Discontinuity in codimension-k manifold decompositions

Much has come in the study of the decompositions of manifolds having manifold fibers from the examination of the discontinuity set of the decomposition map. The main result of this paper gives a limitation to the extent of the discontinuity set. Let π:M → B be a codimension-k manifold decomposition of M, an (n + k)-manifold, into sets having the shape of closed oriented n-manifolds. Suppose that the dimension of B is finite. Then D, the discontinuity set of π, does not locally separate B. The result is proved using the Leray-Grothendieck spectral sequence for the map π along with careful manipulation of subsets of the target space B. Crucial to the proof is the manipulation of the coefficient modules used in the cohomology of both M and B. Under the assumption that the dimension of B is finite, we go on to prove that if the point preimages of the map π have trivial first homology, the dimension of D is less than or equal to k − 2 for k ⩾ 2. The same conclusion is reached if the point preimages have the shape of S1.