# Compact, connected 2-manifold is a connected sum of another manifold and $P^2$ under some condition

A question from Introduction to topological manifold exercise 5-4

Exe5.4: Suppose M is a compact, connected 2-manifold that contains a subset $$B\subset M$$ that is homeomorphic to the Möbius band. Show that there is a compact 2-manifold M' such that M is homeomorphic to a connected sum $$M'$$#$$P^2$$.

I know if M is a compact, connected 2-manifold, then M has a very good surface presentation. But I have no geometric intuition on the question and can't figure out how to rigorously apply the surface presentation.

• Hint: $\mathbb{RP}^2$ is given by gluing a unit ball along the boundary of a mobius band (see here). Jun 2 at 16:13