The following picture depicts a 2-manifold \(X\): a closed surface in \(\RR^3\) drawn as a circular chain of five tori, each torus joined to its two neighbours, so that the surface shows five small holes, one in each link, arranged around one large central hole.
We know that every 2-manifold is homeomorphic to one of
\[S^2,\ T_1,\ T_2,\ \ldots,\ \RR P^2,\ \RR P^2\#\RR P^2,\ \RR P^2\#\RR P^2\#\RR P^2,\ \ldots\]
Determine which of these 2-manifolds is homeomorphic to \(X\).
The source gives \(X\) only by the picture; the description in the statement is a reading of that picture.