|
Given a manifold M, one can study the configuration space of n points on the manifold, which is the subspace of M^n in which two points cannot be in the same position. The study of these spaces from a homotopical perspective is of interest in very distinct areas such as algebraic topology or quantum field theory. However, even if we started with a simple manifold M, despite the apparent simplicity such configuration spaces are remarkably complicated; even the homology of these spaces is reasonably unknown, let alone their homotopy type. In this talk, I will give an introduction to the problem of understanding configuration spaces and present an algebraic model of these spaces using "Graph Complexes": Chain complexes whose elements are linear combinations of combinatorial graphs with a differential given by some combinatorial rule such as contraction of edges.
I will explain how these models give us new tools to approach some long standing problems and show how they allow us to answer fundamental questions about the dependence on the homotopy type of M.
|
