The%20Topology%20of%20Graph%20Configuration%20Spaces

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The Topology of Graph Configuration Spaces

• David G.C. Handron
• Carnegie Mellon University
• handron_at_andrew.cmu.edu

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The Topology of Graph Configuration Spaces
1. Configuration Spaces 2. Graphs 3. Topology 4.
Morse Theory 5. Results
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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

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Other Configuration Spaces

• Cyclic Configuration Spaces

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Other Configuration Spaces

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

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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.

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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

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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.

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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

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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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• 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,
  ...

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Subgraphs

• 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 graph
  GV' includes all the edges of G joining
  vertices in V'.

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Contractions
We can contract a graph with respect to an edge
... ...by identifying the vertices joined by
that edge
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Contractions, cont.
We can contract with respect to a set of
edges. Simply identify each pair of vertices.
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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.

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Partitions and Induced Graphs

• For each partition P of a graph G, there is a
  corresponding contraction
• contract all the edges in GP.

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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.

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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.

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Euler Characteristic

• The Euler characteristic of a polyhedron is
  commonly describes as v-ef.

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Euler Characteristic

• The Euler characteristic of a polyhedron is
  commonly describes as v-ef.
• Cube 8-1262

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Euler Characteristic

• The Euler characteristic of a polyhedron is
  commonly describes as v-ef.
• Cube 8-1262
• Tetrahedron 4-642

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Euler Characteristic

• The Euler characteristic of a polyhedron is
  commonly describes as v-ef.
• Cube 8-1262
• Tetrahedron 4-642
• Both topologically equivalent (homeomorphic) to a
  sphere.

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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

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Morse Theory

• A Morse function is a smooth function from a
  manifold M to R
• which has non-degenerate critical points.

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Non-Degenerate Critical Points

• A point p in M is a critical point if df0. In
  coordinates this means
  
  
  
• A critical point is non-degenerate if the Hessian
  matrix of second partial derivatives has nonzero
  determinate.

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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...

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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.