  We say that a tropical fan is homologically smooth if each of its open
subsets verify tropical Poincare duality. A tropical homology manifold is a
tropical variety that is locally modelled by open subsets of homologically
smooth tropical fans.
  We show that homological smoothness is a T-stable property in the category of
tropical fans. This implies in particular that quasilinear fans are
homologically smooth, and tropical varieties locally modelled by them are
tropical homology manifolds. Previously, this was known only for locally
matroidal tropical varieties.
  In order to show the above results, we prove a tropical analogue of the
Deligne weight spectral sequence for homologically smooth tropical fans. This
allows to describe the cohomology of tropical modifications, and will be of
importance in our companion work which develops a Hodge theory in the tropical
setting.
