### Configuration spaces on punctured manifolds

DOI: http://dx.doi.org/10.12775/TMNA.2002.023

#### 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, < i> Geometry and Topology of Configuration Spaces< /i> , 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.

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, < i> Geometry and Topology of Configuration Spaces< /i> , 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.

#### Keywords

Configuration spaces; manifolds; fiberwise homotopy trivial

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