  Given a simplicial complex with $n$ simplices, we consider the Connected
Subsurface Recognition (c-SR) problem of finding a subcomplex that is
homeomorphic to a given connected surface with a fixed boundary. We also study
the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead
search for a surface whose boundary, number of connected components, and total
genus are given.
  For both of these problems, we give parameterized algorithms with respect to
the treewidth $k$ of the Hasse diagram that run in $2^{O(k \log k)}n^{O(1)}$
time. For the SoG problem, we also prove that our algorithm is optimal assuming
the exponential-time hypothesis. In fact, we prove the stronger result that our
algorithm is ETH-tight even without restriction on the total genus.
