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), 135138
MSC:
Primary 54F45; Secondary 55M10, 58B05
MathSciNet review:
647912
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A relationship is established between the homological codimension of the point inverses of a map and the dimension of its image. An infinitedimensional version leads to the conclusion that the image of a proper map defined on Hilbert space cannot be countable dimensional. A finitedimensional version yields: if is a proper map, is a orientable manifold without boundary, and , then there is a point and an integer such that .
 [1]
F. D. Ancel, On ANR's and celllike maps, Seminar Notes, University of Oklahoma, 1980.
 [2]
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 0205212
(34 #5045), http://dx.doi.org/10.1090/S00029947196702052123
 [3]
T.
A. Chapman, Lectures on Hilbert cube manifolds, American
Mathematical Society, Providence, R. I., 1976. Expository lectures from the
CBMS Regional Conference held at Guilford College, October 1115, 1975;
Regional Conference Series in Mathematics, No. 28. MR 0423357
(54 #11336)
 [4]
T.
A. Chapman, Dense sigmacompact subsets of
infinitedimensional manifolds, Trans. Amer.
Math. Soc. 154
(1971), 399–426. MR 0283828
(44 #1058), http://dx.doi.org/10.1090/S00029947197102838287
 [5]
Robert
J. Daverman, Detecting the disjoint disks property, Pacific J.
Math. 93 (1981), no. 2, 277–298. MR 623564
(82k:57007)
 [6]
Robert
J. Daverman and John
J. Walsh, Čech homology characterizations of
infinitedimensional manifolds, Amer. J. Math. 103
(1981), no. 3, 411–435. MR 618319
(83k:57008), http://dx.doi.org/10.2307/2374099
 [7]
Witold
Hurewicz and Henry
Wallman, Dimension Theory, Princeton Mathematical Series, v.
4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
(3,312b)
 [8]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [9]
I.
A. Vaiĭnšteĭn, On closed mappings of metric
spaces, Doklady Akad. Nauk SSSR (N.S.) 57 (1947),
319–321 (Russian). MR 0022067
(9,153b)
 [1]
 F. D. Ancel, On ANR's and celllike maps, Seminar Notes, University of Oklahoma, 1980.
 [2]
 R. D. Anderson, Topological properties of the Hilbert cube and the infinite product of open intervals, Trans. Amer. Math. Soc. 126 (1967), 200216. MR 0205212 (34:5045)
 [3]
 T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R.I., 1976. MR 0423357 (54:11336)
 [4]
 , Dense sigmacompact subsets of infinitedimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399426. MR 0283828 (44:1058)
 [5]
 R. J. Daverman, Detecting the disjoint disks property, Pacific J. Math. 93 (1981), 277298. MR 623564 (82k:57007)
 [6]
 R. J. Daverman and J. J. Walsh, Čech homology characterizations of infinite dimensional manifolds, Amer. J. Math. 103 (1981), 411435. MR 618319 (83k:57008)
 [7]
 W. Hurewicz and H. Wallmann, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1948. MR 0006493 (3:312b)
 [8]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
 [9]
 I. A. Vaĭnšteĭn, On closed mappings of metric spaces, Dokl. Akad. Nauk SSSR 57 (1947), 319321. MR 0022067 (9:153b)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
54F45,
55M10,
58B05
Retrieve articles in all journals
with MSC:
54F45,
55M10,
58B05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198206479120
PII:
S 00029939(1982)06479120
Keywords:
set,
infinite codimension,
countable dimensional,
homological embedding properties,
dimension,
Hilbert space,
Hilbert cube
Article copyright:
© Copyright 1982
American Mathematical Society
