Obstructions to Compatible Splittings

Obstructions to Shiftedness

In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to shiftedness and prove there are finitely many i...

Obstructions to Homotopy Invariance

Obstructions to Shellability

We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable 2-dimensional simplicial complex contains a nonshellable induced subcomplex with less than 8 vertices. We also establish CL-shellability of interval orders and as a consequence obtain a fo...

Topological Obstructions to Graph Colorings

For any two graphs G and H Lovász has defined a cell complex Hom (G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4....

New obstructions to symplectic embeddings

In this paper we establish new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces. These restrictions are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, we also show that they are sharp.