicial complex for the solid torus, so my idea was to take a bunch of cubes and break them up into 5 tetrahedra and then glue all the cubes together and remove one from the center.
But if I break two cubes into 5 tetrahedra and put them next to each other then their simplexes have to intersect at another simplex. So I'm wondering does that have to be a simplex that is already in the structure or can they intersect in any triangle (since that is a simplex). So will something like my picture be allowed? If we put the cubes beside each other, the 3-simplexes intersect at some triangles which are 2-simplexes, is that allowed or must they intersect at an n-simplex which is already in the structure?
Two n-simplices in a simplicial complex must either not intersect or intersect in a (n-1)-simplex that is contained in your simplicial complex. In your case, I believe you can just rotate the cube before you glue it to have them line up correctly.