A classification scheme for cellular decompositions of manifolds

AbstractGeneral position properties are introduced which measure the complexity of 2-cell images inherent in decomposition spaces which are not manifolds. It is shown that cell-like upper semicontinuous decompositions of n-manifolds, n ⩾ 5, are cellular if and only if they satisfy one of these properties. A classification scheme for cellular decompositions of manifolds is obtained by using these properties along with the intrinsic dimension of the decomposition. This scheme relies on topological properties of the decomposition space, rather than on the particular decomposition giving rise to it. A principal result obtained is that a cellular decomposition of as n-manifold, n ⩾ 5, is not secretly d-dimensional only if there are (d + 3) 2-cell images in the decomposition space which intersect in an “essential” manner. This generalizes one implication of the Cell-like Approximation Theorem of R.D. Edwards