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.

 

[tacman]

Betti numbers and Euler characteristic are important topological invariants that help us understand the structure and properties of simplicial complexes. The Betti numbers of a simplicial complex are a sequence of integers that describe the number of holes in the complex at different dimensions, while the Euler characteristic is a single number that summarizes the topological information of the complex.

In the case described, where the simplicial complex is made only of 2-simplices and all of them are adjacent to a single edge, we have a book shape that can only be embedded in dimensions 3+. This means that the complex has no 0-dimensional holes (connected components) and no 1-dimensional holes (cycles). However, it does have a 2-dimensional hole, as it cannot be embedded in 2 dimensions without self-intersections.

Thus, the second Betti number (representing the number of 2-dimensional holes) of this complex would be 1. This is the same as the second Betti number of a tetrahedron, which also has a single 2-dimensional hole. However, the Euler characteristic of the book shape would be different from that of a tetrahedron. The Euler characteristic of a tetrahedron is 0, while the book shape has an Euler characteristic of -1. This is because the book shape has one more 3-dimensional simplex (the "cover" of the book) than the tetrahedron.

In conclusion, while the book shape and the tetrahedron may have the same second Betti number, their Euler characteristics are different, highlighting the importance of considering both invariants to fully understand the topological properties of a simplicial complex.