Lectures on the Geometry and Topology of Configuration Spaces

1. Lectures on the Geometry and Topology of Configuration Spaces SadokKallel American University of Sharjah (UAE)

2. Introduction If you type ``configuration spaces" on the archives, you get nearly 300 references. You get many more on Google. Configuration Spaces have been used in Topology at least since the early 50’s (particularly as applied to embedding problems). There are configuration spaces in geometry and topology, with applications to physics and various other parts of mathematics. This will be our focus. There are configuration spaces of mechanical systems.

3. PART 1:CONSTRUCTIONS AND EXAMPLES Theorems change. Examples Remain. Marshall Hall

4. The configuration space (or C-Space) is the set of all admissible positions of the object. CONFIGURATION SPACES IN MECHANICS

5. The configuration space of all two legged arms with revolute joints is

6. Configuration Space Obstacle

8. Planar Linkages

11. Configuration Spaces in Topology • . • . • .

12. The topology of F(X,n) is very rich, even for low values of n

13. Examples

16. Show that

17. Unordered or Unlabeled Configurations

20. B(C,3) and the trefoil knot

24. Fadell-NeuwirthFibrations

28. Proof of the Fadell-Neuwirth Bundle Theorem

30. Applications

33. Configurations and Braid Groups

37. The Dirac Problem

44. PART 2: APPLICATIONS

45. Configurations on Graphs

46. 2 AGV’s on T-graph

47. Other computations How do weunderstand the homotopy type of F(G,2) in general ?