Algorithmic topology and classification of 3-manifolds:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2007
|
| Edition: | 2.ed. |
| Series: | Algorithms and computation in mathematics
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 467 - 473 |
| Physical Description: | XIV, 492 S. Ill., graph. Darst. |
| ISBN: | 9783540458982 |
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MARC
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| 100 | 1 | |a Matveev, Sergej V. |d 1947- |e Verfasser |0 (DE-588)124843956 |4 aut | |
| 245 | 1 | 0 | |a Algorithmic topology and classification of 3-manifolds |c Sergei Matveev |
| 250 | |a 2.ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XIV, 492 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algorithms and computation in mathematics |v 9 | |
| 500 | |a Literaturverz. S. 467 - 473 | ||
| 650 | 4 | |a Low-dimensional topology | |
| 650 | 4 | |a Three-manifolds (Topology) | |
| 650 | 0 | 7 | |a Algorithmus |0 (DE-588)4001183-5 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Algorithms and computation in mathematics |v 9 |w (DE-604)BV011131286 |9 9 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016553028 | ||
Record in the Search Index
| _version_ | 1804137736711438336 |
|---|---|
| adam_text | Contents
Simple and Special Polyhedra ............................. 1
1.1
Spines of
З-МашѓоМѕ
.................................... 1
1.1.1
Collapsing
........................................ 1
1.1.2
Spines
........................................... 2
1.1.3
Simple and Special Polyhedra
....................... 4
1.1.4
Special Spines
.................................... 5
1.1.5
Special Polyhedra and Singular
Triangulations
........ 10
1.2
Elementary Moves on Special Spines
....................... 13
1.2.1
Moves on Simple Polyhedra
......................... 14
1.2.2
2-Cell Replacement Lemma
......................... 19
1.2.3
Bubble Move
..................................... 22
1.2.4
Marked Polyhedra
................................. 25
1.3
Special Polyhedra Which are not Spines
.................... 30
1.3.1
Various Notions of Equivalence for Polyhedra
......... 31
1.3.2
Moves on Abstract Simple Polyhedra
................ 35
1.3.3
How to Hit the Target Without Inverse
ÍZ-Turns
...... 43
1.3.4
Zeeman s Collapsing Conjecture
..................... 46
Complexity Theory of 3-Manifolds
......................... 59
2.1
What is the Complexity of a 3-Mamfold?
................... 60
2.1.1
Almost Simple Polyhedra
........................... 60
2.1.2
Definition and Estimation of the Complexity
.......... 62
2.2
Properties of Complexity
................................. 67
2.2.1
Converting Almost Simple Spines into Special Ones
.... 67
2.2.2
The Finiteness Property
............................ 70
2.2.3
The Additivity Property
........................... 71
2.3
Closed Manifolds of Small Complexity
..................... 72
2.3.1
Enumeration Procedure
............................ 72
2.3.2
Simplification Moves
............................... 74
2.3.3
Manifolds of Complexity
< 6........................ 76
XII Contents
2.4
Graph Manifolds of Waldhausen
........................... 83
2.4.1
Properties of Graph Manifolds
...................... 83
2.4.2
Manifolds of Complexity <8
........................ 89
2.5
Hyperbolic Manifolds
.................................... 97
2.5.1
Hyperbolic Manifolds of Complexity
9 ............... 97
2.6
Lower Bounds of the Complexity
..........................100
2.6.1
Logarithmic Estimates
............................101
2.6.2
Complexity of Hyperbolic 3-Manifolds
...............104
2.6.3
Manifolds Having Special Spines with One 2-Cell
......105
3 Haken
Theory of Normal Surfaces
.........................107
3.1
Basic Notions and Haken s Scheme
........................107
3.2
Theory of Normal Curves
.................................110
3.2.1
Normal Curves and Normal Equations
...............110
3.2.2
Fundamental Solutions and Fundamental Curves
......114
3.2.3
Geometric Summation
.............................115
3.2.4
An Alternative Approach to the Theory of Normal
Curves
...........................................119
3.3
Normal Surfaces in 3-Manifolds
...........................123
3.3.1
Incompressible Surfaces
............................123
3.3.2
Normal Surfaces in 3-Manifolds with Boundary Pattern
126
3.3.3
Normalization Procedure
...........................127
3.3.4
Fundamental Surfaces
..............................134
3.3.5
Geometric Summation
.............................135
3.4
Normal Surfaces in Handle Decompositions
................138
4
Applications of the Theory of Normal Surfaces
............147
4.1
Examples of Algorithms Based on Haken s Theory
...........147
4.1.1
Recognition of Splittable Links
......................148
4.1.2
Getting Rid of Clean Disc Patches
...................150
4.1.3
Recognizing the Unknot and Calculating the Genus
of a Circle in the Boundary of a 3-Manifold
...........157
4.1.4
Is M3 Irreducible and Boundary Irreducible?
.........160
4.1.5
Is a Proper Surface Incompressible and Boundary
Incompressible?
...................................163
4.1.6
Is M3 Sufficiently Large?
...........................166
4.2
Cutting 3-Manifolds along Surfaces
........................176
4.2.1
Normal Surfaces and Spines
........................176
4.2.2
Triangulations
vs. Handle Decompositions
............188
5
Algorithmic Recognition of S3
.............................191
5.1
Links in a 3-Ball
........................................192
5.1.1
Compressing Discs and One-legged Crowns
...........192
5.1.2
Thin Position of Links
.............................195
Contents XIII
5.2
The Rubinstein Theorem
.................................199
5.2.1
2-Normal Surfaces
.................................199
5.2.2
Proof of the Rubinstein Theorem
....................203
5.2.3
The Algorithm
....................................209
Classification of
Haken
3-Manifolds
........................213
6.1
Main Theorem
..........................................213
6.2
The
Waldhausen
Theorem
................................216
6.2.1
Deforming Homotopy Equivalences of Surfaces
........217
6.2.2
Deforming Homotopy Equivalences of 3-Manifolds
to Homeomorphisms
...............................218
6.3
Finiteness Properties for Surfaces
..........................224
6.3.1
Two Reformulations of the Recognition Theorem
......224
6.3.2
Abstract Extension Moves
..........................227
6.3.3
First Finiteness Property and a Toy Form
of the Second
.....................................228
6.3.4
Second Finiteness Property for Simple 3-Manifolds
.... 231
6.4
Jaco-Shalen-Johannson Decomposition
....................240
6.4.1
Improving
Isotopy
that Separates Surfaces
............241
6.4.2
Does M3 Contain Essential Tori and
Annuli?.........245
6.4.3
Different Types of Essential Tori and
Annuli..........248
6.4.4
JSJ-Decomposition Exists and is Unique
.............261
6.4.5 Seifert
and /-Bundle Chambers
.....................264
6.4.6
Third Finiteness Property
..........................271
6.5
Extension Moves
........................................273
6.5.1
Description of General Extension Moves
..............273
6.5.2
Structure of Chambers
.............................281
6.5.3
Special Extension Moves: Easy Case
.................286
6.5.4
Difficult Case
.....................................294
6.5.5
Recognition of Simple Staffings Manifolds
with Periodic Monodromy
..........................298
6.5.6
Recognition of Simple Staffings Manifolds
with Nonperiodic Monodromy
......................303
6.5.7
Recognition of Quasi-Stallings Manifolds
.............307
6.5.8
Subdivision of Solid Tori
...........................312
6.5.9
Proof of the Recognition Theorem
...................320
З-МапігоМ
Recognizer
.....................................327
7.1
Computer Presentation of 3-Manifolds
.....................327
7.1.1
Cell Complexes
...................................328
7.1.2
S-Manifolds as Thickened Spines
....................330
7.2
Simplifying Manifolds and Spines
..........................332
7.2.1
Coordinate Systems on Tori
........................332
7.2.2
Reduction of Cell Structures
........................334
7.2.3
Collapses
.........................................335
XIV Contents
7.2.4
Surgeries
.........................................336
7.2.5
Disc Replacement Moves
...........................343
7.3
Labeled Molecules
.......................................347
7.3.1
What is a Labeled Molecule?
.......................347
7.3.2
Creating a Labeled Molecule
........................349
7.3.3
Assembling
Seifert
Atoms
..........................351
7.4
The Algorithm
..........................................354
7.5
Tabulation
.............................................355
7.5.1
Comments on the Table
............................357
7.5.2
Hyperbolic Manifolds up to Complexity
12...........358
7.5.3
Why the Table Contains no Duplicates?
..............360
7.6
Other Applications of the 3-Manifold Recognizer
............362
7.6.1
Enumeration of Heegaard Diagrams of Genus
2 .......362
7.6.2
З
-Manifolds
Represented by Crystallizations
with
< 32
Vertices
................................365
7.6.3
Classification of Crystallizations of Genus
2...........367
7.6.4
Recognition of Knots and Unknots
..................370
7.7
Two-Step Enumeration of 3-Manifolds
.....................371
7.7.1
Relative Spines and Relative Complexity
.............372
7.7.2
Assembling
.......................................377
7.7.3
Modified Enumeration of Manifolds and Spines
........380
8
The Turaev-Viro Invariants
...............................383
8.1
The Turaev-Viro Invariants
...............................383
8.1.1
The Construction
.................................383
8.1.2
Turaev-Viro Type Invariants of Order
r
< 3..........386
8.1.3
Construction and Properties of the
ε
-Invariant
........
392
8.1.4
Turaev-Viro Invariants of Order
r
> 3...............395
8.1.5
Computing Turaev-Viro Invariants
..................402
8.1.6
More on
e-Invariant
...............................407
8.2
3-Manifolds Having the Same Invariants
of Turaev-Viro Type
.....................................409
A Appendix
..................................................421
A.I Manifolds of Complexity
< 6..............................421
A.2 Minimal Spines of Manifolds up to Complexity
6............426
A.3 Minimal Spines of Some Manifolds of Complexity
7..........454
A.4 Tables of Turaev-Viro Invariants
..........................461
References
.....................................................481
Index
..........................................................489
|
| adam_txt |
Contents
Simple and Special Polyhedra . 1
1.1
Spines of
З-МашѓоМѕ
. 1
1.1.1
Collapsing
. 1
1.1.2
Spines
. 2
1.1.3
Simple and Special Polyhedra
. 4
1.1.4
Special Spines
. 5
1.1.5
Special Polyhedra and Singular
Triangulations
. 10
1.2
Elementary Moves on Special Spines
. 13
1.2.1
Moves on Simple Polyhedra
. 14
1.2.2
2-Cell Replacement Lemma
. 19
1.2.3
Bubble Move
. 22
1.2.4
Marked Polyhedra
. 25
1.3
Special Polyhedra Which are not Spines
. 30
1.3.1
Various Notions of Equivalence for Polyhedra
. 31
1.3.2
Moves on Abstract Simple Polyhedra
. 35
1.3.3
How to Hit the Target Without Inverse
ÍZ-Turns
. 43
1.3.4
Zeeman's Collapsing Conjecture
. 46
Complexity Theory of 3-Manifolds
. 59
2.1
What is the Complexity of a 3-Mamfold?
. 60
2.1.1
Almost Simple Polyhedra
. 60
2.1.2
Definition and Estimation of the Complexity
. 62
2.2
Properties of Complexity
. 67
2.2.1
Converting Almost Simple Spines into Special Ones
. 67
2.2.2
The Finiteness Property
. 70
2.2.3
The Additivity Property
. 71
2.3
Closed Manifolds of Small Complexity
. 72
2.3.1
Enumeration Procedure
. 72
2.3.2
Simplification Moves
. 74
2.3.3
Manifolds of Complexity
< 6. 76
XII Contents
2.4
Graph Manifolds of Waldhausen
. 83
2.4.1
Properties of Graph Manifolds
. 83
2.4.2
Manifolds of Complexity <8
. 89
2.5
Hyperbolic Manifolds
. 97
2.5.1
Hyperbolic Manifolds of Complexity
9 . 97
2.6
Lower Bounds of the Complexity
.100
2.6.1
Logarithmic Estimates
.101
2.6.2
Complexity of Hyperbolic 3-Manifolds
.104
2.6.3
Manifolds Having Special Spines with One 2-Cell
.105
3 Haken
Theory of Normal Surfaces
.107
3.1
Basic Notions and Haken's Scheme
.107
3.2
Theory of Normal Curves
.110
3.2.1
Normal Curves and Normal Equations
.110
3.2.2
Fundamental Solutions and Fundamental Curves
.114
3.2.3
Geometric Summation
.115
3.2.4
An Alternative Approach to the Theory of Normal
Curves
.119
3.3
Normal Surfaces in 3-Manifolds
.123
3.3.1
Incompressible Surfaces
.123
3.3.2
Normal Surfaces in 3-Manifolds with Boundary Pattern
126
3.3.3
Normalization Procedure
.127
3.3.4
Fundamental Surfaces
.134
3.3.5
Geometric Summation
.135
3.4
Normal Surfaces in Handle Decompositions
.138
4
Applications of the Theory of Normal Surfaces
.147
4.1
Examples of Algorithms Based on Haken's Theory
.147
4.1.1
Recognition of Splittable Links
.148
4.1.2
Getting Rid of Clean Disc Patches
.150
4.1.3
Recognizing the Unknot and Calculating the Genus
of a Circle in the Boundary of a 3-Manifold
.157
4.1.4
Is M3 Irreducible and Boundary Irreducible?
.160
4.1.5
Is a Proper Surface Incompressible and Boundary
Incompressible?
.163
4.1.6
Is M3 Sufficiently Large?
.166
4.2
Cutting 3-Manifolds along Surfaces
.176
4.2.1
Normal Surfaces and Spines
.176
4.2.2
Triangulations
vs. Handle Decompositions
.188
5
Algorithmic Recognition of S3
.191
5.1
Links in a 3-Ball
.192
5.1.1
Compressing Discs and One-legged Crowns
.192
5.1.2
Thin Position of Links
.195
Contents XIII
5.2
The Rubinstein Theorem
.199
5.2.1
2-Normal Surfaces
.199
5.2.2
Proof of the Rubinstein Theorem
.203
5.2.3
The Algorithm
.209
Classification of
Haken
3-Manifolds
.213
6.1
Main Theorem
.213
6.2
The
Waldhausen
Theorem
.216
6.2.1
Deforming Homotopy Equivalences of Surfaces
.217
6.2.2
Deforming Homotopy Equivalences of 3-Manifolds
to Homeomorphisms
.218
6.3
Finiteness Properties for Surfaces
.224
6.3.1
Two Reformulations of the Recognition Theorem
.224
6.3.2
Abstract Extension Moves
.227
6.3.3
First Finiteness Property and a Toy Form
of the Second
.228
6.3.4
Second Finiteness Property for Simple 3-Manifolds
. 231
6.4
Jaco-Shalen-Johannson Decomposition
.240
6.4.1
Improving
Isotopy
that Separates Surfaces
.241
6.4.2
Does M3 Contain Essential Tori and
Annuli?.245
6.4.3
Different Types of Essential Tori and
Annuli.248
6.4.4
JSJ-Decomposition Exists and is Unique
.261
6.4.5 Seifert
and /-Bundle Chambers
.264
6.4.6
Third Finiteness Property
.271
6.5
Extension Moves
.273
6.5.1
Description of General Extension Moves
.273
6.5.2
Structure of Chambers
.281
6.5.3
Special Extension Moves: Easy Case
.286
6.5.4
Difficult Case
.294
6.5.5
Recognition of Simple Staffings Manifolds
with Periodic Monodromy
.298
6.5.6
Recognition of Simple Staffings Manifolds
with Nonperiodic Monodromy
.303
6.5.7
Recognition of Quasi-Stallings Manifolds
.307
6.5.8
Subdivision of Solid Tori
.312
6.5.9
Proof of the Recognition Theorem
.320
З-МапігоМ
Recognizer
.327
7.1
Computer Presentation of 3-Manifolds
.327
7.1.1
Cell Complexes
.328
7.1.2
S-Manifolds as Thickened Spines
.330
7.2
Simplifying Manifolds and Spines
.332
7.2.1
Coordinate Systems on Tori
.332
7.2.2
Reduction of Cell Structures
.334
7.2.3
Collapses
.335
XIV Contents
7.2.4
Surgeries
.336
7.2.5
Disc Replacement Moves
.343
7.3
Labeled Molecules
.347
7.3.1
What is a Labeled Molecule?
.347
7.3.2
Creating a Labeled Molecule
.349
7.3.3
Assembling
Seifert
Atoms
.351
7.4
The Algorithm
.354
7.5
Tabulation
.355
7.5.1
Comments on the Table
.357
7.5.2
Hyperbolic Manifolds up to Complexity
12.358
7.5.3
Why the Table Contains no Duplicates?
.360
7.6
Other Applications of the 3-Manifold Recognizer
.362
7.6.1
Enumeration of Heegaard Diagrams of Genus
2 .362
7.6.2
З
-Manifolds
Represented by Crystallizations
with
< 32
Vertices
.365
7.6.3
Classification of Crystallizations of Genus
2.367
7.6.4
Recognition of Knots and Unknots
.370
7.7
Two-Step Enumeration of 3-Manifolds
.371
7.7.1
Relative Spines and Relative Complexity
.372
7.7.2
Assembling
.377
7.7.3
Modified Enumeration of Manifolds and Spines
.380
8
The Turaev-Viro Invariants
.383
8.1
The Turaev-Viro Invariants
.383
8.1.1
The Construction
.383
8.1.2
Turaev-Viro Type Invariants of Order
r
< 3.386
8.1.3
Construction and Properties of the
ε
-Invariant
.
392
8.1.4
Turaev-Viro Invariants of Order
r
> 3.395
8.1.5
Computing Turaev-Viro Invariants
.402
8.1.6
More on
e-Invariant
.407
8.2
3-Manifolds Having the Same Invariants
of Turaev-Viro Type
.409
A Appendix
.421
A.I Manifolds of Complexity
< 6.421
A.2 Minimal Spines of Manifolds up to Complexity
6.426
A.3 Minimal Spines of Some Manifolds of Complexity
7.454
A.4 Tables of Turaev-Viro Invariants
.461
References
.481
Index
.489 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Matveev, Sergej V. 1947- |
| author_GND | (DE-588)124843956 |
| author_facet | Matveev, Sergej V. 1947- |
| author_role | aut |
| author_sort | Matveev, Sergej V. 1947- |
| author_variant | s v m sv svm |
| building | Verbundindex |
| bvnumber | BV023369741 |
| callnumber-first | Q - Science |
| callnumber-label | QA612 |
| callnumber-raw | QA612.14 |
| callnumber-search | QA612.14 |
| callnumber-sort | QA 3612.14 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 300 |
| ctrlnum | (OCoLC)175223709 (DE-599)BVBBV023369741 |
| dewey-full | 514.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514.2 |
| dewey-search | 514.2 |
| dewey-sort | 3514.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2.ed. |
| format | Book |
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| id | DE-604.BV023369741 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:12:11Z |
| indexdate | 2024-07-09T21:17:02Z |
| institution | BVB |
| isbn | 9783540458982 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016553028 |
| oclc_num | 175223709 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XIV, 492 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Algorithms and computation in mathematics |
| series2 | Algorithms and computation in mathematics |
| spelling | Matveev, Sergej V. 1947- Verfasser (DE-588)124843956 aut Algorithmic topology and classification of 3-manifolds Sergei Matveev 2.ed. Berlin [u.a.] Springer 2007 XIV, 492 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithms and computation in mathematics 9 Literaturverz. S. 467 - 473 Low-dimensional topology Three-manifolds (Topology) Algorithmus (DE-588)4001183-5 gnd rswk-swf Niederdimensionale Topologie (DE-588)4280826-1 gnd rswk-swf Niederdimensionale Topologie (DE-588)4280826-1 s Algorithmus (DE-588)4001183-5 s DE-604 Algorithms and computation in mathematics 9 (DE-604)BV011131286 9 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016553028&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Matveev, Sergej V. 1947- Algorithmic topology and classification of 3-manifolds Algorithms and computation in mathematics Low-dimensional topology Three-manifolds (Topology) Algorithmus (DE-588)4001183-5 gnd Niederdimensionale Topologie (DE-588)4280826-1 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4280826-1 |
| title | Algorithmic topology and classification of 3-manifolds |
| title_auth | Algorithmic topology and classification of 3-manifolds |
| title_exact_search | Algorithmic topology and classification of 3-manifolds |
| title_exact_search_txtP | Algorithmic topology and classification of 3-manifolds |
| title_full | Algorithmic topology and classification of 3-manifolds Sergei Matveev |
| title_fullStr | Algorithmic topology and classification of 3-manifolds Sergei Matveev |
| title_full_unstemmed | Algorithmic topology and classification of 3-manifolds Sergei Matveev |
| title_short | Algorithmic topology and classification of 3-manifolds |
| title_sort | algorithmic topology and classification of 3 manifolds |
| topic | Low-dimensional topology Three-manifolds (Topology) Algorithmus (DE-588)4001183-5 gnd Niederdimensionale Topologie (DE-588)4280826-1 gnd |
| topic_facet | Low-dimensional topology Three-manifolds (Topology) Algorithmus Niederdimensionale Topologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016553028&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011131286 |
| work_keys_str_mv | AT matveevsergejv algorithmictopologyandclassificationof3manifolds |