This book discusses the necessary tools to compute the lower algebraic K-theory of the integral group ring for the pure braid groups on the 2- sphere and on the real projective plane. We begin with the statement of the fibered isomorphism conjecture of Farrell-Jones through the definitions of all necessary ingredients for the actual computations. We illustrate the defnitions with specific examples used later on to discuss the proof of the main results of this work. Consider the 2- sphere or the real projective plane and let PBn(M) and Bn(M) be the pure and the full braid groups on n-strands...