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David G.C. Handron Carnegie Mellon University handron@andrew.cmu.edu 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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The Topology of Graph Configuration Spaces

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David g c handron carnegie mellon university handron@andrew cmu edu l.jpg

David G.C. Handron

Carnegie Mellon University

handron@andrew.cmu.edu

The Topology of Graph Configuration Spaces


Slide2 l.jpg

The Topology of Graph Configuration Spaces

1. Configuration Spaces

2. Graphs

3. Topology

4. Morse Theory

5. Results


Configuration spaces l.jpg

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 Spaces


Cyclic configuration spaces l.jpg

Cyclic Configuration Spaces:

Other Configuration Spaces


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Cyclic Configuration Spaces:

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

Other Configuration Spaces


Other configuration spaces l.jpg

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


Other configuration spaces7 l.jpg

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


Other configuration spaces8 l.jpg

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 l.jpg

Graph Configuration Spaces

  • Configuration of points in a manifold

  • One point for each vertex of a graph

  • Points corresponding to adjacent vertices must be distinct


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  • Cyclic configuration spaces correspond to graphs that form a loop

  • Path configuration spaces correspond to graphs that form a continuous path


Graph theory l.jpg

A graphG 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


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  • V(G)={v1, v2, v3, v4, v5, v6}

  • E(G)={{v1, v3}, {v2, v3},...}


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  • V(G)={v1, v2, v3, v4, v5, v6}

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


Subgraphs l.jpg

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'.

Subgraphs


Contractions l.jpg

Contractions

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

...by identifying the vertices joined by that edge


Contractions cont l.jpg

Contractions, cont.

We can contract with respect to a set of edges.

Simply identify each pair of vertices.


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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


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

contract all the edges in G[P].

Partitions and Induced Graphs


Examples l.jpg

Examples

  • 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 l.jpg

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


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Euler Characteristic

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


Euler characteristic22 l.jpg

Euler Characteristic

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

  • Cube: 8-12+6=2


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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 characteristic24 l.jpg

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 l.jpg

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-complex


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

which has non-degenerate critical points.

Morse Theory


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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 l.jpg

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


Morse theory results l.jpg

(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


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