Geometry and Topology of Configuration Spaces [Paperback]

With applications in mind, this self-contained monograph provides a coherent and thorough treatment of the configuration spaces of Euclidean spaces and spheres, making the subject accessible to researchers and graduates with a minimal background in classical homotopy theory and algebraic topology.

I. The Homotopy Theory of Configuration Spaces.- I. Basic Fibrations.- 1 The Projection projk, r : $$\mathbb{F}_k (M) \to \mathbb{F}_r (M)$$.- 2 Relations to Homogeneous Spaces G/H.- 3 The Pull-back to On+1, r.- 4 $$\mathbb{F}_{k - 1,1} (\mathbb{R}^{n + 1} )$$ Restricted to On+1,r.- 5 Historical Remarks.- II. Configuration Space of ?n+1, n < 1.- 1 Filtration of $$\mathbb{F}_k (\mathbb{R}^{n + 1} )$$.- 2 Action of ?k.- 3 The Y-B Relations.- 4 Filtration of $$\pi _* (\mathbb{F}_k (\mathbb{R}^{n + 1} ))$$.- 5 When Are the Canonical Fibrations Trivial?.- 6 Historical Remarks.- III. Configuration Spaces of Sn+1, n < 1.- 1 Filtration of $$\pi _* (\mathbb{F}_{k + 1} (S^{n + 1} )),n > 1$$.- 2 Relation with $$\mathbb{F}_k (\mathbb{R}^{n + 1} )$$.- 3 The Groups ?n,?n+1, (n + 1) Odd.- 4 Symmetry Invariance of ?k+1.- 5 The Y-B Relations, (n + 1) Odd.- 6 The Dirac Class ?k+1.- 7 The Lie Algebra $$\pi _* (\mathbb{F}_r (S^{n + 1} ))$$, n < 1.- 8 Are The Canonical Fibrations Trivial?.- 9 Historical Remarks.- IV. The Two Dimensional Case.- 1 Asphericity of $$\mathbb{F}_k (\mathbb{R}^2 )$$.- 2 Generators for $$\pi _1 (\mathbb{F}_k (\mathbb{R}^2 ),q)$$.- 3 The Action of $$\mathbb{F}_k (\mathbb{R}^2 )$$.- 4 The Y-B Relations.- 5 A Presentation of $$\pi _1 (\mathbb{F}_k (\mathbb{R}^2 ),q)$$.- 6 When Are the Canonical Fibrations Trivial?.- 7 The Group $$\pi _1 (\mathbb{F}_{k + 1} (S^2 ),q^e )$$.- 8 Historical Remarks.- II. Homology and Cohomology of $$(\mathbb{F}_k (M)$$.- V. The Algebra $$H^* (\mathbb{F}_k (M);\mathbb{Z})$$.- 1 The Group $$H^* (\mathbb{F}_k (\mathbb{R}^{n + 1} );\mathbb{Z})$$.- 2 Invariance Under ?k.- 3 The Cohomological Y-B Relations.- 4 The Strulþ