betti numbers and euler characterstic?

[Coolphreak]

Let's say you have some type of simplicial complex that is made only of 2 simplices. What happens if all those 2 simplices are adjacent to a single edge (creating a type of book shape), so that this complex can only be embedded in dimensions 3+? Would this complex have the same 2nd betti number as a tetrahedron?

[masnevets]

If I am understanding your description correctly, it seems to be that your book complex is a contractible space, and hence has trivial reduced homology. So the 2nd Betti number is 0.

The tetrahedron is also contractible, so the same thing happens. If you mean the boundary of a tetrahedron, then that's homotopy equivalent to S^2, which has 2nd Betti number 1.