Partially acyclic manifold decompositions yielding generalized manifolds

Certain classes of upper semicontinuous decompositions (uscd's) are investigated. The methods established in the literature of manifold decompositions are compared with methods arising from the theory of sheaves. A wide range of results from the theories of sheaf cohomology, generalized manifolds, and ANRs are discussed in order to develop proofs of the theorems below. Examples and applications are given.

Definition : An upper semicontinuous decomposition G of a (n+k)-manifold is called a codimension-k manifold decomposition if each element of G is [a continuum having the shape of] a closed orientable n-manifold. We say G is j-acyclic if the reduced Čech homology of each element of G is trivial up through dimension j.

Theorem. Every non-degenerate (k-2)-acyclic codimension-k manifold decomposition G of an orientable (n+k)-manifold M (3≤k≤n+1) yields a k-manifold B as its decomposition space, if B is finite dimensional and if the set E (which consists of those points of B at which the Leray sheaf in dimension n of the decomposition map is not locally constant) does not locally separate B. Moreover, the set E is actually a locally finite subset of B.

Addendum: In the case k = n+1, the assumptions of non-degeneracy of G and of orientability of M are not necessary.