Cohomological obstructions to Nielsen realization

On the generalized Nielsen realization problem

The main goal of this paper is to give the first examples of equivariant aspherical Poincaré complexes, that are not realized by group actions on closed aspherical manifolds M . These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of ...

The Diffeomorphism Group of a K3 Surface and Nielsen Realization

The Nielsen Realization problem asks when the group homomorphism Diff(M) → π0Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed ...

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...