The e invariant and transfer map Abstract: In this thesis we study a refined Borel regulator from algebraic K theory of complex numbers to topological K theory with coefficients in C Z which is constructed by Karoubi Jones and Westbury. <br><br>We investigate the homotopy properties of this regulator and its relation with the xi invariant of flat vector bundles. We prove that their construction is equivalent to the one given by Atiyah Patodi and Singer and explain how this regulator generalizes the Adams e invariant. The possible homotopy theoretic constructions of this regulator are also discussed. In particular we show that there is a unique infinite loop space map from algebraic K theory of complex numbers to topological K theory with coefficients in C Z whose induced homomorphism between homotopy groups is the same as this regulator. The result implies an index theorem analogous to the Bismut Lott index theorem