  The tropical fundamental class of a rational balanced polyhedral fan induces
cap products between tropical cohomology and tropical Borel-Moore homology.
When all these cap products are isomorphisms, the fan is said to be a tropical
Poincar\'e duality space. If all the stars of faces also are such spaces, such
as for fans of matroids, the fan is called a local tropical Poincar\'e duality
space.
  In this article, we first give some necessary conditions for fans to be
tropical Poincar\'e duality spaces and a classification in dimension one. Next,
we prove that tropical Poincar\'e duality for the stars of all faces of
dimension greater than zero and a vanishing condition implies tropical
Poincar\'e duality of the fan. This leads to necessary and sufficient
conditions for a fan to be a local tropical Poincar\'e duality space. Finally,
we use such fans to show that certain abstract balanced polyhedral spaces
satisfy tropical Poincar\'e duality.
