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E-raamat: Ordered Groups and Topology

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Ordered Groups and TopologySuurem pilt
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This book deals with the connections between topology and ordered groups. It begins with a self-contained introduction to orderable groups and from there explores the interactions between orderability and objects in low-dimensional topology, such as knot theory, braid groups, and 3-manifolds, as well as groups of homeomorphisms and other topological structures. The book also addresses recent applications of orderability in the studies of codimension-one foliations and Heegaard-Floer homology. The use of topological methods in proving algebraic results is another feature of the book.

The book was written to serve both as a textbook for graduate students, containing many exercises, and as a reference for researchers in topology, algebra, and dynamical systems. A basic background in group theory and topology is the only prerequisite for the reader.

Arvustused

The book finds a good balance between being a resource for researchers and a graduate textbook." - Sebastian Wolfgang Hensel, Mathematical Reviews

"The diligent and disciplined reader of this thin (<150 pp) book will be rewarded by a lot more than knowledge of ordered groups." - Lee P. Neuwirth, Zentralblatt MATH

"Given the huge popularity enjoyed by low dimensional topology these days, and all for good reason, it should make a very positive impact. The book is easy to read and deals with very pretty mathematics." - Michael Berg, MAA Reviews

Preface ix
Chapter 1 Orderable groups and their algebraic properties
1(20)
§1.1 Invariant orderings
2(1)
§1.2 Examples
3(3)
§1.3 Bi-orderable groups
6(1)
§1.4 Positive cone
7(1)
§1.5 Topology and the spaces of orderings
8(3)
§1.6 Testing for orderability
11(2)
§1.7 Characterization of left-orderable groups
13(2)
§1.8 Group rings and zero divisors
15(1)
§1.9 Torsion-free groups which are not left-orderable
16(5)
Chapter 2 Holder's theorem, convex subgroups and dynamics
21(10)
§2.1 Holder's theorem
21(3)
§2.2 Convex subgroups
24(2)
§2.3 Bi-orderable groups are locally indicable
26(1)
§2.4 The dynamic realization of a left-ordering
27(4)
Chapter 3 Free groups, surface groups and covering spaces
31(12)
§3.1 Surfaces
31(2)
§3.2 Ordering free groups
33(3)
§3.3 Ordering surface groups
36(3)
§3.4 A Theorem Of Farrell
39(4)
Chapter 4 Knots
43(22)
§4.1 Review of classical knot theory
43(4)
§4.2 The Wirtinger presentation
47(2)
§4.3 Knot groups are locally indicable
49(2)
§4.4 Bi-ordering certain knot groups
51(9)
§4.5 Crossing changes: A theorem of Smythe
60(5)
Chapter 5 Three-dimensional manifolds
65(12)
§5.1 Ordering 3-manifold groups
66(1)
§5.2 Surgery
67(4)
§5.3 Branched coverings
71(4)
§5.4 Bi-orderability and surgery
75(2)
Chapter 6 Foliations
77(14)
§6.1 Examples
77(4)
§6.2 The leaf space
81(1)
§6.3 Seifert fibered spaces
81(6)
§6.4 R-covered foliations
87(2)
§6.5 The universal circle
89(2)
Chapter 7 Left-orderings of the braid groups
91(24)
§7.1 Orderability properties of the braid groups
93(2)
§7.2 The Dehornoy ordering of Bn
95(2)
§7.3 Thurston's orderings of Bn
97(11)
§7.4 Applications of the Dehornoy ordering to knot theory
108(7)
Chapter 8 Groups of homeomorphisms
115(6)
§8.1 Homeomorphisms of a space
115(1)
§8.2 PL homeomorphisms of the cube
116(2)
§8.3 Proof of Theorem 8.6
118(1)
§8.4 Generalizations
119(1)
§8.5 Homeomorphisms of the cube
120(1)
Chapter 9 Conradian left-orderings and local indicability
121(10)
§9.1 The defining property of a Conradian ordering
122(4)
§9.2 Characterizations of Conradian left-orderability
126(5)
Chapter 10 Spaces of orderings
131(16)
§10.1 The natural actions on LO(G)
132(1)
§10.2 Orderings of Zn and Sikora's theorem
133(2)
§10.3 Examples of groups without isolated orderings
135(2)
§10.4 The space of orderings of a free product
137(2)
§10.5 Examples of groups with isolated orderings
139(1)
§10.6 The number of orderings of a group
140(4)
§10.7 Recurrent orderings and a theorem of Witte-Morris
144(3)
Bibliography 147(6)
Index 153
Adam Clay, University of Manitoba, Winnipeg, MB, Canada.

Dale Rolfsen, University of British Columbia, Vancouver, BC, Canada.
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