The classification of the virtually cyclic subgroups of the sphere braid groups Daciberg Lima Gonçalves, John Guaschi

Format:

Book

Author/Creator:

Gonçalves, Daciberg Lima, author.

Guaschi, John, author.

Series:

SpringerBriefs in mathematics 2191-8198

Language:

English

Subjects (All):

Braid theory.

Algebra.

Mathematics.

Group Theory and Generalizations.

Algebraic Topology.

algebra.

Local Subjects:

Mathematics.

Group Theory and Generalizations.

Algebraic Topology.

Algebra.

Physical Description:

1 online resource

Place of Publication:

Cham Springer 2013

System Details:

PDF

text file

Summary:

This manuscript is devoted to classifying the isomorphism classes of the virtually cyclic subgroups of the braid groups of the 2-sphere. As well as enabling us to understand better the global structure of these groups, it marks an important step in the computation of the K-theory of their group rings. The classification itself is somewhat intricate, due to the rich structure of the finite subgroups of these braid groups, and is achieved by an in-depth analysis of their group-theoretical and topological properties, such as their centralisers, normalisers and cohomological periodicity. Another important aspect of our work is the close relationship of the braid groups with mapping class groups. This manuscript will serve as a reference for the study of braid groups of low-genus surfaces, and is addressed to graduate students and researchers in low-dimensional, geometric and algebraic topology and in algebra

Contents:

Introduction and statement of the main results Virtually cyclic groups: generalities, reduction and the mapping class group Realisation of the elements of V1(n) and V2(n) in Bn(S2) Appendix: The subgroups of the binary polyhedral groups

Machine generated contents note: References

2.1. Virtually Cyclic Groups: Generalities

2.2. Centralisers and Normalisers of Some Maximal Finite Subgroups of Bn(S2)

2.3. Reduction of Isomorphism Classes of Fx θZ Via Out (F)

2.4. Reduction of Isomorphism Classes of Fx θZ Via Conjugacy Classes

2.5. Reduction of Isomorphism Classes of Fx θZ Via Periodicity

2.5.1. Homotopy Type of the Configuration Spaces Fn(S2) and Dn(S2)

2.5.2. Cohomological Condition for the Realisation of Type I Virtually Cyclic Groups

2.6. Necessity of the Conditions on V1(n) and V2(n)

2.6.1. Necessity of the Conditions on V1(n)

2.6.2. Necessity of the Conditions on V2(n) References

3.1. Type I Subgroups of Bn(S2) of the form F x Z with F Cyclic

3.1.1. Type I Subgroups of the form Zq x Z

3.1.2. Type I Subgroups of the form Zq x pZ

3.2. Type I Subgroups of Bn(S2) of the form F x Z with F Dicyclic, F is not = to Q8

3.3. Type I Subgroups of Bn(S2) of the form Q8 x Z

3.4. Type I Subgroups of Bn(S2) of the form F x Z with F = T*, O*, I*

3.4.1. Type I Subgroups of Bn(S2) of the form F x Z with F = T*, O*, I*

3.4.2. Realisation of T* x ωZ

3.5. Proof of the Realisation of the Elements of V1(n) in Bn(S2)

3.6. Realisation of the Elements of V2(n) in Bn(S2)

3.6.1. Realisation of the Elements of V2(n) with Cyclic or Dicyclic Factors

3.6.2. Realisation of O* *T* O* in Bn(S2)

3.7. Proof of the Realisation of Elements of V2(n) in Bn(S2)

3.8. Isomorphism Classes of Virtually Cyclic Subgroups of Bn(S2) of Type II

3.9. Classification of the Virtually Cyclic Subgroups of MCG(S2,n) References

Notes:

Includes bibliographical references

Online resource; title from PDF title page (SpringerLink, viewed September 9, 2013)

Other Format:

Print version Gonçalves, Daciberg Lima. Classification of the virtually cyclic subgroups of the sphere braid groups

ISBN:

9783319002576

3319002570

3319002562

9783319002569

OCLC:

857765890

Access Restriction:

Restricted for use by site license