Control and relaxation over the circle

We formulate and prove a geometric version of the Fundamental Theorem of Al gebraic K-Theory which relates the K-theory of the Laurent polynomial exten sion of a ring to the K-thcory of the ring. The geometric version relates t he higher simple homotopy theory of the product of a finite complex and a c ircle with that of the complex. By using methods of controlled topology, we also obtain a geometric version of the Fundamental Theorem of Lower Algebr aic K-Theory. The main new innovation is a geometrically defined Nil space.