The Classification of the Virtually Cyclic Subgroups of the Sphere Braid GroupsThis 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 isaddressed to graduate students and researchers in low-dimensional, geometric and algebraic topology and in algebra. |
Contents
| 1 | |
Generalities Reduction and the Mapping Class Group | 15 |
3 Realisation of the Elements of v1n and v2n in bn st | 51 |
Appendix
The Subgroups of the Binary Polyhedral Groups | 99 |
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Common terms and phrases
Abelianisation algebraic K-theory amalgamated products Aut F Aut(H automorphism binary polyhedral groups Bn S2 Bn(S BnðS2Þ braid groups centraliser classification commutative diagram commute pairwise completes the proof conjugacy classes conjugate defined Dehn twist denote Dica Dicam dicyclic group dicyclic subgroups Dihs element of order elements of V2(n exists F G2 finite subgroups follows form F G is isomorphic G1 and G2 given by Definition Gonçalves Guaschi Hence Hom(Z homomorphism implies infinite order infinite virtually cyclic isomorphic to Q3 isomorphism classes Lemma Let G Let H mapping class group Math maximal finite normal subgroup obtain polyhedral groups proof of Proposition Proposition 11 realised as subgroups resp result Sect short exact sequence strict divisor subgroup isomorphic subgroup of Bn(S2 subgroup of order subgroups of Bn subgroups of MCG suppose surjective Type I subgroups universal covering space virtually cyclic groups virtually cyclic subgroups Z-factor