I am confused by a statement: in Wikipedia,
In topological graph theory the first Betti number of a graph G with n vertices, m edges and k connected components equals $$m - n + k.$$
I am verifying two simple examples, a disk and an annulus. An annulus is the same as a cylinder.
Ex 1: I suppose that a disk can be represented by a triangle (black region of the figure).
For a disk, we have $$b_1=m - n + k=3-3+1=1.$$
Question 1: So is the first Betti number $b_1=1?$ Shouldn't a disk have the first Betti number $b_1=0?$
Ex 2: I suppose that an anulus can be represented by a triangle with a hole triangle (black region of the figure).
For an annulus, we have
$$b_1=m - n + k=6-6+1=1.$$
Question 2: So is the first Betti number $b_1=1?$ This is the correct computation for an annulus--- the first Betti number $b_1=1?$ Correct?
$\endgroup$ 01 Answer
$\begingroup$As is written in Wikipedia, the equality $b_1 = m - n + k$ makes sense and is true for topological graphs, not arbitrary spaces. A full triangle and an annulus are not graphs, so the equality isn't true (the fact that you get the right result for the annulus is more of a coincidence than anything).
For an arbitrary CW complex, the Euler characteristic $\chi$ is equal to the alternated sum of the number of cells in each dimension. For a topological graph, $\chi$ is equal to $b_0 - b_1$ where $b_0 = k$ is the number of connected components, hence the equality. As you can see this proof cannot work at all for arbitrary complexes.
$\endgroup$ 5
