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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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  • [1] R. H. Bing, A hereditarily infinite-dimensional space, General Topology and its Relation to Modern Analysis and Algebra. II (Proc. Second Prague Topological Sympos., 1966), Academia, Prague, 1967, pp. 56-62. MR 0233336 (38:1658)
  • [2] D. W. Henderson, An infinite-dimensional compactum with no positive-dimensional compact subset--a simpler construction, Amer. J. Math. 89 (1967), 105-121. MR 0210072 (35:967)
  • [3] -, Each strongly infinite-dimensional compactum contains a hereditarily infinite-dimensional compact subset, Amer. J. Math. 89 (1967), 122-123. MR 0210073 (35:968)
  • [4] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1941. MR 0006493 (3:312b)
  • [5] V. I. Kuz'minov, Homological dimension theory, Russian Math. Surveys 23 (1968), no. 5, 1-45. MR 0240813 (39:2158)
  • [6] L. Rubin, R. Schori and J. Walsh, New dimension-theory techniques for constructing infinite-dimensional examples, General Topology and Appl. 10 (1979), 93-102. MR 519716 (80e:54049)
  • [7] R. Schori and J. Walsh, Examples of hereditarily strongly infinite-dimensional compacta, Topology Proc. 3 (1978), 495-506. MR 540508 (80m:54051)
  • [8] J. J. Walsh, A class of spaces with infinite cohomological dimension, Michigan Math. J. 27 (1980), 215-222. MR 568642 (82j:55001)
  • [9] -, Infinite dimensional compacta containing no $ n$-dimensional $ (n \geqslant 1)$ subsets, Topology 18 (1979), 91-95. MR 528239 (80e:54050)
  • [10] A. V. Zarelua, On hereditarily infinite dimensional spaces, Theory of Sets and Topology (Memorial volume in honor of Felix Hausdorff), edited by G. Asser, J. Glachsmeyer and W. Rinow, VEB Deutscher Verlag der Wissenschaften, Berlin, 1971, pp. 509-525. (Russian) MR 0343252 (49:7996)
  • [11] -, Construction of strongly infinite-dimensional compacta by means of rings of continuous functions, Soviet Math. Dokl. 15 (1974), 106-110.

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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

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