Replacing two cross-caps by a handle

Question

For a non-orientable surface, we can replace a handle by two cross-caps. Can we do the opposite i.e replace any two cross-caps by a handle?

Any help is appreciated!!

Answer 1

This is Dyck's Theorem.

You need either to have a third crosscap available, or be careful to use a crossed handle (i.e. a Klein bottle). Otherwise you'd be able to convert a sphere with two crosscaps (which is non-orientable) into a torus (which is orientable).

Answer 2

The short answer is, yes, you can do that. Basically you can show that $$\mathbb{RP}^2\#\mathbb{RP}^2\#\mathbb{RP}^2\cong \mathbb{T}\#\mathbb{RP}^2.$$ This is a homeomorphism, hence reversible. This relates to your question since $\mathbb{RP}^2$ is a cross-cap, $\mathbb{T}$ is a torus (handle in your terminology) and every nonorientable compact surface has $\mathbb{RP}^2$ as a connect-summand.

In fact, usually one uses this opposite direction to prove the classification of surfaces. Using various techniques you show a surface must be a connect sum of some number of copies of $\mathbb{T}$ and $\mathbb{RP}^2$, and then uses the fact that the presence of a single $\mathbb{RP}^2$ allows you to convert all $\mathbb{T}$'s into pairs of cross-caps, showing every compact nonorientable surface is a connect-sum of some number of cross-caps.