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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Partially acyclic manifold decompositions yielding generalized manifolds
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by David F. Snyder PDF
Trans. Amer. Math. Soc. 325 (1991), 531-571 Request permission

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
  • © Copyright 1991 American Mathematical Society
  • 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