The Topology of Graph Configuration Spaces

David G.C. Handron

Carnegie Mellon Universityhandron@andrew.cmu.edu

The Topology of Graph Configuration Spaces

1. Configuration Spaces2. Graphs3. Topology4. Morse Theory5. Results

Configuration Spaces

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

Cyclic Configuration Spaces:

Other Configuration Spaces

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

Other Configuration Spaces

Cyclic Configuration Spaces: Not 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 Spaces

Cyclic Configuration Spaces: Not 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 Spaces

Cyclic Configuration Spaces: Not 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 Spaces

Graph Configuration Spaces

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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

Graph Theory

A graph G 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.

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G:

V(G)={v1, v2, v3, v4, v5, v6}

E(G)={{v1, v3}, {v2, v3},...}

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G:

V(G)={v1, v2, v3, v4, v5, v6}

E(G)={{v1, v3}, {v2, v3},...} = {e13, e23, e34, ...}

Subgraphs

A subgraph of a graph G is a graph Hsuch 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 graph G[V'] includes all the edges of G joining vertices in V'.

Contractions

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G:

G/e34:

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

...by identifying the vertices joined by that edge

Contractions, cont.

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G:

G/{e13, e34, e46}:

We can contract with respect to a set of edges.

Simply identify each pair of vertices.

Partitions

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 and Induced Graphs

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

contract all the edges in G[P].

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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.

Topology

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.

Euler Characteristic

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

Euler Characteristic

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

Cube: 8-12+6=2

Euler Characteristic

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

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.

Euler Characteristic of a CW-complex

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

Morse Theory

A Morse function is a smooth function from a manifold M to R

which has non-degenerate critical points.

Non-Degenerate Critical Points

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.

Index of a Non-Degenerate Critical Point

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...

Morse Theory Results

(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.