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Weinberger S. — The topological classification of stratified spaces
Weinberger S. — The topological classification of stratified spaces



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Название: The topological classification of stratified spaces

Автор: Weinberger S.

Аннотация:

This book provides the theory for stratified spaces, along with important examples and applications, that is analogous to the surgery theory for manifolds. In the first expository account of this field, Weinberger provides topologists with a new way of looking at the classification theory of singular spaces with his original results.

Divided into three parts, the book begins with an overview of modern high-dimensional manifold theory. Rather than including complete proofs of all theorems, Weinberger demonstrates key constructions, gives convenient formulations, and shows the usefulness of the technology. Part II offers the parallel theory for stratified spaces. Here, the topological category is most completely developed using the methods of "controlled topology." Many examples illustrating the topological invariance and noninvariance of obstructions and characteristic classes are provided. Applications for embeddings and immersions of manifolds, for the geometry of group actions, for algebraic varieties, and for rigidity theorems are found in Part III.

This volume will be of interest to topologists, as well as mathematicians in other fields such as differential geometry, operator theory, and algebraic geometry.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1

Год издания: 1995

Количество страниц: 298

Добавлена в каталог: 01.11.2010

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Предметный указатель
$-\infty$ decoration      133
$K^{-i}$      124 156 166 187-88
$K^{BQ}_{0}$      127
$K_{0}$      21
$K_{1}$      24
$\alpha$-approximation theorem (of Chapman - Ferry)      161-62
$\Delta$ set      71
$\Delta$(X)      64-65 131 135 223-25
$\mathbb{Z}_{n}$-manifolds      64
$\mathbb{Z}_{n}$-surgery obstruction      64
$\pi - \pi$ theorem      54-58 140 144 145
$\pi - \pi$ theorem, and the existence of surgery theory      55-56
Acyclic representation      26
Adams conjecture      67
Alexander polynomial (and its relation to torsions)      27
Alexander trick (or lemma)      40 41 158
Algebraic Poincar$\acute{e}$ complex      75-79
Algebraic Poincar$\acute{e}$ complex, equivariant      223
Algebraic Poincar$\acute{e}$ complex, quadratic      76
Algebraic Poincar$\acute{e}$ complex, relation to surgery      77
Algebraic Poincar$\acute{e}$ complex, symmetric      76 83
Algebraic Poincar$\acute{e}$ complex, visible      76 84
Almost flat bundles      169-71
Almost flat manifolds      175-76
Almost flat manifolds, and the classification of lens spaces      109
Annulus conjecture      158-59
Application to the Novikov conjecture      255-56
Approximate fibrations      162-64 188-89 263-64
Aspherical space      9 100 102 103 242-45 254- 60 263-64
Assembly map      73-75 78 80 81 83-85 100-101 105 130 137 147-51 157 160 168 175 179 184 210 231 238
Atiyah - Bott numbers      see "Nonlinear similarities"
Atiyah - Patodi - Singer invariant      109 (see also "Rho invariant"
Atiyah - Patodi - Singer invariant, and the classification of lens spaces      109
Atiyah - Patodi - Singer invariant, homotopy invariance of      211
Atiyah - Patodi - Singer invariant, homotopy invariance of, up to rationale      212
Atiyah - Patodi - Singer invariant, homotopy noninvariance of      109
Atiyah - Patodi - Singer invariant, role in classification of manifolds      109
Atiyah - Singer classes      239
Atiyah - Singer Index theorem      65 169
Atiyah - Singer Index theorem, G-signature formula      103 104-5
Atiyah - Singer Index theorem, G-signature formula, and topological analogue      223-24
Attaching region (of a handle)      29
Base change      235-36
Bass - Heller - Swan formula      38 43 61 101 123 168 171
Bieberbach’s theorem      111 175 242
Block bundle      73-75 113 120 166
Block bundle, fibration obstruction theory      74
Borel conjecture      100-103 156 168
Borel conjecture, proof for flat and almost flat manifolds      175-76
Borel conjecture, the method of Farrell and Jones      258-59
Bounded propagation speed operators, algebra of      172
Boundedly constant fundamental group      154
Browder - Casson - Haefliger - Sullivan - Wall theorem      93 197;
Browder - Livesay invariant      108
Browder - Novikov theorem      2 91
Browder - Quinn theory      139-41
Browder’s splitting theorem      91
Bubble quotient      235
Cappeirs splitting theorem      92 94 102 208 254-55
CE maps, Edwards’ approximation theorem = Disjoint Disks theorem      178
CE maps, Siebenmann’s approximation theorem      161-62
Cell-like set      178
Characteristic class formulae, Capell - Shanesor projection formula      214-16
Characteristic class formulae, Cheeger’s formula      206
Characteristic class formulae, circle actions      229
Characteristic class formulaefor bundles (formula)      227
Characteristic classes, of algebraic varieties      215 (see also "Novikov conjecture " "Signature operator")
Characteristic classes, of arbitrary stratified spaces      131 135
Characteristic classes, of G-manifolds      223-25
Characteristic classes, of homotopy equivalences      61-69
Characteristic classes, of manifolds      64-65
Characteristic classes, of Witt spaces      209-10
Characteristic variety      63
Characteristic variety theorem      61-64 78 88
Classification of degree one normal maps      57
Classification of degree one normal maps, of embeddings      93-95 196-99 200-201
Classification of degree one normal maps, of fake CP’s      87-90
Classification of degree one normal maps, of fake lens spaces      104-7
Classification of degree one normal maps, of fake tori      98
Classification of degree one normal maps, of free involutions on the sphere      108
Classification of degree one normal maps, of groups which act freely on the sphere      110-11
Classification of degree one normal maps, of homology manifolds      178-80
Classification of degree one normal maps, of lens spaces      27
Classification of degree one normal maps, of semifree actions on the sphere      237-40
Classification of degree one normal maps, of semifree semilinear actions on the sphere      42
Coarse homology, cohomology, K-theory, and bounded propagation speed operators      171-73
Cocore      29
Collar detection theorem      35
Complex, bounded chain      155
Complex, controlled chain      155
Complex, finitely dominated      21
Complex, Intersection chain      204-10
Complex, of sheaves      206
Complex, Poincar$\acute{e}$ (and simple Poincare complex)      46
Control      148-49
Control, approximate      157 158
Control, bounded      150
Control, continuous at infinity      149 151
Controlled, h-cobordism      155
Controlled, h-cobordism theorem      155
Controlled, surgery obstruction      155
Core      29
Corona      172
Cosheaf      136
Cosheaf homology      136
Crigid      249
Crigid, spaces with crigid holinks      260-61
Cyclic cohomology      258 263
Cylindrical rigidity      244 263
Davis numerical surgery obstruction      240
Decomposition theorem of BBDG      214
Decomposition theorem of BBDG, for L-cosheaves      198 199 213 222
decorations      59
Descent and the Novikov conjecture      255-58
Destabilization theorem for structures      135-87
Desuspension theorem for homotopy lens spaces      104-7 225
Desuspension theorem for homotopy lens spaces, for odd order group actions on the sphere      225
Dictionary between surgery theory and operator theory      167-75
Double suspension theorem      98-99 178
Dress induction      111
Dynamical/foliated control method      258-60
Eilenberg swindle      20 34 122-23 155
Elementary collapse      22
Elementary expansion      22
Elementary matrix      24 34
Elliptic operator, K-homology class associated to      46
Embeddings, codimension 1      200
Embeddings, codimension 2      200-201
Embeddings, higher codimensions      93 197
Embeddings, pure embeddings      196
END      32
End, fundamental group of      32
End, neighborhood of      32
End, obstruction to completing      32
End, tame      32
End, theorem      33 164
Engulfing      35 162 189
Equivariant Bott periodicity and the desuspension theorem      225
Equivariant handlebody structures      220
Equivariant transversality      220
Eta invariant      see "Atiyah - Patodi - Singer invariant" "Rho
Euler characteristic      22
Existence of nonlinear similarities      see "Nonlinear similarities"
Exotic products      226
Exotic spheres      3 40
Exponential isomorphism      75
f      76
F/O      67-68
F/PL      65-67
F/Top      61-65
Farrell - Hsiang theorem on rigidity of flat and almost flat manifolds      175-
Farrell - Jones      258-59
Farrell fibering theorem      97 244
Ferry’s theorem on $\alpha$-dominations      161
Filtered category      115-16
Finiteness obstruction      19-22
Finiteness obstruction, Ferry’s geometric version      38 153
Finiteness obstruction, Vanishing for ANR’s (West’s theorem)      165
Finiteness obstruction, Wall’s original      21
Finiteness of ANR’s      19 42 165
Flattening      236
For surgery obstructions      240
Gap hypothesis      13 221-22 245-47
Generalized Poincar$\acute{e}$ conjecture      2 39
Geometric algebra      153-56
Geometric group      see "Geometric algebra"
Geometric module      see "Geometric algebra"
Geometric morphism      see "Geometric algebra"
Germ neighborhood      188-89
Germ of a MAF      163
Gromov - Lawson - Rosenberg conjecture      102 168-75
Grove - Peterson - Wu finiteness theorem      176-77
h-cobordism theorem      33-35
H-cobordism theorem, bounded and controlled      155
H-cobordism theorem, equivariant      219
H-cobordism theorem, PL stratified      127
H-cobordism theorem, proper      35-36
H-cobordism theorem, topological stratified      133 182-84
Haefliger knots      40 177
Handles      28 29
Handles, equivariant (theorem of Steinberger - West)      220
Handles, handlebody structures      29 30
Harmonic maps      253
Haspherical space      248
Hauptvermutung, counterexamples      122
Hauptvermutung, relation to Rochlin’s theorem      65-67
Hauptvermutung, triangulation      65-67 165-67
Higher      262-63
Hirzebruch signature formula      2 89 91
Hodge conjecture      211
Holink (=homotopy link)      119
Homologically trivial group actions      95 103
Homology collars, and converses to Smith theory      95 237-40
Homology collars, extension across      95
Homology manifolds      177-81
Homology manifolds, conjectures regarding their geometry      181
Homology manifolds, nonresolvable      179
Homology manifolds, surgery theory for      179
Homology propagation of group actions      95
Homotopically stratified space      119
Homotopy equivalence, bounded and controlled      151
Homotopy equivalence, simple      23
Homotopy link      119
Hughes sectioning theorem      see "Approximate fibrations"
Hyperbolic form      53
Hyperbolic group      258
Hyperbolic manifold      169 242 255-59
I-cosheaf      130
Immersions      53 199
INDEX      29 65 102-3 167-75
Induction theory      see "Dress induction"
Intermediate L-groups      240 (see also "Decorations")
Intersection form      53
Intersection homology      204-9
Intersection homology, and $L^{2}$ cohomology      206
Intersection homology, and small resolutions      213
Intersection homology, BBDG decomposition theorem      214
Intersection homology, definition      204
Intersection homology, Kunneth gormula      206
Intersection homology, L-classes and signature classes      206
Intersection homology, self duality      209
Intersection homology, sheaf description (Deligne construction)      208
Intersection number      30 52
Involution on Wh      39 127
Isovariant category      13 221-25
Isovariant surgery      221-25
J-homomorphism      48 67-68
Jacob’s ladder      31
Kervaire manifold      88-89
Kirby - Siebenmann obstruction      66-67 130
L -space      72
L-group = surgery obstruction group      see also "Surgery algebraic" "Poincar
L-group = surgery obstruction group, for compact manifolds      53
L-group = surgery obstruction group, for manifolds with boundary      54
L-group = surgery obstruction group, for noncompact manifolds      59-60
L-group = surgery obstruction group, for stratified spaces      129 133-34 139-40
L-group = surgery obstruction group, of finite groups      105 105-7 224 240
L-spectrum      73
Lens spaces      27
Lens spaces, classification of      27 104-8 231
Levine’s unknotting criterion      40
Lichnerowicz’s theorem      102
Lipschitz category      125 160 256
Local flatness, homotopy characterization of      196
Local holink      119
localization      62
Localization theorem of Ranicki      78
Madsen - Thomas - Wall theorem      110-11
Main theorem of controlled topology      156
Main theorem of controlled topology, applications of      161-67
Main theorem of controlled topology, for K-theory      156
Main theorem of controlled topology, for L-theory      157
Main theorem of controlled topology, parallel in operator theory      167-75
Manifold approximate fibration (MAF)      162
Manifold stratified group action      219
Manifold stratified space      119
Mapping cylinder neighborhood      119 165 183
Mapping torus      38
Mather’s trick      38
Mayer - Viet sequences      102 136-37 255
Mazur’s (stable classification) theorem      38
Milnor duality formula      39
Milnor manifold      88-89
Mittag - Leffler condition      31
Modules, bounded and controlled, geometric      153-55
Modules, bounded and controlled, projective      20
NIL      38 101 123 149 244
Nonlinear similarities      12 124-25 230 231-33
Normal invariant      51
Normally smooth map      139
Novikov additivity      105
Novikov additivity, via Witt space cobordism      211-12
Novikov conjecture      99-103 170-71 255- 58
Novikov conjecture, equivariant      224
Novikov conjecture, Gromov - Lawson - Rosenberg conjecture      168-75
Novikov conjecture, index theory methods      102 170-71
Novikov conjecture, splitting methods      102
Novikov conjecture, stratified      252
Novikov conjecture, the method of descent      255-58
Numerical formulae for $K_{0}$ obstructions      22
Odd order group theorem      see "Nonlinear similarities"
Operator      64-65
Orientation true map      82
Periodicity of L-groups      54 77 197 226
Petrie conjecture      203
PL stratified space      117
PL stratified structure sequence      130 141-47
PL weakly stratified space      117
Plumbing      55-56
Poincar$\acute{e}$, complex      46
Poincar$\acute{e}$, complex, geometric and algebraic      76-77
Poincar$\acute{e}$, conjecture      3 39
Poincar$\acute{e}$, duality      46
Poincar$\acute{e}$, embedding      9 93 197
Poincar$\acute{e}$, surgery      144
Positive scalar curvature      see "Gromov - Lawson - Rosenberg conjecture"
Product formula, for surgery obstructions      77-78
Product formula, for torsion      37
Proper actions, Novikov conjecture for      261
Proper h-cobordism theorem      35-36
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