ISBN: 3540666699
TITLE: Geometry and Topology of Configuration Spaces
AUTHOR: Fadell, Edward R.; Husseini, Sufian Y.
TOC:

Preface vii
Part I. The Homotopy Theory of Configuration Spaces
Introduction 3
I. Basic Fibrations 5
1 The Projection proj_k,r: F_k(M) -> F_r(M) 6
2 Relations to Homogeneous Spaces G/H 9
3 The Pull-back to O_n+1,r 10
4 F_k-1,1 (R^n+1) Restricted to O_n+1,r 11
5 Historical Remarks 12
II. Configuration Space of R^n+1, n>1 13
1 Filtration of F_k(R^n+1) 14
2 Action of Sigma_k 18
3 The Y-B Relations 20
4 Filtration of pi_*(F_k(R^n+1)) 21
5 When Are the Canonical Fibrations Trivial? 24
6 Historical Remarks 28
III. Configuration Spaces of S^n+1, n>1 29
1 Filtration of pi_*(F_k+1(S^n+1)), n>1 31
2 Relation with F_k(R^n+1) 33
3 The Groups pi_n, pi_n+1, (n +1) Odd 37
4 Symmetry Invariance of delta_k+1 38
5 The Y-B Relations, (n +1) Odd 41
6 The Dirac Class Delta_k+1 45
7 The Lie Algebra pi_*(F_r(S^n+1)), n>1 49
8 Are The Canonical Fibrations Trivial? 52
9 Historical Remarks 55
IV. The Two Dimensional Case 57
1 Asphericity of F_k(R^2) 58
2 Generators for pi_1(F_k(R^2), q) 59
3 The Action of Sigma_k on F_k(R^2) 63
4 The Y-B Relations 69
5 A Presentation of pi_1(F_k(R^2), q) 73
6 When Are the Canonical Fibrations Trivial? 81
7 The Group pi_1(F_k+1(S^2), q^epsilon) 84
8 Historical Remarks 89
Part II. Homology and Cohomology of F_k(M)
Introduction 93
V. The Algebra H*(F_k(M); Z) 95
1 The Group H *(F_k(R^n+1); Z) 96
2 Invariance Under Sigma_k 100
3 The Cohomological Y-B Relations 101
4 The Structure of H*(F_k(R^n+1)) 103
5 The group H^n(F_k+1(S^n+1)) 105
6 H*(F_k+1(S^n+1)) as an H*(F_3(S^n+1))-Module 111
7 The Algebra H*(F_k+1(S^n+1)), n+1 Even 112
8 The Algebra H*(F_k+1(S^n+1)), n+1 Odd 113
9 Historical Remarks 115
VI. Cellular Models 117
1 A Model for F_3(R^n+1) 121
2 The Twisted-Product Structure on H_*(F_k-r,r) 126
3 Perturbation and Affine Maps 127
4 An Illustrated Example 133
5 Multispherical Cycles 134
6 Twisted Products in H_*(F_k+1(S^n+1)), n+1 Odd 137
7 Twisted Products in H_*(F_k+1(S^n+1)), n+1 Even 141
8 The Cellular Structure of F_k(R^n+1), n>1 143
9 The Cellular Structure of F_k+1(S^n+1) 147
10 The Cellular Structure of F_k(R^2) 149
11 The Cellular Structure for F_k+1(S^2) 151
12 Historical Remarks 151
VII. Cellular Chain Models 153
1 Cellular Chain Coalgebras 153
2 The Coalgebra of F_k(R^n+1) 155
3 The Coalgebra of F_k+1(S^n+1), (n+1) Odd 157
4 The Coalgebra C _*(Y), Y ~ F_k+1(S^n+1), (n+1) Even 161
Part III. Homology and Cohomology of Loop Spaces
Introduction 167
VIII. The Algebra H_*(Omega F_k(M))) 171
1 The Coalgebra H_*(Omega F_k-r,r) 172
2 The Primitives in H_*(Omega F_k-r,r) 174
3 The Hopf Algebra H_*(Omega F_k-r,r) 177
4 The Algebra H_*(Omega F_k+1(S^n+1)), (n+1) Odd 180
5 The Algebra H_*(Omega F_k+1(S^n+1)), (n+1) Even 182
6 Historical Remarks 185
IX. RPT-Constructions 187
1 RPT-Models for Omega(X) 188
2 Homotopy Inverse for M(X) 194
3 An RPT-Model for Lambda(X) 196
4 An RPT-Model for Lambda_sigma(X) 201
5 A Cellular Spectral Sequence 204
6 Historical Remarks 206
X. Cellular Chain Algebra Models 207
1 The Adams-Hilton Algebra 207
2 An RPT-model for Omega(Pi^m _i=1 S_i) 210
3 C_*(M(X_k-r,r)), X_k-r,r ~ F_k-r,r 215
4 C_*(M(Y_k+1), Y_k+1 ~ F_k+1(S^n+1), (n+1) Odd 217
5 C_*(M(Y)), Y ~ F_k+1(S^n+1), (n+1) Even 219
6 The Eilenberg-Moore Spectral Sequence of Lambda(M) 220
XI. The Serre Spectral Sequence 225
1 The Case of F_k-r,r, n>1 225
2 The Case of F_k+1(S^(n+1)), (n+1) Odd 234
3 The Case of F_k+1(S^n+1), (n+1) Even 240
XII. Computation of H_*(Lambda(M)) 243
1 Splitting of H_*(Lambda F_k(R^n+1); Z_2) 244
2 Coproducts in H_*(Lambda F_3(R^n+1); Z_2) 254
3 The Growth of H_*(Lambda (R^n+1 _k-1) 257
4 The Growth of H_*(Lambda F_k(R^n+1)) 261
5 Cup Length in H_*(Lambda (F_k-r,r); Z_2) 265
6 Historical Remarks 267
XIII. Gamma-Category and Ends 269
1 Relative Category 270
2 Ends in W^1,2_T(R^3(n+1)) 274
3 Gamma-category 283
4 Strongly Admissible Sets 284
5 Historical Notes 290
XIV. Problems of k-body Type 293
1 Analytic Ends 294
2 The First Example 296
3 The Second Example 299
4 Historical Remarks 303
References 305
Index 311
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