Configuration spaces on punctured manifolds

Abstract

The object here is to study the following question in the homotopy theory of configuration spaces of a general manifold $M$: When is the fibration $\mathbb F_{k+1}(M)\rightarrow\mathbb F_r(M)$, $r< k+1$, fiber homotopically trivial? The answer to this question for the special cases when $M$ is a sphere or euclidean space is given in [E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, Springer, New York, 2001]. The key to the solution of the problem for compact manifolds $M$ is the study of an associated question for the punctured manifold $M-q$, where $q$ is a point of $M$. The fact that $M-q$ admits a nonzero vector field plays a crucial role. Also required are investigations into the Lie algebra $\pi_*(\mathbb F_{k+1}(M))$, with special attention to the punctured case $\pi_*(\mathbb F_k(M-q))$. This includes the so-called Yang-Baxter equations in homotopy, taking into account the homotopy group elements of $M$ itself as well as the classical braid elements.