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Brown K.S. — Cohomology of Groups
Brown K.S. — Cohomology of Groups



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Название: Cohomology of Groups

Автор: Brown K.S.

Аннотация:

As a second year graduate textbook, Cohomology of Groups introduces students to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology. The basics of the subject are given (along with exercises) before the author discusses more specialized topics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1996

Количество страниц: 324

Добавлена в каталог: 10.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\delta$-functor      78
$\partial$-functor      75
Abelianization      36
Abeliar groups (calculation of homology of)      121ff
Abutment      163
Acyclic chain complex      5
Acyclic chain complex, contractibility of      24 30 132 274
Acyclic cover      168 265
Acyclic models theorem      25
Acyclic module (with respect to a homology theory)      72
Acyclic space      15
Additive category      25
Additive functor      33
Admissible $\Gamma$-complex      258
Admissible acyclic chain complex      129
Admissible exact sequence      129
Admissible injection      129
Admissible product      118
Alexander — Whitney map      108 286
Amalgamation (or amalgamated free product)      49ff
Amalgamation (or amalgamated free product), tree associated to      53
Anti-commutati vity      111 118
Arithmetic groups      217 218
Arithmetic groups, Euler characteristic of      254ff
Aspherical space      1 15
Associated graded module      162
Augmentation ideal and map      12
Augmentation-preserving chain map      23 132
Automorphism group      102
Automorphism group, outer      104
Bar resolution      19
Bernoulli numbers      255
Binary icosahedral group      47 155
Binary octahedral group      155
Binary tetrahedral group      155
Boundaries      4
Boundary operator      4
Cap product      112—113 141
Cartan — Leray spectral sequence      173
Chain complex      4
Chain map      5
Chevalley group      254
Co-compact      217
Co-effaceable      73
Co-extension of scalars      63
Co-free module      66
Co-induced module      67
Co-invariants      34
Coboundaries      4
Cocycles      4
Coefficient system      167
Cohomological dimension      184
Cohomological dimension, virtual      226
Cohomological functor      78
Cohomologically trivial module      148 287
Cohomology of a cochain complex      4
Cohomology of a group      57 134 170 277
Cohomology of a group, equivariant      172 180ff 281ff
Cohomology with compact supports      209
Combinatorial path      42
Commutator pairing      97
Compact supports      209
Complete diagonal approximation      138 275
Complete resolution      132 273
Complete tensor product      137
Composition product      115 143
Congruence subgroup      40
Conjugation action      48 79 80
Connecting homomorphism      6 71 75
Connecting homomorphism, interpretation in terms of group extensions      95
Contractible, contracting homotopy      5 (see also “Acyclic chain complex”)
Convergence of a spectral sequence      163
Corestriction map      80 136
Cross product      109
Crossed homomorphism      45 88
Crossed module      102
Cup product      109 130ff 278
Cup product in cyclic groups      114
Cycles      4
Cyclic group (resolution, homology, etc.)      16 21 35 58 114
Deck transformation      31
Degree      4
Derivation      45 88
Derivation, principal      60 89
Diagonal action      55 56
Diagonal approximation (or map)      108 138 275 286
Differential      4
Dihedral group      98
Dimension, Cohomological      184
Dimension, geometric      185 205
Dimension, homological      204
Dimension, projective      152 184 287
Dimension, virtual cohomological      226
Dimension-shifting      74 136
Direct limits and homology      121 195
Discrete subgroups of Lie groups      38ff 217ff
Divided polynomial algebra      119 125
Divided powers      124
Divisible module      66
Double complex      164
Double coset formula      69 82
Dual      28 133 145
Duality group      221
Duality group, Poincare      222
Duality group, virtual      229
Duality pairing      145
Duality theorem      28 144ff
Dualizing module      221 276
Effaceable      72
Eilenberg trick      187—188
Eilenberg — Ganea theorem      205
Eilenberg — MacLane complex (or space)      15 40
Eilenberg — MacLane complex (or space), existence of      18 19 205
Elementary abelian p-group      156 267 271
ends      223
Equivariant Euler characteristic      249—250
Equivariant homology and cohomology      172 180ff 281ff
Euler characteristic      164 243ff
Euler characteristic of a group      247 249
Evaluation maps and pairings      9 113 145 234
Exact functor      8
Ext      60
Extension (group)      47 86ff 171 248 257 281
Extension of scalars      62
Exterior algebra, exterior power      97 119 121
F-isomorphism      286
Factor set      91
Farrell cohomology      277ff
Filtration      162
Finite chain complex, resolution      243
Finite homological type      246
Finitely dominated space      200
Finitely presented module      193
Finiteness conditions      184ff
Finiteness conditions, FL      199
Finiteness conditions, FP, $FP_n$, $FP_{\infty}$      193 195 197
Finiteness conditions, VFL      226
Finiteness conditions, VFP      226
Finiteness conditions, WFL      226
Five-term exact sequence      47 171 175
Fixed-point theorems      21 181 246 261
Fixed-point-free representation      154
Flat module      29 56
Free derivative      45 90
Free G-set, G-complex      13
Free group (homology, resolution, etc.)      16—17 37 247
Function complex      5
Fundamental class      145 204
Fundamental group of a graph of groups      179
Fundamental lemma of homological algebra      22 130
G-complex (or G-CW-complex)      14
G-module      13
G-set      13
Gauss — Bonnet measure      253
General linear group      38
Generalized quaternion group      98 155
Geometric dimension      185 205
Gottlieb’s theorem      252
Graded module      4
Graph      52
Graph of groups      179
Group extension      47 86ff 171 248 257 281
Group ring (or algebra)      12 14
Hatton — Stallings rank      233
Hirsch number      186
HNN extension      179
Hochschild — Serre spectral sequence      171 281
Homological dimension      204
Homological functor      75
Homology of a chain complex      4
Homology of a group      1 35 56 135 168 172
Homology of a group, equivariant      172 180ff 281
Homotopy, homotopy equivalence      5ff 29
Hopf’s theorems      1 41
Hopf’s theorems, explicit formula for      46
Horn      56
Idempotent      27 234 238
Induced module      60 67ff
Injective module      26 65ff
Invariant cohomology class      84
Invariants      27
Invertible ideal      27
Irdecomposable      231 235 236
K(G, l)-complex      15 (see also “Eilenberg — MacLane complex”)
Knol groups      212
Krull dimension of H*(G)      159
Kummer’s criterion      270
Kunneth formula      7 109 120
Lefschetz fixed-point theorem      21 246 261
Length of a resolution      11
Lie groups (discrete subgroups of)      38ff 217ff
Long exact homology sequence      6 71 75
Lyndon’s theorem      37 44 185 228
Mapping cone      6
Mayer — Vietoris sequence      51 74 178ff
Nakayama’s Lemma      150 236
Nerve of a covering      166
Nilpotent group (cohomological dimension of)      186 213
Norm element      20 58
Norm map      58
Normalized bar resolution      19
One-relator group      37 44 185 228
Ordered simplicial complex      227
Outer automorphism group      104
p-groups with a cyclic subgroup of index p      97ff
p-groups with a unique subgroup of order p      99
p-groups with every abelian normal subgroup cyclic      101
p-groups, calculation of homology via      84
Partially ordered set (topological concepts applied to)      261 262 291
Perfect group      46 96 198
Periodic cohomology and resolution      20 133 153 158 180 288
Permutation module      13
Poincare duality group      222
Pontryagin product      117
Primary decomposition of homology groups      84 141
Principal derivation      60 89
Product, cap      112—113 141
Product, composition      115 143
Product, cross      109
Product, cup      109 130ff 278
Product, of ordered simplicial complexes      227
Product, shuffle      118
Product, tensor      7 10 55 107 137
Projective dimension      152 184 287
Projective module      21 26ff 56
Projective module over a group ring      27 149 152 201
Projective module over a local ring      235
Projective module, rank of      230ff
Projective module, stably free      201
Proper $\Gamma$-complex      226
Proper action      39
Pull-back      94
Quaternion group      98 155
Quillen’s theorem      159
Rank of a finitely generated abelian group      242
Rank of a nilpotent group      186
Rank of a projective module      230fF
Reduced homology      211
Regular cover      31
Regular cover, spectral sequence of      173
Relation module      43 44 90 198
Relative homological algebra      25.129ff
Relative injective module      129
Relative injective resolution      131
Representable factor      25
Resolution      10
Resolution, bar      19
Resolution, complete      132 273
Resolution, finite      199
Resolution, of finite type      193
Resolution, periodic      20 133 153
Resolution, standard      18
Resolution, uniqueness of      24
Resolution, via topology      14ff
Restriction map      80 136
Restriction of scalars      62 69
Rim’s theorem      152 287
Schanuel’s Lemma      192 193
Semi-direct product      87
Serre’s theorem      190
Shapiro’s Lemma      73 136
Shu file product      118
Simple module      149
Simplicial complex (ordered)      227
Simplicial complex (ordered), associated to a partially ordered set      261—262
Simplicial product      227
Solomon — Tits theorem      270
Spec A      235
Special linear group      39 157ff 213ff 229 255
Spectral sequence      162ff
Split extension      87
Stably free      201
Stallings — Swan theorem      185 223
Standard resolution      18
Strict anti-commutativity      118
Strict upper triangular group      38 185 213
Suspension      5
Swan’s theorem      240—241
Sylow subgroups (calculation of homology via)      84
Symmetric group      48 85 114
Symplectic group      255 269
Tate homology and cohomology      134ff 170 180ff
Tensor product      7 10 55 107 137
Tor      60
Trace      231—232
Transfer map      80 83ff
TREE      53
Type FL, FP, etc.      see “Finiteness conditions”
Universal central extension      96
Universal coefficient theorem      8 127 170 198 202
Upper triangular group      38 185 213
Virtual notions      225fT
Weak equivalence      5ff 29
Yoneda’s Lemma      25
Zariski topology      236—237
Zeta function      254
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