Question about the Betti numbers

Question

Definition of Betti number at http://en.wikipedia.org/wiki/Betti_number

The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the maximum amount of cuts that can be made before separating a surface into two pieces"

Is it true for all $n$ or just for $n=1$? What does it mean by "separating a surface into two pieces"? Is it related to connected components?

Answer 1

"Cutting a surface into two pieces" corresponds to the case of a $2$-manifold (= surface) and $n=1$, in more generality it's the rest of the sentence: "two pieces or 0-cycles, 1-cycles, etc."

In the case of a surface and $n=1$, the first Betti number $b_1$ does correspond to how many "cuts" you can make before getting more than one connected components.

• With a sphere, $b_1 = 0$ and you can't make any cuts (in the form of a closed curve, basically a circle) without separating the sphere into two connected components.
• With a torus, $b_1 = 2$ and you can make two cuts (in the form of a circle) and still keep only one connected component (try to visualize it). If you make three, you'll get at least two components.

Keep in mind that this is only an informal explanation, and with increasing $n$ it's harder and harder to understand what $b_n$ "means" (how many times you can cut your space with an $n$-cycle and still keep one $(n-1)$-"piece").