Why do $P^2$#$T$ and $P^2$#$P^2$#$P^2$ have different genus?

Question

I have seen that the genus of the connected sum of closed surfaces is suppose to be sum of the genus of the individual surfaces thus $P^2$#$T$(connected sum of torus and real projective plane) is suppose to have genus $2$ and $P^2$#$P^2$#$P^2$ is suppose to have genus $3$. However the two spaces are also suppose to be homeomorphic isn't that a contradiction or is genus not a topological invariant?

Answer 1

The concept of "genus" is well established and commonly used for orientable surfaces such as connected sums of toruses.

On the other hand, the concept of "genus" for nonorientable surfaces, such as $P^2$, $P^2 \# T$, and $P^2 \# P^2 \# P^2$, is less well established and less commonly used. Perhaps one reason for this is that the orientable and nonorientable versions do not mix quite as well as one would naively expect, which is what you have discovered.

To explain, let me first state the definitions precisely. Every closed oriented surface can be expressed as a connected sum of $0$ or more toruses, and the number of such such summands is called the "orientable genus" (a connected sum of zero toruses being, by convention, the 2-sphere). Also, every closed non-oriented surface can be expressed as a connected sum of $1$ or more projective planes, and the number of those summands is called the "non-orientable genus".

Orientable genus has the property that it is additive with respect to connected sum. Nonorientable genus has the same property.

However, it is not true that orientable and nonorientable genus are jointly additive. In other words, if $A$ is a closed orientable surface and $B$ is a closed nonorientable surface, then $A \# B$ is a closed nonorientable surface and $$(\text{nonorientable genus})(A \# B) \ne (\text{orientable genus})(A) + (\text{nonorientable genus})(B) $$ On the other hand, what is true is $$(\text{nonorientable genus})(A \# B) = 2*(\text{orientable genus})(A) + (\text{nonorientable genus})(B) $$

Answer 2

The non-orientable genus k and genus g are not the same and cannot be mixed in calculations. In terms of Euler characteristic, for some surface $S$. $$\chi(S) = 2-k$$ whereas $$\chi(S) = 2-2g$$ (see https://en.wikipedia.org/wiki/Genus_(mathematics)). This is why the non-orientable genus and genus are not additive over connect sums (by using the formula for the Euler characteristic for a connect for example).