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Partially acyclic manifold decompositions yielding generalized manifolds


Author: David F. Snyder
Journal: Trans. Amer. Math. Soc. 325 (1991), 531-571
MSC: Primary 57N15; Secondary 55M25, 55N30, 57P05
DOI: https://doi.org/10.1090/S0002-9947-1991-1074150-2
MathSciNet review: 1074150
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1074150-2
Keywords: Upper semicontinuous decomposition, Leray sheaf, spectral sequence, homology sphere
Article copyright: © Copyright 1991 American Mathematical Society

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