So let $S$ be a closed, orientable surface, and let $P$ be some point on Surface $S.$ I want to introduce two paths $a$ and $b$ such that:
-both $a$ and $b$ start and end at point $P$
-$a$ and $b$ do not cross itself anywhere
-$a$ and $b$ do not meet anywhere except the start/end point $P$
-(most importantly) The intersection of $a$ and $b$ at point $P$ is transverse and not tangential.
With these conditions, how would one prove that the surface $S$ is of the form $Torus \# U$ for some closed surface $U?$