  For a tropical manifold of dimension n we show that the tropical homology
classes of degree (n-1, n-1) which arise as fundamental classes of tropical
cycles are precisely those in the kernel of the eigenwave map. To prove this we
establish a tropical version of the Lefschetz (1, 1)-theorem for rational
polyhedral spaces that relates tropical line bundles to the kernel of the wave
homomorphism on cohomology. Our result for tropical manifolds then follows by
combining this with Poincar\'e duality for integral tropical homology.
