  This paper studies the homotopy type of the moduli space of compact
n-manifold thickenings of a finite complex. The main result computes the
homotopy fibers of the stabilization map from n-thickenings to
(n+1)-thickenings in a wide range. In particular, we obtain an EHP-type
sequence for thickenings extending work of C.T.C. Wall.
