Topological and bivariant K-theory:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2007
|
| Schriftenreihe: | Oberwolfach Seminars
36 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 262 S. graph. Darst. |
| ISBN: | 9783764383985 3764383984 |
Internformat
MARC
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| 016 | 7 | |a 983417040 |2 DE-101 | |
| 020 | |a 9783764383985 |c Pb. : EUR 31.99 (freier Pr.), sfr 49.90 |9 978-3-7643-8398-5 | ||
| 020 | |a 3764383984 |c Pb. : EUR 31.99 (freier Pr.), sfr 49.90 |9 3-7643-8398-4 | ||
| 024 | 3 | |a 9783764383985 | |
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| 100 | 1 | |a Cuntz, Joachim |d 1948- |e Verfasser |0 (DE-588)137444311 |4 aut | |
| 245 | 1 | 0 | |a Topological and bivariant K-theory |c Joachim Cuntz ; Ralf Meyer ; Jonathan M. Rosenberg |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XI, 262 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oberwolfach Seminars |v 36 | |
| 650 | 7 | |a Algebraïsche topologie |2 gtt | |
| 650 | 7 | |a K-theorie |2 gtt | |
| 650 | 4 | |a Algebraic topology | |
| 650 | 4 | |a K-theory | |
| 650 | 0 | 7 | |a Topologische K-Theorie |0 (DE-588)4334283-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Topologische K-Theorie |0 (DE-588)4334283-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Meyer, Ralf |e Verfasser |4 aut | |
| 700 | 1 | |a Rosenberg, Jonathan |e Verfasser |4 aut | |
| 830 | 0 | |a Oberwolfach Seminars |v 36 |w (DE-604)BV019806550 |9 36 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015858829&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-015858829 | ||
Datensatz im Suchindex
| _version_ | 1804136830779523072 |
|---|---|
| adam_text | Contents
Preface
ix
1
The elementary algebra of K-theory
1
1.1
Projective
modules, idempotents, and vector bundles
........ 2
1.1.1
General properties
....................... 4
1.1.2
Similarity of idempotents
................... 5
1.1.3
Relationship to vector bundles
................ 5
1.2
Passage to K-theory
.......................... 8
1.2.1
Euler
characteristics of finite
projective
complexes
.....· 9
1.2.2
Definition of
Ko
for non-unital rings
............. 10
1.3
Exactness properties of K-theory
................... 12
1.3.1
Half-exactness of Ko
...................... 12
1.3.2
Invertible elements and the index map
............ 13
1.3.3 Nilpotent
extensions and local rings
............. 15
2
Functional calculus and
topologica!
K-theory
19
2.1
Bomological analysis
.......................... 19
2.1.1
Spaces of continuous maps
................... 22
2.1.2
Bornological tensor products
................. 23
2.1.3
Local Banach algebras and functional calculus
....... 24
2.2
Homotopy
invariance
and exact sequences for local Banach algebras
27
2.2.1
Homotopy
invariance
of K-theory
............... 28
2.2.2
Higher K-theory
........................ 30
2.2.3
The
Puppe
exact sequence
................... 31
2.2.4
The Mayer-Vietoris sequence
................. 32
2.2.5
Projections and idempotents in C*-algebras
......... 34
2.3
Invariance
of K-theory for
isoradial subalgebras........... 36
2.3.1 Isoradial homomorphisms................... 36
2.3.2
Nearly idempotent elements
.................. 38
2.3.3
The
invariance
results
..................... 39
2.3.4
Continuity and stability
.................... 41
vi
Contents
3
Homotopy
invariance
of stabilised algebraic K-theory
45
3.1
Ingredients in the proof
........................ 46
3.1.1
Split-exact functors and quasi-homomorphisms
....... 46
3.1.2
Inner automorphisms and stability
.............. 49
3.1.3
A convenient stabilisation
................... 51
3.1.4
Holder continuity
........................ 53
3.2
The homotopy
invariance
result
.................... 54
3.2.1
A key lemma
.......................... 54
3.2.2
The main results
........................ 57
3.2.3
Weak versus Ml stability
................... 60
4
Bott
periodicity
63
4.1
Toeplitz algebras
............................ 63
4.2
The proof of
Bott
periodicity
..................... 65
4.3
Some K-theory computations
..................... 69
4.3.1
The Atiyah-Hirzebruch spectral sequence
.......... 72
5
The K-theory of crossed products
75
5.1
Crossed products for a single automorphism
............. 75
5.1.1
Crossed Toeplitz algebras
................... 77
5.2
The Pimsner-Voiculescu exact sequence
............... 79
5.2.1
Some consequences of the Pimsner-Voiculescu Theorem
. . 83
5.3
A glimpse of the
Baum-
Connes conjecture
.............. 83
5.3.1
Toeplitz cones
......................... 88
5.3.2
Proof of the decomposition theorem
............. 89
6
Towards bivariant K-theory: how to classify extensions
91
6.1
Some tricks with smooth homotopies
................. 91
6.2
Tensor algebras and classifying maps for extensions
......... 94
6.3
The suspension-stable homotopy category
.............. 99
6.3.1
Behaviour for infinite direct sums
............... 105
6.3.2
An alternative approach
.................... 106
6.4
Exact triangles in the suspension-stable homotopy category
.... 108
6.5
Long exact sequences in triangulated categories
........... 113
6.6
Long exact sequences in the suspension-stable homotopy category
. 116
6.7
The universal property of the suspension-stable homotopy category
119
7
Bivariant K-theory for bornological algebras
123
7.1
Some tricks with stabilisations
.................... 124
7.1.1
Comparing stabilisations
.................... 124
7.1.2
A general class of stabilisations
................ 125
7.1.3
Smooth stabilisations everywhere
............... 128
7.2
Definition and basic properties
.................... 129
7.3
Bott
periodicity and related results
.................. 132
Contents
vii
7.4 K-theory versus bivariant K-theory.................. 135
7.4.1
Comparison with other
topological K-theories........ 137
7.5
The Weyl algebra
............................ 139
8
A survey of bivariant K-theories
141
8.1
K-Theory with coefficients
....................... 143
8.2
Algebraic dual K-theory
........................ 146
8.3
Homotopy-theoretic KK-theory
.................... 148
8.4
Brown-Douglas-Fillmore extension theory
.............. 149
8.5
Bivariant K-theories for C*-algebras
................. 152
8.5.1
Adapting our machinery
.................... 152
8.5.2
Another variant related to E-theory
............. 156
8.5.3
Comparison with Kasparov s definition
............ 157
8.5.4
Some remarks on the Kasparov product
........... 164
8.6
Equivariant bivariant K-theories
................... 171
9
Algebras of continuous trace, twisted K-theory
173
9.1
Algebras of continuous trace
...................... 173
9.2
Twisted K-theory
............................ 182
10
Crossed products by
R
and Connes Thorn Isomorphism
185
10.1
Crossed products and Takai Duality
................. 185
10.2
Connes Thom Isomorphism Theorem
................ 189
10.2.1
Connes original proof
..................... 189
10.2.2
Another proof
.......................... 191
11
Applications to physics
195
11.1
K-theory in physics
........................... 195
11.2
T-duality
................................ 197
12
Some connections with index theory
203
12.1
Pseudo-differential operators
..................... 204
12.1.1
Definition of pseudo-differential operators
.......... 204
12.1.2
Index problems from pseudo-differential operators
..... 207
12.1.3
The Dolbeault operator
.................... 208
12.2
The index theorem of
Baum,
Douglas, and Taylor
.......... 210
12.2.1
Toeplitz operators
....................... 210
12.2.2
A formula for the boundary map
............... 212
12.2.3
Application to the Dolbeault operator
............ 215
12.3
The index theorems of Kasparov and Atiyah-Singer
........ 216
12.3.1
The Thom isomorphism and the Dolbeault operator
.... 220
12.3.2
The Dolbeault element and the topological index map
. . 222
viii
Contente
13
Localisation of triangulated categories
225
13.1
Examples of localisations
....................... 229
13.1.1
The Universal Coefficient Theorem
.............. 230
13.1.2
The Baum-Connes assembly map via localisation
...... 233
13.2
The Octahedral Axiom
......................... 234
Bibliography
241
Notation and Symbols
249
Index
257
|
| adam_txt |
Contents
Preface
ix
1
The elementary algebra of K-theory
1
1.1
Projective
modules, idempotents, and vector bundles
. 2
1.1.1
General properties
. 4
1.1.2
Similarity of idempotents
. 5
1.1.3
Relationship to vector bundles
. 5
1.2
Passage to K-theory
. 8
1.2.1
Euler
characteristics of finite
projective
complexes
.· 9
1.2.2
Definition of
Ko
for non-unital rings
. 10
1.3
Exactness properties of K-theory
. 12
1.3.1
Half-exactness of Ko
. 12
1.3.2
Invertible elements and the index map
. 13
1.3.3 Nilpotent
extensions and local rings
. 15
2
Functional calculus and
topologica!
K-theory
19
2.1
Bomological analysis
. 19
2.1.1
Spaces of continuous maps
. 22
2.1.2
Bornological tensor products
. 23
2.1.3
Local Banach algebras and functional calculus
. 24
2.2
Homotopy
invariance
and exact sequences for local Banach algebras
27
2.2.1
Homotopy
invariance
of K-theory
. 28
2.2.2
Higher K-theory
. 30
2.2.3
The
Puppe
exact sequence
. 31
2.2.4
The Mayer-Vietoris sequence
. 32
2.2.5
Projections and idempotents in C*-algebras
. 34
2.3
Invariance
of K-theory for
isoradial subalgebras. 36
2.3.1 Isoradial homomorphisms. 36
2.3.2
Nearly idempotent elements
. 38
2.3.3
The
invariance
results
. 39
2.3.4
Continuity and stability
. 41
vi
Contents
3
Homotopy
invariance
of stabilised algebraic K-theory
45
3.1
Ingredients in the proof
. 46
3.1.1
Split-exact functors and quasi-homomorphisms
. 46
3.1.2
Inner automorphisms and stability
. 49
3.1.3
A convenient stabilisation
. 51
3.1.4
Holder continuity
. 53
3.2
The homotopy
invariance
result
. 54
3.2.1
A key lemma
. 54
3.2.2
The main results
. 57
3.2.3
Weak versus Ml stability
. 60
4
Bott
periodicity
63
4.1
Toeplitz algebras
. 63
4.2
The proof of
Bott
periodicity
. 65
4.3
Some K-theory computations
. 69
4.3.1
The Atiyah-Hirzebruch spectral sequence
. 72
5
The K-theory of crossed products
75
5.1
Crossed products for a single automorphism
. 75
5.1.1
Crossed Toeplitz algebras
. 77
5.2
The Pimsner-Voiculescu exact sequence
. 79
5.2.1
Some consequences of the Pimsner-Voiculescu Theorem
. . 83
5.3
A glimpse of the
Baum-
Connes conjecture
. 83
5.3.1
Toeplitz cones
. 88
5.3.2
Proof of the decomposition theorem
. 89
6
Towards bivariant K-theory: how to classify extensions
91
6.1
Some tricks with smooth homotopies
. 91
6.2
Tensor algebras and classifying maps for extensions
. 94
6.3
The suspension-stable homotopy category
. 99
6.3.1
Behaviour for infinite direct sums
. 105
6.3.2
An alternative approach
. 106
6.4
Exact triangles in the suspension-stable homotopy category
. 108
6.5
Long exact sequences in triangulated categories
. 113
6.6
Long exact sequences in the suspension-stable homotopy category
. 116
6.7
The universal property of the suspension-stable homotopy category
119
7
Bivariant K-theory for bornological algebras
123
7.1
Some tricks with stabilisations
. 124
7.1.1
Comparing stabilisations
. 124
7.1.2
A general class of stabilisations
. 125
7.1.3
Smooth stabilisations everywhere
. 128
7.2
Definition and basic properties
. 129
7.3
Bott
periodicity and related results
. 132
Contents
vii
7.4 K-theory versus bivariant K-theory. 135
7.4.1
Comparison with other
topological K-theories. 137
7.5
The Weyl algebra
. 139
8
A survey of bivariant K-theories
141
8.1
K-Theory with coefficients
. 143
8.2
Algebraic dual K-theory
. 146
8.3
Homotopy-theoretic KK-theory
. 148
8.4
Brown-Douglas-Fillmore extension theory
. 149
8.5
Bivariant K-theories for C*-algebras
. 152
8.5.1
Adapting our machinery
. 152
8.5.2
Another variant related to E-theory
. 156
8.5.3
Comparison with Kasparov's definition
. 157
8.5.4
Some remarks on the Kasparov product
. 164
8.6
Equivariant bivariant K-theories
. 171
9
Algebras of continuous trace, twisted K-theory
173
9.1
Algebras of continuous trace
. 173
9.2
Twisted K-theory
. 182
10
Crossed products by
R
and Connes' Thorn Isomorphism
185
10.1
Crossed products and Takai Duality
. 185
10.2
Connes' Thom Isomorphism Theorem
. 189
10.2.1
Connes' original proof
. 189
10.2.2
Another proof
. 191
11
Applications to physics
195
11.1
K-theory in physics
. 195
11.2
T-duality
. 197
12
Some connections with index theory
203
12.1
Pseudo-differential operators
. 204
12.1.1
Definition of pseudo-differential operators
. 204
12.1.2
Index problems from pseudo-differential operators
. 207
12.1.3
The Dolbeault operator
. 208
12.2
The index theorem of
Baum,
Douglas, and Taylor
. 210
12.2.1
Toeplitz operators
. 210
12.2.2
A formula for the boundary map
. 212
12.2.3
Application to the Dolbeault operator
. 215
12.3
The index theorems of Kasparov and Atiyah-Singer
. 216
12.3.1
The Thom isomorphism and the Dolbeault operator
. 220
12.3.2
The Dolbeault element and the topological index map
. . 222
viii
Contente
13
Localisation of triangulated categories
225
13.1
Examples of localisations
. 229
13.1.1
The Universal Coefficient Theorem
. 230
13.1.2
The Baum-Connes assembly map via localisation
. 233
13.2
The Octahedral Axiom
. 234
Bibliography
241
Notation and Symbols
249
Index
257 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Cuntz, Joachim 1948- Meyer, Ralf Rosenberg, Jonathan |
| author_GND | (DE-588)137444311 |
| author_facet | Cuntz, Joachim 1948- Meyer, Ralf Rosenberg, Jonathan |
| author_role | aut aut aut |
| author_sort | Cuntz, Joachim 1948- |
| author_variant | j c jc r m rm j r jr |
| building | Verbundindex |
| bvnumber | BV022652840 |
| callnumber-first | Q - Science |
| callnumber-label | QA612 |
| callnumber-raw | QA612.33 |
| callnumber-search | QA612.33 |
| callnumber-sort | QA 3612.33 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 230 |
| classification_tum | MAT 552f |
| ctrlnum | (OCoLC)141385291 (DE-599)DNB983417040 |
| dewey-full | 514.23 512/.66 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology 512 - Algebra |
| dewey-raw | 514.23 512/.66 |
| dewey-search | 514.23 512/.66 |
| dewey-sort | 3514.23 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022652840 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:22:09Z |
| indexdate | 2024-07-09T21:02:38Z |
| institution | BVB |
| isbn | 9783764383985 3764383984 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015858829 |
| oclc_num | 141385291 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-29T DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-91G DE-BY-TUM DE-29T DE-355 DE-BY-UBR DE-11 |
| physical | XI, 262 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Oberwolfach Seminars |
| series2 | Oberwolfach Seminars |
| spelling | Cuntz, Joachim 1948- Verfasser (DE-588)137444311 aut Topological and bivariant K-theory Joachim Cuntz ; Ralf Meyer ; Jonathan M. Rosenberg Basel [u.a.] Birkhäuser 2007 XI, 262 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oberwolfach Seminars 36 Algebraïsche topologie gtt K-theorie gtt Algebraic topology K-theory Topologische K-Theorie (DE-588)4334283-8 gnd rswk-swf Topologische K-Theorie (DE-588)4334283-8 s DE-604 Meyer, Ralf Verfasser aut Rosenberg, Jonathan Verfasser aut Oberwolfach Seminars 36 (DE-604)BV019806550 36 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015858829&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cuntz, Joachim 1948- Meyer, Ralf Rosenberg, Jonathan Topological and bivariant K-theory Oberwolfach Seminars Algebraïsche topologie gtt K-theorie gtt Algebraic topology K-theory Topologische K-Theorie (DE-588)4334283-8 gnd |
| subject_GND | (DE-588)4334283-8 |
| title | Topological and bivariant K-theory |
| title_auth | Topological and bivariant K-theory |
| title_exact_search | Topological and bivariant K-theory |
| title_exact_search_txtP | Topological and bivariant K-theory |
| title_full | Topological and bivariant K-theory Joachim Cuntz ; Ralf Meyer ; Jonathan M. Rosenberg |
| title_fullStr | Topological and bivariant K-theory Joachim Cuntz ; Ralf Meyer ; Jonathan M. Rosenberg |
| title_full_unstemmed | Topological and bivariant K-theory Joachim Cuntz ; Ralf Meyer ; Jonathan M. Rosenberg |
| title_short | Topological and bivariant K-theory |
| title_sort | topological and bivariant k theory |
| topic | Algebraïsche topologie gtt K-theorie gtt Algebraic topology K-theory Topologische K-Theorie (DE-588)4334283-8 gnd |
| topic_facet | Algebraïsche topologie K-theorie Algebraic topology K-theory Topologische K-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015858829&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019806550 |
| work_keys_str_mv | AT cuntzjoachim topologicalandbivariantktheory AT meyerralf topologicalandbivariantktheory AT rosenbergjonathan topologicalandbivariantktheory |