Cohomology of groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | Undetermined |
| Published: |
New York [u.a.]
Springer
[1997]
|
| Edition: | 2. corr. print., [Nachdr.] |
| Series: | Graduate texts in mathematics
87 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 306 S. |
| ISBN: | 0387906886 3540906886 |
Staff View
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| 245 | 1 | 0 | |a Cohomology of groups |c Kenneth S. Brown |
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Record in the Search Index
| _version_ | 1804135041208418304 |
|---|---|
| adam_text | Contents
Introduction
1
CHAPTER I
Some Homological Algebra
4
0.
Review of Chain Complexes
4
1.
Free Resolutions
10
2.
Group Rings
12
3.
G-Modules
13
4.
Resolutions of
Z
Over
Zť?
via Topology
14
5.
The Standard Resolution
18
6.
Periodic Resolutions via Free Actions on Spheres
20
7.
Uniqueness of Resolutions
21
8.
Projective
Modules
26
Appendix. Review of Regular Coverings
31
CHAPTER II
The Homology of a Group
33
1.
Generalities
33
2.
Co-invariants
34
3.
The Definition of H+G
35
4.
Topological Interpretation
36
5. Hopf
s
Theorems
41
6.
Functoriality
48
7.
The Homology of Amalgamated Free Products
49
Appendix. Trees and Amalgamations
52
CHAPTER III
Homology and Cohomoiogy with Coefficients
55
0.
Preliminaries on ®G and Homo
55
1.
Definition of Ht(G, M) and H*(G, M)
56
vii
viii Contents
2.
Tor and Ext
60
3.
Extension and Co-extension of
Sealars 62
4.
Injective Modules
65
5.
Induced and Co-induced Modules
67
6. #„,
and H* as Functors of the Coefficient Module
71
7.
Dimension Shifting
74
8.
Я»
and
Я*
as Functors of Two Variables
78
9.
The Transfer Map
80
10.
Applications of the Transfer
83
CHAPTER IV
Low Dimensional Cohomology and Group Extensions
86
1.
Introduction
86
2.
Split Extensions
87
3.
The Classification of Extensions with Abelian Kernel
91
4.
Application
:
/»-Groups with a Cyclic Subgroup of Index
ρ
97
5.
Crossed Modules and
Я3
(Sketch)
102
6.
Extensions With Non-Abelian Kernel (Sketch)
104
CHAPTER V
Products
107
1.
The Tensor Product of Resolutions
107
2.
Cross-products
108
3.
Cup and Cap Products
109
4.
Composition Products
114
5.
The Pontryagin Product
117
6.
Application
:
Calculation of the Homology of an Abelian Group
121
CHAPTER VI
Cohomology Theory of Finite Groups
128
1.
Introduction
128
2.
Relative Homological Algebra
129
3.
Complete Resolutions
131
4.
Definition of H*
134
5.
Properties of
Й*
136
6.
Composition Products
142
7.
A Duality Theorem
144
8.
Cohomologically Trivial Modules
148
9.
Groups with Periodic Cohomology
153
CHAPTER
VII
Equivariant Homology and Spectral Sequences
161
1.
Introduction
161
2.
The Spectral Sequence of a Filtered Complex
161
Contents
¡γ
ІЛ.
3. Double
Complexes
164
4.
Example: The Homology
of
a Union
166
5.
Homology
of
a Group
with
Coefficients in a Chain
Complex
168
6.
Example: The Hochschild-Serre Spectral
Sequence
171
7.
Equivariant Homology
172
8.
Computation
oíd1
175
9.
Example: Amalgamations
178
10.
Equivariant
Tate Cohomology
180
CHAPTER
VIII
Finiteness Conditions
183
1.
Introduction
183
2.
Cohomological Dimension
184
3.
Serre s Theorem
190
4.
Resolutions of Finite Type
191
5.
Groups of
Type FPn
197
6.
Groups of
Type
FP
and
Л.
199
7.
Topological
Interpretation
205
8.
Further
Topological
Results
210
9.
Further Examples
213
10.
Duality Groups
219
11.
Virtual Notions
225
CHAPTER IX
Euler
Characteristics
230
1.
Ranks of
Projective
Modules
:
Introduction
230
2.
The Hattori-Stallings Rank
231
3.
Ranks Over Commutative Rings
235
4.
Ranks Over Group Rings; Swan s Theorem
239
5.
Consequences of Swan s Theorem
242
6.
Euler
Characteristics of Groups: The Torsion-Free Case
246
7.
Extension to Groups with Torsion
249
8.
Euler
Characteristics and Number Theory
253
9.
Integrality Properties of
χ(Γ)
257
10.
Proof of Theorem
9.3 ;
Finite Group Actions
258
11.
The Fractional Part of
χ(Τ)
261
12.
Acyclic Covers
;
Proof of Lemma
11.2 265
13.
The/»-Fractional
Partorir)
266
14.
A Formula for
%T(si)
270
CHAPTER X
Farrell Cohomology Theory
273
1.
Introduction
. 273
2.
Complete Resolutions
273
3.
Definition and Properties of
Я*(Г)
277
Contents
4. Equivariant Farrell Cohomology 28
1
5. Cohomologically Trivial Modules 287
6.
Groups with Periodic
Cohomology 288
7.
Я*(Г)
and the Ordered Set of Finite Subgroups of
Γ
291
References
295
Notation Index
301
Index
303
|
| adam_txt |
Contents
Introduction
1
CHAPTER I
Some Homological Algebra
4
0.
Review of Chain Complexes
4
1.
Free Resolutions
10
2.
Group Rings
12
3.
G-Modules
13
4.
Resolutions of
Z
Over
Zť?
via Topology
14
5.
The Standard Resolution
18
6.
Periodic Resolutions via Free Actions on Spheres
20
7.
Uniqueness of Resolutions
21
8.
Projective
Modules
26
Appendix. Review of Regular Coverings
31
CHAPTER II
The Homology of a Group
33
1.
Generalities
33
2.
Co-invariants
34
3.
The Definition of H+G
35
4.
Topological Interpretation
36
5. Hopf
s
Theorems
41
6.
Functoriality
48
7.
The Homology of Amalgamated Free Products
49
Appendix. Trees and Amalgamations
52
CHAPTER III
Homology and Cohomoiogy with Coefficients
55
0.
Preliminaries on ®G and Homo
55
1.
Definition of Ht(G, M) and H*(G, M)
56
vii
viii Contents
2.
Tor and Ext
60
3.
Extension and Co-extension of
Sealars 62
4.
Injective Modules
65
5.
Induced and Co-induced Modules
67
6. #„,
and H* as Functors of the Coefficient Module
71
7.
Dimension Shifting
74
8.
Я»
and
Я*
as Functors of Two Variables
78
9.
The Transfer Map
80
10.
Applications of the Transfer
83
CHAPTER IV
Low Dimensional Cohomology and Group Extensions
86
1.
Introduction
86
2.
Split Extensions
87
3.
The Classification of Extensions with Abelian Kernel
91
4.
Application
:
/»-Groups with a Cyclic Subgroup of Index
ρ
97
5.
Crossed Modules and
Я3
(Sketch)
102
6.
Extensions With Non-Abelian Kernel (Sketch)
104
CHAPTER V
Products
107
1.
The Tensor Product of Resolutions
107
2.
Cross-products
108
3.
Cup and Cap Products
109
4.
Composition Products
114
5.
The Pontryagin Product
117
6.
Application
:
Calculation of the Homology of an Abelian Group
121
CHAPTER VI
Cohomology Theory of Finite Groups
128
1.
Introduction
128
2.
Relative Homological Algebra
129
3.
Complete Resolutions
131
4.
Definition of H*
134
5.
Properties of
Й*
136
6.
Composition Products
142
7.
A Duality Theorem
144
8.
Cohomologically Trivial Modules
148
9.
Groups with Periodic Cohomology
153
CHAPTER
VII
Equivariant Homology and Spectral Sequences
161
1.
Introduction
161
2.
The Spectral Sequence of a Filtered Complex
161
Contents
¡γ
ІЛ.
3. Double
Complexes
164
4.
Example: The Homology
of
a Union
166
5.
Homology
of
a Group
with
Coefficients in a Chain
Complex
168
6.
Example: The Hochschild-Serre Spectral
Sequence
171
7.
Equivariant Homology
172
8.
Computation
oíd1
175
9.
Example: Amalgamations
178
10.
Equivariant
Tate Cohomology
180
CHAPTER
VIII
Finiteness Conditions
183
1.
Introduction
183
2.
Cohomological Dimension
184
3.
Serre's Theorem
190
4.
Resolutions of Finite Type
191
5.
Groups of
Type FPn
197
6.
Groups of
Type
FP
and
Л.
199
7.
Topological
Interpretation
205
8.
Further
Topological
Results
210
9.
Further Examples
213
10.
Duality Groups
219
11.
Virtual Notions
225
CHAPTER IX
Euler
Characteristics
230
1.
Ranks of
Projective
Modules
:
Introduction
230
2.
The Hattori-Stallings Rank
231
3.
Ranks Over Commutative Rings
235
4.
Ranks Over Group Rings; Swan's Theorem
239
5.
Consequences of Swan's Theorem
242
6.
Euler
Characteristics of Groups: The Torsion-Free Case
246
7.
Extension to Groups with Torsion
249
8.
Euler
Characteristics and Number Theory
253
9.
Integrality Properties of
χ(Γ)
257
10.
Proof of Theorem
9.3 ;
Finite Group Actions
258
11.
The Fractional Part of
χ(Τ)
261
12.
Acyclic Covers
;
Proof of Lemma
11.2 265
13.
The/»-Fractional
Partorir)
266
14.
A Formula for
%T(si)
270
CHAPTER X
Farrell Cohomology Theory
273
1.
Introduction
. 273
2.
Complete Resolutions
273
3.
Definition and Properties of
Я*(Г)
277
Contents
4. Equivariant Farrell Cohomology 28
1
5. Cohomologically Trivial Modules 287
6.
Groups with Periodic
Cohomology 288
7.
Я*(Г)
and the Ordered Set of Finite Subgroups of
Γ
291
References
295
Notation Index
301
Index
303 |
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| id | DE-604.BV021264661 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:42:52Z |
| indexdate | 2024-07-09T20:34:12Z |
| institution | BVB |
| isbn | 0387906886 3540906886 |
| language | Undetermined |
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| physical | X, 306 S. |
| publishDate | 1997 |
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| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Brown, Kenneth S. Verfasser aut Cohomology of groups Kenneth S. Brown 2. corr. print., [Nachdr.] New York [u.a.] Springer [1997] X, 306 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 87 Kohomologie (DE-588)4031700-6 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Homologie (DE-588)4141951-0 gnd rswk-swf Kohomologietheorie (DE-588)4164610-1 gnd rswk-swf Kohomologietheorie (DE-588)4164610-1 s Gruppentheorie (DE-588)4072157-7 s DE-604 Homologie (DE-588)4141951-0 s 1\p DE-604 Kohomologie (DE-588)4031700-6 s 2\p DE-604 Graduate texts in mathematics 87 (DE-604)BV000000067 87 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014585857&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Brown, Kenneth S. Cohomology of groups Graduate texts in mathematics Kohomologie (DE-588)4031700-6 gnd Gruppentheorie (DE-588)4072157-7 gnd Homologie (DE-588)4141951-0 gnd Kohomologietheorie (DE-588)4164610-1 gnd |
| subject_GND | (DE-588)4031700-6 (DE-588)4072157-7 (DE-588)4141951-0 (DE-588)4164610-1 |
| title | Cohomology of groups |
| title_auth | Cohomology of groups |
| title_exact_search | Cohomology of groups |
| title_exact_search_txtP | Cohomology of groups |
| title_full | Cohomology of groups Kenneth S. Brown |
| title_fullStr | Cohomology of groups Kenneth S. Brown |
| title_full_unstemmed | Cohomology of groups Kenneth S. Brown |
| title_short | Cohomology of groups |
| title_sort | cohomology of groups |
| topic | Kohomologie (DE-588)4031700-6 gnd Gruppentheorie (DE-588)4072157-7 gnd Homologie (DE-588)4141951-0 gnd Kohomologietheorie (DE-588)4164610-1 gnd |
| topic_facet | Kohomologie Gruppentheorie Homologie Kohomologietheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014585857&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT brownkenneths cohomologyofgroups |