Homological embedding properties of the fibers of a map and the dimension of its image
HTML articles powered by AMS MathViewer
- by John J. Walsh PDF
- Proc. Amer. Math. Soc. 85 (1982), 135-138 Request permission
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 $\check {H}^i (g^{-1}(y);G) \ne 0$.References
-
F. D. Ancel, On ANR’s and cell-like maps, Seminar Notes, University of Oklahoma, 1980.
- R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200–216. MR 205212, DOI 10.1090/S0002-9947-1967-0205212-3
- T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357
- T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399–426. MR 283828, DOI 10.1090/S0002-9947-1971-0283828-7
- Robert J. Daverman, Detecting the disjoint disks property, Pacific J. Math. 93 (1981), no. 2, 277–298. MR 623564
- Robert J. Daverman and John J. Walsh, Čech homology characterizations of infinite-dimensional manifolds, Amer. J. Math. 103 (1981), no. 3, 411–435. MR 618319, DOI 10.2307/2374099
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- I. A. Vaiĭnšteĭn, On closed mappings of metric spaces, Doklady Akad. Nauk SSSR (N.S.) 57 (1947), 319–321 (Russian). MR 0022067
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 135-138
- MSC: Primary 54F45; Secondary 55M10, 58B05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0647912-0
- MathSciNet review: 647912