Let \(X\) be the 2-manifold drawn as a circular chain of five tori, each torus joined to its two neighbours, and let the classification list be
\[S^2,\ T_1,\ T_2,\ \ldots,\ \RR P^2,\ \RR P^2\#\RR P^2,\ \RR P^2\#\RR P^2\#\RR P^2,\ \ldots\]
Determine which of these 2-manifolds is homeomorphic to the space \(Y\) which is constructed similarly to \(X\), but where two of the tori making up the “links” in the circular chain are replaced by \(\RR P^2\)’s.
The source states this problem as “Repeat the above problem for the space \(Y\) which is constructed similarly to \(X\), but where two of the tori making up the ‘links’ in the circular chain are replaced by \(\RR P^2\)’s,” referring to practice problem 6.