Which theorem of homotopy theory states that if two objects have different genus then they are not homotopy equivalent?

Question

I'm quite new inexperienced in the field but from what I see two objects with different genus are not homotopy equivalent.

Question: Which theorem of homotopy theory states that if two objects have different genus then they are not homotopy equivalent?

Answer 1

So, for closed surfaces of genus $g$, the Euler characteristic is uniquely determined:

\begin{equation} \chi=2-2g. \end{equation}

Now, the Euler characteristic can be defined as the alternating sum of the Betti numbers (i.e. the ranks of the singular homology groups):

$$\chi = b_0 - b_1 + b_2 + \cdots $$

These homology groups are isomorphic for any two homotopy-equivalent spaces, and thus the Betti numbers coincide. Hence the Euler characteristic is a homotopy invariant, and we also obtain that the genus $g$ must be the same for all homotopy-equivalent spaces.