### Control and Relaxation over the Circle, by Bruce Hughes and Stratos Prassidis

We formulate and proove a geometric version of the Fundamental Theorem
of Algebraic K-Theory which relates the K-theory of the Laurent
polynomial extension of a ring to the K-theory of the ring.
The geometric version relates the higher simple homotopy theory of
the product of a finite complex and a circle with that of the complex.
By using methods of controlled topology, we also obtain a geometric
version of the Fundamental Theorem of Lower Algebraic K-Theory.
The main new innovation is a geometrically defined Nil space.

Bruce Hughes <hughes@math.vanderbilt.edu>

Stratos Prassidis <prassie@math.vanderbilt.edu>