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 be a cohomologically stable mapping defined from a compactum to the , let be the projection, and let and be opposite faces of . If is a separator or a continuum-wise separator of and in , then 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 .

**[1]**R. H. Bing,*A hereditarily infinite dimensional space*, General Topology and its Relations to Modern Analysis and Algebra, II (Proc. Second Prague Topological Sympos., 1966) Academia, Prague, 1967, pp. 56–62. MR**0233336****[2]**David W. Henderson,*An infinite-dimensional compactum with no positive-dimensional compact subsets—a simpler construction*, Amer. J. Math.**89**(1967), 105–121. MR**0210072**, https://doi.org/10.2307/2373100**[3]**David W. Henderson,*Each strongly infinite-dimensional compactum contains a hereditarily infinite-dimensional compact subset*, Amer. J. Math.**89**(1967), 122–123. MR**0210073**, https://doi.org/10.2307/2373101**[4]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[5]**V. I. Kuz′minov,*Homological dimension theory*, Uspehi Mat. Nauk**23**(1968), no. 5 (143), 3–49 (Russian). MR**0240813****[6]**Leonard R. Rubin, R. M. Schori, and John J. Walsh,*New dimension-theory techniques for constructing infinite-dimensional examples*, General Topology Appl.**10**(1979), no. 1, 93–102. MR**519716****[7]**R. M. Schori and John J. Walsh,*Examples of hereditarily strongly infinite-dimensional compacta*, Proceedings of the 1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), II, 1978, pp. 495–506 (1979). MR**540508****[8]**John J. Walsh,*A class of spaces with infinite cohomological dimension*, Michigan Math. J.**27**(1980), no. 2, 215–222. MR**568642****[9]**John J. Walsh,*Infinite-dimensional compacta containing no 𝑛-dimensional (𝑛≥1) subsets*, Topology**18**(1979), no. 1, 91–95. MR**528239**, https://doi.org/10.1016/0040-9383(79)90017-X**[10]**A. V. Zarelua,*Hereditarily infinite-dimensional spaces*, Theory of sets and topology (in honour of Felix Hausdorff, 1868–1942), VEB Deutsch. Verlag Wissensch., Berlin, 1972, pp. 509–525 (Russian). MR**0343252****[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