arXiv subjects
Publications and source records attributed to Lukas Katthän.
Let $P_n$ denote the $n$-th linear ordering polytope. We define projections from $P_n$ to the $n$-th permutahedron and to the $(n-1)$-st linear ordering polytope. Both projections are equivariant with respect to the natural $\Sn$-action and they project to orthogonal subspaces. In particular the second projection defines an $S_n$-action in $P_{n-1}$.
Let Δbe a (d-1)-dimensional homology sphere on n vertices with m minimal non-faces. We consider the invariant α:= m - (n-d) and prove that for a given value of α, there are only finitely many homology spheres that cannot be obtained through one-point suspension and suspension from another. Moreover, we describe all homology spheres with αup to four and, as a corollary, all homology spheres with up to eight minimal non-faces. To prove these results we consider the nerve of the minimal non-faces of Δ.