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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Detecting cohomologically stable mappings


Author: Philip L. Bowers
Journal: Proc. Amer. Math. Soc. 86 (1982), 679-684
MSC: Primary 54F45; Secondary 55M10
DOI: https://doi.org/10.1090/S0002-9939-1982-0674105-3
MathSciNet review: 674105
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Abstract: Let $ f$ be a cohomologically stable mapping defined from a compactum $ X$ to the $ (n + 1)[ - {\text{cell}}{I^{n + 1}}$, let $ \pi :{I^{n + 1}} \to {I^n}$ be the projection, and let $ A = {I^n} \times \{ 1\} $ and $ B = {I^n} \times \{ - 1\} $ be opposite faces of $ {I^{n + 1}}$. If $ S$ is a separator or a continuum-wise separator of $ {f^{ - 1}}(A)$ and $ {f^{ - 1}}(B)$ in $ X$, then $ \pi f \vert S$ is cohomologically stable. This result is used to extend certain computations of cohomological dimension that are due to Walsh, who considered only the special case of the identity mapping on $ {I^{n + 1}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0674105-3
Keywords: Cohomological dimension, cohomologically stable mapping, Eilenberg-Mac Lane space
Article copyright: © Copyright 1982 American Mathematical Society