  Let $\Scr A$ be a unital C*-algebra. We describe \it K-skeleton
factorizations \rm of all invertible operators on a Hilbert C*-module $\Scr
H_{\Scr A}$, in particular on $\Scr H=l^2$, with the Fredholm index as an
invariant. We then outline the isomorphisms $K_0(\Scr A) \cong \pi
_{2k}([p]_0)\cong \pi _{2k} ({GL}^p_r(\Scr A))$ and $K_1(\Scr A)\cong \pi
_{2k+1}([p]_0)\cong \pi _{2k+1}(GL^p_r(\Scr A))$ for $k\ge 0 $, where $[p]_0$
denotes the class of all compact perturbations of a projection $p$ in the
infinite Grassmann space ${Gr}^{\infty }(\Scr A)$ and $GL^p_r(\Scr A)$ stands
for the group of all those invertible operators on $\Scr H_{\Scr A}$
essentially commuting with $p$.
