Acyclic 2-dimensional complexes and Quillen’s conjecture

Polyakov Conjecture and 2+1 Dimensional Gravity

After briefly reviewing the hamiltonian approach to 2+1 dimensional gravity in absence of matter on closed universes, we consider 2+1 dimensional gravity coupled to point particles in an open universe. We show that the hamiltonian structure of the theory is the result of a conjecture put forward by Polyakov in a different context. A proof is given of such a conjecture. Finally we give the exact...

Serre’s Conjecture on 2-dimensional Galois representations

Topology of 2-dimensional Complexes

We present results concerning geometric topology of 2-dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3-manifolds (and thickenings), the problem of resolving arbitrary 2-polyhedra by fake surfaces (including an application to reduce the classical Whitehead asphericity conjecture to special polyhedra) and existence of nonhomeomorphic 3-manifolds with eq...

Graphs of Acyclic Cubical Complexes

It is well known that chordal graphs are exactly the graphs of acyclic simplicial complexes . In this note we consider the analogous class of graphs associated with acyclic cubical complexes . These graphs can be characterized among median graphs by forbidden convex subgraphs . They possess a number of properties (in analogy to chordal graphs) not shared by all median graphs . In particular , w...

Pd4-complexes and 2-dimensional Duality Groups

This paper is a synthesis and extension of three earlier papers on PD4-complexes X with fundamental group π such that c.d.π = 2 and π has one end. Our goal is to show that the homotopy types of such complexes are determined by π, the Stiefel-Whitney classes and the equivariant intersection pairing on π2(X). We achieve this under further conditions on π. It remains an open problem to give a homo...