Abstract: We study a natural family of graphs
associated to symplectic manifolds with torus actions. These graphs encode
information about the manifold, in particular they are useful for computations
in equivariant cohomology. For this reason, it is important to understand
the combinatorics and combinatorial significance of the structures inherited
from the geometric setting. Accordingly, we can start with
a graph G, and define combinatorial analogues
of geometric objects, including Morse functions and Betti numbers. We will
examine several examples, and describe their properties, some motivated
by differential geometry and topology, and some purely combinatorial.
In particular, we will examine the graphs associated with compact homogeneous
spaces, and discuss the geometric properties one can study combinatorially
in terms of the graphs. I will not assume any background in symplectic
geometry. |
