Face enumeration—from spheres to manifolds
Ed Swartz
Cornell University, Ithaca, United States
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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 [47], allow us to give a complete characterization of the -vectors of arbitrary simplicial triangulations of , surfaces, and . 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.
Cite this article
Ed Swartz, Face enumeration—from spheres to manifolds. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 449–485
DOI 10.4171/JEMS/156