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Birman J.S. - Braids, Links, and Mapping Class Groups. :: Электронная библиотека попечительского совета мехмата МГУ
 
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Birman J.S. - Braids, Links, and Mapping Class Groups.
Birman J.S. - Braids, Links, and Mapping Class Groups.

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Название: Braids, Links, and Mapping Class Groups.

Автор: Birman J.S.

Аннотация:

This manuscript is based upon lectures given at Princeton University during the fall semester of 1971-72. The central theme is Artin's braid group, and the many ways that the notion of a braid has proved to be important in low dimensional topology.
Chapter 1 is concerned with the concept of a braid as a group of motions of points in a manifold. Structural and algebraic properties of tht braid groups of two manifolds are studied, and systems of defining relations are derived for the braid groups of the plane and sphere. Chapter 2 focuses on the connections between the classical braid group and the classical knot problem. This is an area of research which has not pro-gressed rapidly, yet there seem to be many interesting questions. The basic results are reviewed, and we then go on to prove an important theorem which was announced by Markov in 1935 but never proved in detail. This is followed by a discussion of a much newer result, Garside's solution to the conjugacy problem in the braid group. The last section of Chapter 2 explores some of the possible implications of the Garside and
Markov theorems.
In Chapter 3 we discuss matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. In Chapter 4, we give an overview of recent results on the connections between braid groups and mapping class groups of surfaces. Finally, in Chapter 5, we discuss briefly the theory of "plats." The Appendix contains a list of problems. All are of a research nature, many of unknown difficulty.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1975

Количество страниц: 228

Добавлена в каталог: 23.06.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$E^{2}$, braid group of      5 17
$E^{2}$, center      28
$E^{2}$, presentation      18
$E^{2}$, pure braid group      20
$E^{2}$, representation in $Aut F_{n]$      25
$S^{2}$, braid group of      34
$S^{2}$, mapping class group of      161 164
Alexander matrix      120
Alexander polynomial of a positive link      101 144-147
Alexander polynomial of closed braid      122 126
Algebraic link problem      38 67 72 94
Algebraic plat problem      202 205
Art in braid group      see '$E^{2}$ braid
Automorphism group of $F_{n}$      see 'Representations of $B_{n}$'
Axis of closed braid      39
Base of diagram of a positive word in $B_{n}$      76
Braid group of $E^{2}$      see '$E^{2}$ braid
Braid group of $S^{2}$      see '$S^{2}$ braid
Braid groups of 2-manifolds      35
Braid groups of manifolds of dimension > 2      15
Braid index      96 200
Braid relations      18
Braid string      6
Branched covering space      181
Bridge representation of a link      197
Burau representation of $B_{n}$      118 121
Burau representation of $B_{n}$, Faithfulness for $n\geq4?$      131-143
Burau representation of $B_{n}$, Faithfulness for n = 3      129
Burau representation of $B_{n}$, Faithfulness for n = 4?      141
Center of $B_{n}$      28
Center of $B_{n}$ of $\pi_{1}B_{0,n}S^{2}$      154
Center of $B_{n}$ of $\pi_{1}F_{0,n}T_{g}$      153
Chain rule of free calculus      105
Classical braid group      see '$E^{2}$ braid
Closed braid      37 41 42
Combinatorial equivalence      39 49
Combing a braid      21-25
Configuration spaces      11
Conjugacy problem in $B_{n}$      38 69
Crookedness of a knot      199
Cyclic diagram      89
Deformation chain      56
Dehn twists      158 165 167 182
Dehn twists, properties of      170 172 175 179
Diagram of a word      73
Elementary braids      10
Elementary deformations of links      39 54
Elementary deformations of links, type $\mathcal{E}$      39
Elementary deformations of links, type $\mathcal{R}$      49
Elementary deformations of links, type $\mathcal{S}$      42
Elementary deformations of links, type $\mathcal{T}$      53
Elementary deformations of links, type $\mathcal{W}$      51
Equivalence of braids      7-10
Exact sequence of fibration $F_{0,n}\rightarrow F_{0,n-1}$      14 23
Exact sequence of fibration $\mathcal{B}_{0}T_{g}\rightarrow B_{0,n}T_{g}$      157
Exact sequence of fibration $\mathcal{F}_{0}T_{g}\rightarrow F_{0,n}T_{g}$      151
Fiber-preserving isotopies      183
Free differential calculus      103
Free group of rank n      23 27 30 48 103 133 139 141 196
Free group of rank n, geometric realization of      32 48 195 196
Fundamental formula of free calculus      106
Fundamental group of the complement of a link      46 48 196
Gassner representation of $P_{n}$      119 121
Gassner representation of $P_{n}$, faithfulness of      129-143
General position      39 52
Generators of M(0,4)      205
Generators of M(0,n)      165
Generators of M(g,0)      169
Generators of M(g,n)      160
Geometric (open) braid      6 21 40
Height of a link      40 55
Homotopy braid group      10
Initial and final routes      79
Isotopy type of a link      39
Knot      38
Lickorish twist      see 'Dehn twist'
Link      38
Linking number      128
Magnus representations of $F_{n}$      110 113
Magnus representations of subgroups of $Aut F_{n}$      116
Mapping class group      148
Mapping class group, generators      160 165 169
Mapping class group, presentation for M(0,n)      165
Mapping class group, presentation for M(2,0)      183
Markov moves      68 95
Markov theorem      37 48 51
Motion groups      5 11
Negative edge      40
Normal matrix      209
Plat      192
Plat index      197 200
Positive edge      40
Positive link      101 144-147
Positive word      70
Power of an element in $B_{n}$      70 77
Prime to $\Delta$      76
Pure braid group      5 20
Pure link      96
Pure mapping class group      148 (see also 'Mapping class group')
Representations of $B_{n}$ in $Aut F_{n}$      25 30 32 46
Sawtooth      42
Semigroup $S_{n}$ embedded in $B_{n}$      74
Spin map      158
Summit power      71 79
Summit power, form      72
Summit power, set      79 82-84
Summit power, tail      71 79
Tail of an element in $B_{n}$      70 78
Unpermuted braid group      see 'Pure braid group'
Word problem in $B_{n}$      24 25 76
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