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Homological embedding properties of the fibers of a map and the dimension of its image

Author: John J. Walsh
Journal: Proc. Amer. Math. Soc. 85 (1982), 135-138
MSC: Primary 54F45; Secondary 55M10, 58B05
MathSciNet review: 647912
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Abstract: A relationship is established between the homological codimension of the point inverses of a map and the dimension of its image. An infinite-dimensional version leads to the conclusion that the image of a proper map defined on Hilbert space cannot be countable dimensional. A finite-dimensional version yields: if $ g:{M^n} \to Y$ is a proper map, $ {M^n}$ is a $ G$-orientable $ n$-manifold without boundary, and $ \dim Y \leqslant k$, then there is a point $ y \in Y$ and an integer $ i \geqslant n - k$ such that $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} ^i}({g^{ - 1}}(y);G) \ne 0$.

References [Enhancements On Off] (What's this?)

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Keywords: $ Z$-set, infinite codimension, countable dimensional, homological embedding properties, dimension, Hilbert space, Hilbert cube
Article copyright: © Copyright 1982 American Mathematical Society

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