The configuration space of a manifold provides the appropriate setting for problems not only in topology but also in other areas such as nonlinear analysis and algebra. With applications in mind, the aim of this monograph is to provide a coherent and thorough treatment of the configuration spaces of Eulidean spaces and spheres which makes the subject accessible to researchers and graduate students with a minimal background in classical homotopy theory and algebraic topology. The treatment regards the homotopy relations of Yang-Baxter type as being fundamental. It also includes a novel and geometric presentation of the classical pure braid group; the cellular structure of these configuration spaces which leads to a cellular model for the associated based and free loop spaces; the homology and cohomology of based and free loop spaces; and an illustration of how to apply the latter to the study of Hamiltonian systems of k-body type.

Klappentext

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.

Inhalt

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$$.- 2 Relation with $$\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})$$.- 2 Invariance Under ?k.- 3 The Cohomological Y-B Relations.- 4 The Structure of $$H^ (\mathbb{F}_k (\mathbb{R}^{n + 1} ))$$.- 5 The group $$H^* (\mathbb{F}{k + 1} (S^{n + 1} ))$$.- 6 $$H^ (\mathbb{F}_{k + 1} (S^{n + 1} ))$$ as an $$H^ (\mathbb{F}3 (S^{n + 1} ))$$-Module.- 7 The Algebra $$H^* (\mathbb{F}{k + 1} (S^{n + 1} ))$$, n + 1 Even.- 8 The Algebra $$H^ (\mathbb{F}{k + 1} (S^{n + 1} ))$$, n + 1 Odd.- 9 Historical Remarks.- VI. Cellular Models.- 1 A Model for $$\mathbb{F}_3 (\mathbb{R}^{n + 1} )$$.- 2 The Twisted-Product Structure on $$H (\mathbb{F}{k - r,r} )$$.- 3 Perturbation and Affine Maps.- 4 An Illustrated Example.- 5 Multispherical Cycles.- 6 Twisted Products in $$H (\mathbb{F}{k + 1} (S^{n + 1} ))$$, n + 1 Odd.- 7 Twisted Products in $$H (\mathbb{F}{k + 1} (S^{n + 1} ))$$, n + 1 Even.- 8 The Cellular Structure of $$\mathbb{F}_k (\mathbb{R}^{n + 1} )$$, n < 1.- 9 The Cellular Structure of $$\mathbb{F}{k + 1} (S^{n + 1} )$$.- 10 The Cellular Structure of $$\mathbb{F}k (\mathbb{R}^2 )$$.- 11 The Cellular Structure for $$\mathbb{F}{k + 1} (S^2 )$$.- 12 Historical Remarks.- VII. Cellular Chain Models.- 1 Cellular Chain Coalgebras.- 2 The Coalgebra of $$\mathbb{F}k (\mathbb{R}^{n + 1} )$$.- 3 The Coalgebra of $$\mathbb{F}{k + 1} (S^{n + 1} )$$, (n+1) Odd.- 4 The Coalgebra C(Y), $$Y, \simeq \mathbb{F}{k + 1} (S^{n + 1} )$$, (n + 1) Even.- III. Homology and Cohomology of Loop Spaces.- VIII. The Algebra $$H (\Omega \mathbb{F}k (M)))$$.- 1 The Coalgebra $$H (\Omega \mathbb{F}{k - r,r} )$$.- 2 The Primitives in $$H (\Omega \mathbb{F}{k - r,r} )$$.- 3 The Hopf Algebra $$H (\Omega \mathbb{F}{k - r,r} )$$.- 4 The Algebra $$H (\Omega \mathbb{F}{k + 1} (S^{n + 1} ))$$, (n + 1) Odd.- 5 The Algebra $$H (\Omega \mathbb{F}{k + 1} (S^{n + 1} ))$$, (n + 1) Even.- 6 Historical Remarks.- IX. RPT-Constructions.- 1 RPT-Models for ?(X).- 2 Homotopy Inverse for M(X).- 3 An RPT-Model for ?(X).- 4 An RPT-Model for ??(X).- 5 A Cellular Spectral Sequence.- 6 Historical Remarks.- X. Cellular Chain Algebra Models.- 1 The Adams-Hilton Algebra.- 2 An RPT-model for $$\Omega (\prod\nolimits{i = 1}^m {Si } )$$.- 3 $$C (M(X{k - r,r} )),,X{k - r,r} \simeq \mathbb{F}{k - r,r} $$.- 4 $$C (M(Y{k + 1} )),,Y{k + 1} \simeq \mathbb{F}{k + 1} (S^{n + 1} )$$, (n + 1) Odd.- 5 $$C (M(Y)),,Y \simeq \mathbb{F}{k + 1} (S^{n + 1} )$$, (n + 1) Even.- 6 The Eilenberg-Moore Spectral Sequence of ?(M).- XI. The Serre Spectral Sequence.- 1 The Case of $$\mathbb{F}{k - r,r} $$, n < 1.- 2 The Case of $$\mathbb{F}{k + 1} (S^{(n + 1)} )$$, (n + 1) Odd.- 3 The Case of $$\mathbb{F}{k + 1} (S^{n + 1} )$$, (n + 1) Even.- XII. Computation of H(?(M)).- 1 Splitting of $$H_ (\Lambda \mathbb{F}k (\mathbb{R}^{n + 1} );\mathbb{Z}_2 )$$.- 2 Coproducts in $$H (\Lambda \mathbb{F}3 (\mathbb{R}^{n + 1} );\mathbb{Z}_2 )$$.- 3 The Growth of $$H (\Lambda (\mathbb{R}{k - 1}^{n + 1} )$$.- 4 The Growth of $$H (\Lambda \mathbb{F}k (\mathbb{R}^{n + 1} ))$$.- 5 Cup Length in $$H (\Lambda (\mathbb{F}_{k - r,r} );\mathbb{Z}_2 )$$.- 6 Historical Remarks.- XIII. ?-Category and Ends.- 1 Relative Category.- 2 Ends in $$W_T^{1,2} (\mathbb{R}^{3(n + 1)} )$$.- 3 ?-category.- 4 Strongly Admissible Sets.- 5 Historical Notes.- XIV. Problems of k-body Type.- 1 Analytic Ends.- 2 The First Example.- 3 The Second Example.- 4 Historical Remarks.- References.