Partially acyclic manifold decompositions yielding generalized manifolds

Abstract:

Let $G$ be an upper semicontinuous decomposition (used) of the $(n + k)$-manifold $M$ into subcontinua having the shape of closed orientable $n$-manifolds $(2 < n,k)$. We define $G$ to be $j$-acyclic if for every element $g$ of $G$ the reduced Čech homology of $g$ vanishes up through dimension $j$. The primary objective of this investigation is to determine the local connectivity properties of the decomposition space $B = M/G$ if $G$ is $(k - 2)$-acyclic and $B$ is finite dimensional. The Leray-Grothendieck spectral sequence of the decomposition map $p$ is analyzed, which relegates the principal part of the investigation to studying the structure of the Leray sheaf of $p$ and its relation to the local cohomology of $B$. Let $E$ denote the subset of $B$ over which the Leray sheaf is not locally constant, $K$ the subset of $E$ over which the Leray sheaf is not locally Hausdorff, and $D = E - K$. Then we get as our main result, which extends work of R. J. Daverman and J. J. Walsh, and generalizes a result of D. S. Coram and P. Duvall as well, Theorem. Let $G$ be a $(k - 2)$-acyclic decomposition of the $(n + k)$-manifold $M$ such that $k < n + 2$, $B = M/G$ is finite dimensional, and the set $E$ does not locally separate $B$. Then $B$ is a generalized $k$-manifold, if either $k = n + 1$, or $k < n + 1$ and $M$ is orientable.