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The Topology of Graph Configuration SpacesPowerPoint Presentation

The Topology of Graph Configuration Spaces

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Carnegie Mellon University

The Topology of Graph Configuration SpacesThe Topology of Graph Configuration Spaces

1. Configuration Spaces

2. Graphs

3. Topology

4. Morse Theory

5. Results

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 of

Configuration SpacesOther Configuration Spaces

Not all points must be distinct, only those whose indices differ by one (mod k).

Other Configuration SpacesNot all points must be distinct, only those whose indices differ by one (mod k).

Used (e.g. by Farber and Tabachnikov) to study periodic billiard paths.

Other Configuration SpacesNot all points must be distinct, only those whose indices differ by one (mod k).

Used (e.g. by Farber and Tabachnikov) to study periodic billiard paths.

Path Configuration Spaces:

Other Configuration SpacesNot all points must be distinct, only those whose indices differ by one (mod k).

Used (e.g. by Farber and Tabachnikov) to study periodic billiard paths.

Path Configuration Spaces:

Used by myself to study non-cyclic billiard paths.

Other Configuration SpacesGraph Configuration Spaces

- Configuration of points in a manifold
- One point for each vertex of a graph
- Points corresponding to adjacent vertices must be distinct

- Cyclic configuration spaces correspond to graphs that form a loop
- Path configuration spaces correspond to graphs that form a continuous path

A a loopgraphG consists of:

(1) a finite set V(G) of vertices, and

(2) a set E(G) of unordered pairs of vertices.

The elements of E(G) are the edges of the graph.

Graph Theory- V(G)={v1, v2, v3, v4, v5, v6}
- E(G)={{v1, v3}, {v2, v3},...}

- V(G)={v1, v2, v3, v4, v5, v6}
- E(G)={{v1, v3}, {v2, v3},...} = {e13, e23, e34, ...}

A subgraph of a graph G is a graph H

such that

(1) Every vertex of H is a vertex of G.

(2) Every edge of H is an edge of G.

If V' is a subset of V(G), the induced graphG[V'] includes all the edges of G joining vertices in V'.

SubgraphsContractions

We can contract a graph with respect to an edge ...

...by identifying the vertices joined by that edge

Contractions, cont.

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.

PartitionsFor each partition P of a graph G, there is a corresponding contraction:

contract all the edges in G[P].

Partitions and Induced GraphsExamples

- Two partitions {{v1,v2,v4},{v3}} and {{v1,v2},{v3},{v4}} may induce the same edge set...
- ...and produce the same quotient.

- A partition is connected if each is a connected graph. It can be shown that connected partitions induce the same subgraphs and partitions.

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.

TopologyEuler Characteristic of a graph-configuration space. This description will involve properties of the graph, and topological properties of the underlying manifold.

- The Euler characteristic of a polyhedron is commonly describes as v-e+f.

Euler Characteristic of a graph-configuration space. This description will involve properties of the graph, and topological properties of the underlying manifold.

- The Euler characteristic of a polyhedron is commonly describes as v-e+f.
- Cube: 8-12+6=2

Euler Characteristic of a graph-configuration space. This description will involve properties of the graph, and topological properties of the underlying manifold.

- The Euler characteristic of a polyhedron is commonly describes as v-e+f.
- Cube: 8-12+6=2
- Tetrahedron: 4-6+4=2

Euler Characteristic of a graph-configuration space. This description will involve properties of the graph, and topological properties of the underlying manifold.

- The Euler characteristic of a polyhedron is commonly describes as v-e+f.
- Cube: 8-12+6=2
- Tetrahedron: 4-6+4=2
- Both topologically equivalent (homeomorphic) to a sphere.

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, then

Euler Characteristic of a CW-complexA Morse function is a smooth function from a manifold out of M to R

which has non-degenerate critical points.

Morse TheoryNon-Degenerate Critical Points out of

- A point p in M is a critical point if df=0. In coordinates this means
- A critical point is non-degenerate if the Hessian matrix of second partial derivatives has nonzero determinate.

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 of negative eigenvalues of the Hessian matrix.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
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