Haken
–manifolds
have been defined and studied by B Foozwell and H Rubinstein
in analogy with the classical Haken manifolds of dimension
,
based upon the theory of boundary patterns developed by K Johannson. The Euler
characteristic of a Haken manifold is analyzed and shown to be equal to the sum of
the Charney–Davis invariants of the duals of the boundary complexes of the
–cells
at the end of a hierarchy. These dual complexes are shown to be flag complexes. It
follows that the Charney–Davis conjecture is equivalent to the Euler characteristic sign
conjecture for Haken manifolds. Since the Charney–Davis invariant of a flag simplicial
–sphere
is known to be nonnegative it follows that a closed Haken
–manifold
has nonnegative Euler characteristic. These results hold as well for generalized Haken
manifolds whose hierarchies can end with compact contractible manifolds rather than
cells.