Face enumeration—from spheres to manifolds

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h -vector of balanced semi-Eulerian complexes and the toric h -vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3-dimensional constructions [47], allow us to give a complete characterization of the f -vectors of arbitrary simplicial triangulations of S^1 \times S^3 , ℂP^2, K3 surfaces, and (S^2 \times S^2) \# (S^2 \times S^2) . We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the g -conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.