A Primer on Mapping Class Groups (Princeton Mathematical) by Benson Farb

By Benson Farb

The examine of the mapping category crew Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce conception. This booklet explains as many very important theorems, examples, and strategies as attainable, fast and without delay, whereas whilst giving complete information and conserving the textual content approximately self-contained. The booklet is appropriate for graduate students.The publication starts off by means of explaining the most group-theoretical houses of Mod(S), from finite iteration by way of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, crucial gadgets and instruments are brought, comparable to the Birman certain series, the advanced of curves, the braid staff, the symplectic illustration, and the Torelli workforce. The publication then introduces Teichmüller house and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston category of floor homeomorphisms. themes comprise the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov thought, and Thurston's method of the type.

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Preface ix
PART ONE 1
Chapter I units AND MAPS 3
I. units and Subsets 3
2. Maps S
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S. the picture research of a Map 10
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10. Ordered units 16
II. Axiom of selection 17
12. items and Sums of units 20
Exercises 23
Chapter 2 MONOIDS AND teams 27
1. Monoids 27
2. Morphisms of Monoids 30
3. specified different types of Morphisms 32
4. Analyses of Morphisms 37
5. Description of Surjective Morphisms 39
6. teams and Morphisms of teams 41
7. Kernels of Morphisms of teams 43
8. teams of Fractions 49
9. The Integers 55
10. Finite and countless units 57
Exercises 64
Chapter three different types 75
1. different types 75
2. Morphisms 79
3. items and Sums 82
Exercises 85
Chapter four jewelry 99
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2. Polynomial earrings 103
3. Analyses of Ring Morphisms 107
4. beliefs 112
5. items of earrings 115
Exercises 116
PART 127
Chapter five particular FACTORIZATION domain names 129
I. Divisibility 130
2. indispensable domain names 133
3. specific Factorization domain names 138
4. Divisibility in UFD\'s 140
5. primary perfect domain names 147
6. issue jewelry of PID\'s 152
7. Divisors 155
8. Localization in critical domain names 159
9. A Criterion for targeted Factorization 164
10. whilst R [X] is a UFD 169
Exercises 171
Chapter 6 basic MODULE thought 176
1. type of Modules over a hoop 178
2. The Composition Maps in Mod(R) 183
3. Analyses of R-Module Morphisms 185
4. distinctive Sequences 193
5. Isomorphism Theorems 201
6. Noetherian and Artinian Modules 206
7. loose R-Modules 210
8. Characterization of department jewelry 216
9. Rank of unfastened Modules 221
10. Complementary Submodules of a Module 224
11. Sums of Modules 231
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13. Torsion Modules over PID\'s 242
14. items of Modules 246
Exercises 248
Chapter 7 SEMISIMPLE earrings AND MODULES 266
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2. Semisimple Modules 271
3. Projective Modules 276
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Chapter eight ARTINIAN earrings 289
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2. loose Submodules of unfastened Modules 355
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Extra info for A Primer on Mapping Class Groups (Princeton Mathematical)

Example text

We can cut S0,3 along α so as to obtain a disk with one marked point (the boundary comes from α, and the marked point comes from r). Since φ preserves the orientations of S0,3 and of α, it follows that φ induces a homeomorphism φ of this disk which is the identity on the boundary (the map φ is the unique set map on the cut-open surface inducing φ). 1 the mapping class group of a once-marked disk is trivial, and so φ is homotopic to the identity. The homotopy induces a homotopy from φ to the identity.

Pairs of simple closed curves {α, β} with i(α, β) = |α ∩ β| = 2 and ˆi(α, β) = 0, and whose union does not separate. 5. Nonseparating simple proper arcs in a surface S that meet the same number of components of ∂S. 6. Chains of simple closed curves. A chain of simple closed curves in a surface S is a sequence α1 , . . , αk with the properties that i(αi , αi+1 ) = 1 for each i and i(αi , αj ) = 0 whenever |i − j| > 1. A chain is nonseparating if the union of the curves does not separate the surface.

K with the properties that i(αi , αi+1 ) = 1 for each i and i(αi , αj ) = 0 whenever |i − j| > 1. A chain is nonseparating if the union of the curves does not separate the surface. Any two nonseparating chains of simple closed curves, with the same number of curves, are topologically equivalent. 43 CURVES AND SURFACES This can be proved by induction. The starting point is the case of nonseparating simple closed curves, and the inductive step is Example 5: cutting along the first few arcs, the next arc becomes a nonseparating arc on the cut surface.

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