  We obtain a weak homotopy equivalence type result between two topological
groups associated with a Kirchberg algebra: the unitary group of the continuous
asymptotic centralizer and the loop group of the automorphism group of the
stabilization. This result plays a crucial role in our subsequent work on the
classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a
special case, we show that the $K$-groups of the continuous asymptotic
centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra.
