The Whitehead Group and the Lower Algebraic K-theory of Braid Groups on S2 and R, Page 2
Let M be the 2-sphere or the 2-projective plane. If G is a braid group of M, we show that G satisfies the Farrell-Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic K-theory for these groups. The main results are that for the 2-sphere the lower algebraic K-groups vanish and for the 2-projective plane also vanishes, except for two cases.
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Braid Groups on S2 and RP2 satisfy the FIC
Ingredients for the Computations
The Whitehead group of PBnS2 and PBnRP2
ˆCa 2-projective 2-sphere action of Q8 algebraic K-theory Binghamton University cartesian square Cayley graph compact connected compute the Whitehead condition conjugation deﬁne deﬁnition denote described discrete group example family of ﬁnite family of virtually Farrell Farrell-Jones Fibered Isomorphism Conjecture finite ﬁnite group ﬁnite subgroups ﬁnitely generated free ﬁrst following diagram following short exact following theorem free group full braid groups fundamental group GLn(R group homomorphism group of n-strands Hence Hi+1/Hi hnite induced map inﬁnite virtually cyclic integral group ring Lemma long exact sequence lower algebraic K-groups Nil groups normal subgroup orbit space PBn(M PBn(RP2 projective plane Proof Proposition pure braid group quaternion group recall the following relative assembly map ri+1 satisfies the FIC short exact sequence spectral sequence spf group strongly poly-free group surface vanishes veriﬁed vertex virtually cyclic groups virtually cyclic subgroups Waldhausen Whitehead group