David G.C. Handron Carnegie Mellon University firstname.lastname@example.org The Topology of Graph Configuration Spaces The Topology of Graph Configuration Spaces 1. Configuration Spaces 2. Graphs 3. Topology 4. Morse Theory 5. Results
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1. Configuration Spaces
4. Morse Theory
Term configuration space is commonly used to refer to the space of configurations of k distinct points in a manifold M
This is a subspace ofConfiguration Spaces
We can contract a graph with respect to an edge ...
...by identifying the vertices joined by that edge
We can contract with respect to a set of edges.
Simply identify each pair of vertices.
The vertices of a graph G can be partitioned into a collection of disjoint subsets.
This partition determines a subgraph of G.
is the induced subgraph of the partition P.Partitions
contract all the edges in G[P].Partitions and Induced Graphs
The goal of this work is to describe topological invarients of a graph-configuration space. This description will involve properties of the graph, and topological properties of the underlying manifold.
Today, we'll be concerned with the Euler characteristic of these configuration spaces.Topology
A CW-complex is similar to a polyhedron. It is constructed out of cells (vertices, edges, faces, etc.) of varying dimension.
Each cell is attached along its edge to cells of one lower dimension.
If n(i) is the number of cells with dimension i, thenEuler Characteristic of a CW-complex
which has non-degenerate critical points.Morse Theory
The index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian matrix.
I'll switch to the whiteboard to explain what this is all about...Index of a Non-Degenerate Critical Point
(1) If f is a Morse function on M, then M is homotopy equivalent to a CW-complex with one cell of dimension i for each critical point of f with index i.
(2) A similar result holds for a stratified Morse function on a stratified space.Morse Theory Results