Face enumeration - from spheres to manifolds
arXiv:0709.3998
Abstract
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 -vector of balanced semi-Eulerian complexes and the toric -vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions \cite{Wal}, allow us to give a complete characterization of the -vectors of arbitrary simplicial triangulations of $S^1 \times S^3, \C P^2,$ 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 -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.
44 pages, 8 figures