A course in topological combinatorics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2013
|
| Schriftenreihe: | Universitext
Mathematics |
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis Abstract |
| Beschreibung: | XII, 238 S. graph. Darst. |
| ISBN: | 1441979093 9781441979094 9781489988263 |
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Datensatz im Suchindex
| _version_ | 1804148921094635520 |
|---|---|
| adam_text | Universitext
Mark de Longueville
A Course in Topological
Combinatorics
Λ
Course in Topological Combinatorics is the first undergraduate textbook on the
field of topological combinatorics, a subject that has become an active and innovative
research area in mathematics over the last thirty years with growing applications in
math, computer science, and other applied areas. Topological combinatorics is concerned
with solutions to combinatorial problems by applying topological tools. In most cases
these solutions are very elegant and the connection between combinatorics and
topology often arises as an unexpected surprise.
The textbook covers topics such as fair division, graph coloring problems, evasiveness
of graph properties, and embedding problems from discrete geometry. The text contains
a large number of figures that support the understanding of concepts and proofs. In
many cases several alternative proofs for the same result are given, and each chapter
ends with a series of exercises. The extensive appendix makes the book completely
self-contained.
The textbook is well suited for advanced undergraduate or beginning graduate
mathematics students. Previous knowledge in topology or graph theory is helpful
but not necessary. The text may be used as a basis for a one-or two-semester course
as well as a supplementary text for a topology or combinatorics class.
Mathematics
ISBN
978-1-4419-7909-4
9 781441
►
springer.com
Contents
1
Fair-Division
Problems...................................................... 1
1.1
Brouwer s Fixed-Point Theorem and Sperner s Lemma
.............. 1
1.2
Envy-Free Fair Division
................................................ 7
1.3
The Borsuk-Ulam Theorem and Tucker s Lemma
................... 11
1.4
A Generalization of Tucker s Lemma
................................. 16
1.5
Consensus ^-Division
.................................................. 21
1.6
The Borsuk-Ulam Property for General Groups
...................... 23
1.7
Consensus ¿--Division
.................................................. 28
Exercises
....................................................................... 31
2
Graph-Coloring Problems
.................................................. 37
2.1
The Kneser Conjecture
................................................. 38
2.2
Lovász s
Complexes
.................................................... 41
2.3
A Conjecture by
Lovász
................................................ 51
2.4
Classes with Good Topological Lower Bounds
for the Chromatic Number
............................................. 63
Exercises
....................................................................... 67
3
Evasiveness of Graph Properties
........................................... 69
3.1
Graph Properties and Their Complexity
............................... 69
3.2
Evasiveness of Monotone Graph Properties
........................... 82
3.3
Karp s Conjecture in the Prime-Power Case
.......................... 89
3.4
The Rivest-Vuillemin Theorem on Set Systems
...................... 92
Exercises
....................................................................... 93
4
Embedding and Mapping Problems
....................................... 97
4.1
The Radon Theorems
................................................... 97
4.2
Deleted Joins and the Z2-Index
........................................ 98
4.3
Bier Spheres
............................................................ 102
4.4
The van Kampen-Flores Theorem
..................................... 104
4.5
The Tverberg Problem
................................................. 106
4.6
An Obstruction to Graph Planarity
.................................... 110
x
Contents
4.7 Conway s Thrackles.................................................... 122
Exercises
.............................................................*.......... 142
5 Appendix
A:
Basic
Concepts
from Graph Theory
....................... 145
A.
1
Graphs
................................................................... 145
A.
2
Graph Invariants
........................................................ 152
A.3 Graph Drawings and Planarity
......................................... 154
A.4 Rotation Systems and Surface Embeddings
........................... 157
Exercises
....................................................................... 161
6
Appendix B: Crash Course in Topology
................................... 163
B.I Some Set-Theoretic Topology
......................................... 163
B.2 Surfaces
................................................................. 172
B.3 Simplicial Complexes
.................................................. 174
B.4 Shellability of Simplicial Complexes
.................................. 178
B.5 Some Operations on Simplicial Complexes
........................... 179
B.6 The Language of Category Theory
.................................... 181
B.7 Some Homological Algebra
........................................... 183
B.8 Axioms for Homology
................................................. 186
B.9 Simplicial Homology
................................................... 188
Exercises
....................................................................... 194
7
Appendix C: Partially Ordered Sets, Order Complexes,
and Their Topology
.......................................................... 199
C.
1
Partially Ordered Sets
.................................................. 199
C.2 Order Complexes
....................................................... 201
C.3 Shellability of Partial Orders
........................................... 203
Exercises
....................................................................... 206
8
Appendix D: Groups and Group Actions
................................. 209
D.I Groups
.................................................................. 209
D.2 Group Actions
.......................................................... 210
D.3 Topological G-Spaces
.................................................. 214
D.4 Simplicial Group Actions
.............................................. 215
Exercises
....................................................................... 217
9
Appendix E: Some Results and Applications from Smith Theory
...... 219
E.
1
The Transfer Homomorphism
......................................... 219
E.2 Transformations of Prime Order
....................................... 221
E.3 A Dimension Estimate and the
Euler
Characteristic
.................. 223
E.4 Homology Spheres and Disks
.......................................... 226
E.5 Cyclic Actions and a Result by Oliver
................................ 227
Exercises
....................................................................... 228
References
......................................................................... 229
Index
............................................................................... 233
A COURSE IN TOPOLOGICAL COMBINATORICS
/ LONGUEVILLE, MARK DE
: 2013
ABSTRACT / INHALTSTEXT
A COURSE IN TOPOLOGICAL COMBINATORICS IS THE FIRST UNDERGRADUATE
TEXTBOOK ON THE FIELD OF TOPOLOGICAL COMBINATORICS, A SUBJECT THAT HAS
BECOME AN ACTIVE AND INNOVATIVE RESEARCH AREA IN MATHEMATICS OVER THE
LAST THIRTY YEARS WITH GROWING APPLICATIONS IN MATH, COMPUTER SCIENCE,
AND OTHER APPLIED AREAS. TOPOLOGICAL COMBINATORICS IS CONCERNED WITH
SOLUTIONS TO COMBINATORIAL PROBLEMS BY APPLYING TOPOLOGICAL TOOLS. IN
MOST CASES THESE SOLUTIONS ARE VERY ELEGANT AND THE CONNECTION BETWEEN
COMBINATORICS AND TOPOLOGY OFTEN ARISES AS AN UNEXPECTED SURPRISE. THE
TEXTBOOK COVERS TOPICS SUCH AS FAIR DIVISION, GRAPH COLORING PROBLEMS,
EVASIVENESS OF GRAPH PROPERTIES, AND EMBEDDING PROBLEMS FROM DISCRETE
GEOMETRY. THE TEXT CONTAINS A LARGE NUMBER OF FIGURES THAT SUPPORT THE
UNDERSTANDING OF CONCEPTS AND PROOFS. IN MANY CASES SEVERAL ALTERNATIVE
PROOFS FOR THE SAME RESULT ARE GIVEN, AND EACH CHAPTER ENDS WITH A
SERIES OF EXERCISES. THE EXTENSIVE APPENDIX MAKES THE BOOK COMPLETELY
SELF-CONTAINED. THE TEXTBOOK IS WELL SUITED FOR ADVANCED UNDERGRADUATE
OR BEGINNING GRADUATE MATHEMATICS STUDENTS. PREVIOUS KNOWLEDGE IN
TOPOLOGY OR GRAPH THEORY IS HELPFUL BUT NOT NECESSARY. THE TEXT MAY BE
USED AS A BASIS FOR A ONE- OR TWO-SEMESTER COURSE AS WELL AS A
SUPPLEMENTARY TEXT FOR A TOPOLOGY OR COMBINATORICS CLASS
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
|
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| spelling | Longueville, Mark de Verfasser aut A course in topological combinatorics Mark de Longueville New York [u.a.] Springer 2013 XII, 238 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Mathematics Combinatorial analysis Algebraic topology Kombinatorische Topologie (DE-588)4137530-0 gnd rswk-swf Kombinatorische Topologie (DE-588)4137530-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4419-7910-0 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024808338&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024808338&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Springer Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024808338&sequence=000005&line_number=0003&func_code=DB_RECORDS&service_type=MEDIA Abstract |
| spellingShingle | Longueville, Mark de A course in topological combinatorics Combinatorial analysis Algebraic topology Kombinatorische Topologie (DE-588)4137530-0 gnd |
| subject_GND | (DE-588)4137530-0 |
| title | A course in topological combinatorics |
| title_auth | A course in topological combinatorics |
| title_exact_search | A course in topological combinatorics |
| title_full | A course in topological combinatorics Mark de Longueville |
| title_fullStr | A course in topological combinatorics Mark de Longueville |
| title_full_unstemmed | A course in topological combinatorics Mark de Longueville |
| title_short | A course in topological combinatorics |
| title_sort | a course in topological combinatorics |
| topic | Combinatorial analysis Algebraic topology Kombinatorische Topologie (DE-588)4137530-0 gnd |
| topic_facet | Combinatorial analysis Algebraic topology Kombinatorische Topologie |
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