Fadell / Husseini Geometry and Topology of Configuration Spaces

2001
ISBN: 978-3-642-56446-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 313 Seiten, eBook

Reihe: Springer Monographs in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Fadell, Edward R.

Husseini, Sufian Y.

Inhaltsverzeichnis

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 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 $$\Omega (\prod\nolimits_{i = 1}^m {S_i } )$$.- 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.