  We study local, global and local-to-global properties of threefolds with
certain singularities. We prove criteria for these threefolds to be rational
homology manifolds and conditions for threefolds to satisfy rational Poincar\'e
duality. We relate the topological Euler characteristic of elliptic Calabi-Yau
threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of
Lie algebras and certain representations, Milnor and Tyurina numbers and other
birational invariants of an elliptic fibration. We give an interpretation in
terms of complex deformations. We state a conjecture on the extension of
Kodaira's classification of singular fibers on relatively minimal elliptic
surfaces to the class of birationally equivalent relatively minimal genus one
fibered varieties and we give results in this direction.
