The Novikov conjecture: geometry and algebra
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2005
|
| Schriftenreihe: | Oberwolfach seminars
33 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Auch als Internetausgabe Includes bibliographical references (p. [237]-254) and index |
| Beschreibung: | XV, 266 S. 24 cm |
| ISBN: | 3764371412 0817671412 |
Internformat
MARC
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| 100 | 1 | |a Kreck, Matthias |d 1947- |e Verfasser |0 (DE-588)108556077 |4 aut | |
| 245 | 1 | 0 | |a The Novikov conjecture |b geometry and algebra |c Matthias Kreck ; Wolfgang Lück |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2005 | |
| 300 | |a XV, 266 S. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oberwolfach seminars |v 33 | |
| 500 | |a Auch als Internetausgabe | ||
| 500 | |a Includes bibliographical references (p. [237]-254) and index | ||
| 650 | 4 | |a Géométrie différentielle non commutative | |
| 650 | 4 | |a K-théorie | |
| 650 | 4 | |a Novikov, Conjecture de | |
| 650 | 4 | |a Topologie différentielle | |
| 650 | 4 | |a Novikov conjecture |v Congresses | |
| 650 | 4 | |a K-theory |v Congresses | |
| 650 | 4 | |a Noncommutative differential geometry |v Congresses | |
| 650 | 4 | |a Differential topology |v Congresses | |
| 650 | 0 | 7 | |a Algebraische Topologie |0 (DE-588)4120861-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Invariantentheorie |0 (DE-588)4162209-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Novikov-Vermutung |0 (DE-588)4402781-3 |2 gnd |9 rswk-swf |
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| 689 | 3 | 0 | |a Differentialtopologie |0 (DE-588)4012255-4 |D s |
| 689 | 3 | |5 DE-604 | |
| 700 | 1 | |a Lück, Wolfgang |e Verfasser |4 aut | |
| 830 | 0 | |a Oberwolfach seminars |v 33 |w (DE-604)BV019806550 |9 33 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-013132328 | ||
Datensatz im Suchindex
| _version_ | 1804133303141269504 |
|---|---|
| adam_text | Contents
Introduction
xi
0
A Motivating Problem
(К.)
1
0.1
Dimensions
< 4............................. 1
0.2
Dimension
6............................... 2
0.3
Dimension
5............................... 4
1
Introduction to the Novikov and the
Borei
Conjecture (L.)
5
1.1
The Original Formulation of the Novikov Conjecture
........ 5
1.2
Invariance
Properties of the L-Class
................. 6
1.3
The
Borei
Conjecture
......................... 8
2
Normal Bordism Groups (K.)
11
2.1
Normal Bordism Groups
........................ 11
2.2
Rational Computation of Normal Bordism Groups
......... 12
2.3
Rational Computation of Oriented Bordism Groups
......... 14
3
The Signature (K.)
17
3.1
The Definition of the Signature
........-... ......... 18
3.2
The Bordism
Invariance
of the Signature
............... 18
3.3
Multiplicativity and other Properties of the Signature
...... . 20
3.4
Geometric Interpretation of Cohomology and the Intersection Form
21
4
The Signature Theorem and the Novikov Conjecture (K.)
25
4.1
The Signature Theorem
........................ 25
4.2
Higher Signatures
............................ 27
4.3
The Novikov Conjecture
........................ 28
4.4
The Pontrjagin Classes are not Homeomorphism Invariants
.... 29
5
The
Projective
Class Group and the Whitehead Group (L.)
33
5.1
The
Projective
Class Group
...................... 33
5.2
The First Algebraic Jr-Group
..................... 35
vi
Contents
5.3
The Whitehead Group
......................... 39
5.4
The Bass- Heller Swan Decomposition
................ 40
6
Whitehead Torsion (L.)
43
6.1
Whitehead Torsion of a Chain Map
.................. 43
6.2
The Cellular Chain Complex of the Universal Covering
....... 47
6.3
The Whitehead Torsion of a Cellular Map
.............. 49
6.4
Simple Homotopy Equivalences
.................... 52
7
The Statement and Consequences of the s-Cobordism Theorem (L.)
55
8
Sketch of the Proof of the s-Cobordism Theorem (L.)
59
8.1
Handlebody Decompositions
...................... 59
8.2
Handlebody Decompositions and CW-Structures
.......... 61
8.3
Reducing the Handlebody Decomposition
.............. 63
8.4
Handlebody Decompositions and Whitehead Torsion
........ 65
9
From the Novikov Conjecture to Surgery (K.)
69
9.1
The Structure Set
........................... 69
9.2
The Assembly Idea
........................... 70
10
Surgery Below the Middle Dimension I: An Example (K.)
75
10.1
Surgery and its Trace
......................... 75
10.2
The Effect on the Fundamental Group and Homology Groups
... 76
10.3
Application to Knottings
....................... 77
11
Surgery Below the Middle Dimension II: Systematically (K.)
79
11.1
The Effect of Surgery in Homology and Homotopy
......... 79
11.2
Surgery below the Middle Dimension
................. 81
11.3
Construction of Certain 6-Manifolds
................. 84
12
Surgery in the Middle Dimension I (K.)
87
12.1
Motivation for the Surgery Obstruction Groups
........... 87
12.2
Unimodular Hermitian Forms
..................... 88
12.3
The L -Groups in Dimensions Am
................... 89
12.4
The L-Groups in Other Dimensions
.................. 90
13
Surgery in the Middle Dimension II (K.)
93
13.1
Equivariant Intersection Numbers
................... 93
13.2
Stably Free Modules
.......................... 94
13.3
The Quadratic Refinement
....................... 95
13.4
The Surgery Obstruction
....................... 97
Contents
vii
14
Surgery in the Middle Dimension III (K.)
99
14.1
Stable- Diffeomorphism Classification
................. 99
14.2
The Surgery Obstruction is a Bordism Invariant
.......... 101
14.3
The Main Result
............................ 101
14.4
Proof of the Main Theorem
...................... 103
14.5
The Exact Surgery Sequence
..................... 106
14.6
Stable Classification of Certain
6-Manifokis
............. 107
15
An Assembly Map (K.)
109
15.1
More on the Definition of the Assembly Map
............ 109
15.2
The Surgery Version of the Novikov Conjecture
........... 112
16
The Novikov Conjecture for Zn (K.)
113
16.1
The Idea of the Proof
......................... 113
16.2
Reduction to Mapping Tori
...................... 113
16.3
The Proof for Rank
1......................... 115
16.4
The Generalization to Higher Rank
.................. 117
17
Роіпсаге
Duality and Algebraic ¿-Groups (L. and Varisco)
119
17.1
Poincaré
duality
............................ 119
17.2
Algebraic L-groups
........................... 124
18
Spectra (L.)
133
18.1
Basic Notions about Spectra
...................... 133
18.2
Homotopy Pushouts and Homotopy Pullbacks for Spaces
...... 135
18.3
Homotopy Pushouts and Homotopy Pullbacks for Spectra
..... 138
18.4
(Co-)Homology Theories Associated to Spectra
........... 139
18.5
.řr-Theory
and ¿-Theory Spectra
................... 141
18.6
The Chern Character for Homology Theories
............ 143
18.7
The Bordism Group Associated to a Vector Bundle
......... 144
18.8
The Thorn Space of a Vector Bundle
................. 145
18.9
The Pontrjagin-Thom Construction
................. 145
18-lOThe Stable Version of the Pontrjagin-Thom Construction
..... 146
18.11
The Oriented Bordism Ring
...................... 148
18.12Stable Homotopy
............................ 149
18.13The Thorn Isomorphism
........................ 150
18.14The Rationalized Oriented Bordism Ring
.............. 150
18.15The Integral Oriented Bordism Ring
................. 151
19
Classifying Spaces of Families (L.)
153
19.1
Basics about
G- C
W-
Complexes
.................... 153
19.2
The Classifying Space for a Family
.................. 156
viii Contents
19.3
Special
Models............................. 157
19.3.1
The Family
of All Subgroups and the Trivial
Family
.... 157
19.3.2
Operator Theoretic Model
................... 157
19.3.3
Discrete Subgroups of Almost Connected Lie Groups
. . . . 157
19.3.4
Simply Connected Non-Positively Curved Manifolds
.... 158
19.3.5
CAT(O)-Spaces
......................... 158
19.3.6
Trees with Finite Isotropy Groups
.............. 158
19.3.7
Amalgamated Products and HNN-Extensions
........ 158
19.3.8
Arithmetic Groups
....................... 159
19.3.9
Outer Automorphism Groups of Free groups
........ 160
19.3.10
Mapping Class groups
.................· · · · 160
19.3.11
One-Relator Groups
...................... 160
19.3.12
Special Linear Groups of (2,2)-Matrices
........... 161
20
Equivariant Homology Theories and the Meta-Conjecture
(Ł.)
163
20.1
The Meta-Conjecture
.......................... 163
20.2
Formulation of the Farrell-Jones and the Baum-Connes Conjecture
164
20.3
Equivariant Homology Theories
.................... 165
20.4
The Construction of Equivariant Homology Theories from Spectra
168
21
The Farrell-Jones Conjecture (L.)
173
21.1
The Bass-Heller-Swan Decomposition in Arbitrary Dimensions
. . 173
21.2
Decorations in L-Theory and the Shaneson Splitting
........ 174
21.3
Changing the Family
.......................... 176
21.4
The Farrell-Jones Conjecture for Torsionfree Groups
........ 177
21.5
The Farrell-Jones Conjecture and the Bore! Conjecture
...... 180
21.6
The Passage from TIN to VQZ
.................... 181
22
The Baum-Connes Conjecture (L.)
185
22.1
Index Theoretic Interpretation of the Baum-Connes Assembly Map
185
22.2
The Baum-Connes Conjecture for Torsionfree Groups
....... 186
22.3
The Trace Conjecture in the Torsionfree Case
............ 187
22.4
The Kadison Conjecture
........................ 187
22.5
The Stable Gromov-Lawson-Rosenberg Conjecture
......... 188
22.6
The Choice of the Family TIN in the Baum-Connes Conjecture
. 189
23
Relating the Novikov, the Farrell-Jones and the Baum-Connes
Conjectures (L.)
191
23.1
The Farrell-Jones Conjecture and the Novikov Conjecture
..... 191
23.2
Relating
Topologici
iŕ-Theory
and L-Theory
............ 195
23.3
The Baum-Connes Conjecture and the Novikov Conjecture
.... 197
Contents ix
24
Miscellaneous (L.)
201
24.1
Status
of the Conjectures
....................... 201
24.2
Methods of Proof
............................ 206
24.3
Computations for Finite Groups
................... 206
24.3.1
Topological if-Theory for Finite Groups
........... 206
24.3.2
Algebraic K-Theory for Finite Groups
............ 207
24.3.3
Algebraic L-Theory for Finite Groups
............ 208
24.4
Rational Computations
........................ 208
24.4.1
Rationalized Topological /f-Theory for Infinite Groups
. . . 209
24.4.2
Rationalized Algebraic /r-Theory for Infinite Groups
.... 210
24.4.3.
Rationalized Algebraic L-Theory for Infinite Groups
.... 211
24.5
Integral Computations
......................... 211
25
Exercises
215
26
Hints to the Solutions of the Exercises
223
References
236
Index
254
Notation
261
Schedules
263
List of Participants
267
|
| any_adam_object | 1 |
| author | Kreck, Matthias 1947- Lück, Wolfgang |
| author_GND | (DE-588)108556077 |
| author_facet | Kreck, Matthias 1947- Lück, Wolfgang |
| author_role | aut aut |
| author_sort | Kreck, Matthias 1947- |
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| building | Verbundindex |
| bvnumber | BV019806676 |
| callnumber-first | Q - Science |
| callnumber-label | QA613 |
| callnumber-raw | QA613 |
| callnumber-search | QA613 |
| callnumber-sort | QA 3613 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 300 SK 350 |
| classification_tum | MAT 460f MAT 552f MAT 576f MAT 180f |
| ctrlnum | (OCoLC)56894472 (DE-599)BVBBV019806676 |
| dewey-full | 516/.07 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.07 |
| dewey-search | 516/.07 |
| dewey-sort | 3516 17 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)1071861417 Konferenzschrift gnd-content |
| genre_facet | Konferenzschrift |
| id | DE-604.BV019806676 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:06:34Z |
| institution | BVB |
| isbn | 3764371412 0817671412 |
| language | English |
| lccn | 2004062360 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013132328 |
| oclc_num | 56894472 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-824 DE-703 DE-384 DE-29T DE-634 DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-824 DE-703 DE-384 DE-29T DE-634 DE-11 DE-188 |
| physical | XV, 266 S. 24 cm |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Oberwolfach seminars |
| series2 | Oberwolfach seminars |
| spelling | Kreck, Matthias 1947- Verfasser (DE-588)108556077 aut The Novikov conjecture geometry and algebra Matthias Kreck ; Wolfgang Lück Basel [u.a.] Birkhäuser 2005 XV, 266 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Oberwolfach seminars 33 Auch als Internetausgabe Includes bibliographical references (p. [237]-254) and index Géométrie différentielle non commutative K-théorie Novikov, Conjecture de Topologie différentielle Novikov conjecture Congresses K-theory Congresses Noncommutative differential geometry Congresses Differential topology Congresses Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Invariantentheorie (DE-588)4162209-1 gnd rswk-swf Novikov-Vermutung (DE-588)4402781-3 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Novikov-Vermutung (DE-588)4402781-3 s DE-604 Algebraische Topologie (DE-588)4120861-4 s Invariantentheorie (DE-588)4162209-1 s Differentialtopologie (DE-588)4012255-4 s Lück, Wolfgang Verfasser aut Oberwolfach seminars 33 (DE-604)BV019806550 33 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013132328&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kreck, Matthias 1947- Lück, Wolfgang The Novikov conjecture geometry and algebra Oberwolfach seminars Géométrie différentielle non commutative K-théorie Novikov, Conjecture de Topologie différentielle Novikov conjecture Congresses K-theory Congresses Noncommutative differential geometry Congresses Differential topology Congresses Algebraische Topologie (DE-588)4120861-4 gnd Invariantentheorie (DE-588)4162209-1 gnd Novikov-Vermutung (DE-588)4402781-3 gnd Differentialtopologie (DE-588)4012255-4 gnd |
| subject_GND | (DE-588)4120861-4 (DE-588)4162209-1 (DE-588)4402781-3 (DE-588)4012255-4 (DE-588)1071861417 |
| title | The Novikov conjecture geometry and algebra |
| title_auth | The Novikov conjecture geometry and algebra |
| title_exact_search | The Novikov conjecture geometry and algebra |
| title_full | The Novikov conjecture geometry and algebra Matthias Kreck ; Wolfgang Lück |
| title_fullStr | The Novikov conjecture geometry and algebra Matthias Kreck ; Wolfgang Lück |
| title_full_unstemmed | The Novikov conjecture geometry and algebra Matthias Kreck ; Wolfgang Lück |
| title_short | The Novikov conjecture |
| title_sort | the novikov conjecture geometry and algebra |
| title_sub | geometry and algebra |
| topic | Géométrie différentielle non commutative K-théorie Novikov, Conjecture de Topologie différentielle Novikov conjecture Congresses K-theory Congresses Noncommutative differential geometry Congresses Differential topology Congresses Algebraische Topologie (DE-588)4120861-4 gnd Invariantentheorie (DE-588)4162209-1 gnd Novikov-Vermutung (DE-588)4402781-3 gnd Differentialtopologie (DE-588)4012255-4 gnd |
| topic_facet | Géométrie différentielle non commutative K-théorie Novikov, Conjecture de Topologie différentielle Novikov conjecture Congresses K-theory Congresses Noncommutative differential geometry Congresses Differential topology Congresses Algebraische Topologie Invariantentheorie Novikov-Vermutung Differentialtopologie Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013132328&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV019806550 |
| work_keys_str_mv | AT kreckmatthias thenovikovconjecturegeometryandalgebra AT luckwolfgang thenovikovconjecturegeometryandalgebra |