The Topology of Graph Configuration Spaces

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

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

3. 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

4. Cyclic Configuration Spaces: Other Configuration Spaces

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

6. 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

7. 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

8. 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

9. 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

10. Cyclic configuration spaces correspond to graphs that form a loop • Path configuration spaces correspond to graphs that form a continuous path

11. 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

12. V(G)={v1, v2, v3, v4, v5, v6} • E(G)={{v1, v3}, {v2, v3},...}

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

14. 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

15. Contractions We can contract a graph with respect to an edge ... ...by identifying the vertices joined by that edge

16. Contractions, cont. We can contract with respect to a set of edges. Simply identify each pair of vertices.

17. 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

18. For each partition P of a graph G, there is a corresponding contraction: contract all the edges in G[P]. Partitions and Induced Graphs

19. 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.

20. 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

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

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

23. 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

24. 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.

25. 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

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

27. 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.

28. 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

29. (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