Sheaf theoretic cohomological dimension and finitistic spaces

Author:
Satya Deo

Journal:
Proc. Amer. Math. Soc. **86** (1982), 545-550

MSC:
Primary 55N30; Secondary 54F45, 55M10

DOI:
https://doi.org/10.1090/S0002-9939-1982-0671233-3

MathSciNet review:
671233

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Abstract | References | Similar Articles | Additional Information

Abstract: For a topological -manifold , we proved earlier [**7**] that , if ; and, for a zero-dimensional manifold (discrete space) we observed that . The question was later raised as to what are those paracompact spaces, besides discrete one, for which . In this paper we prove that there is none, i.e., if is not discrete then . Another question which cropped up only recently in the cohomological theory of topological transformation groups is whether or not there exists a finitistic space which is not of finite (sheaf theoretic) integral cohomological dimension. We show that this question is related to a famous unsolved problem of cohomological dimension theory.

**[1]**P. S. Alexandroff,*Some old problems in homological dimension theory*, Proc. Internat. Sympos. Topology Appl., Herceg-Novi, 1968, pp. 38-42. MR**0273607 (42:8485)****[2]**G. E. Bredon,*Sheaf theory*, Mc-Graw Hill, New York, 1967. MR**0221500 (36:4552)****[3]**-,*Introduction to compact transformation groups*, Academic Press, New York, 1972. MR**0413144 (54:1265)****[4]**A. Borel, et al.,*Seminar on transformation groups*, Ann. of Math. Studies, No. 46, Princeton Univ. Press, Princeton, N. J., 1960. MR**0116341 (22:7129)****[5]**H. Cartan, Seminar E.N.S., 1950-51.**[6]**H. Cohen,*A cohomological definition of dimension for locally compact spaces*, Duke Math. J.**21**(1954), 209-224. MR**0066637 (16:609b)****[7]**S. Deo,*Cohomological dimension of a**-manifold is*, Pacific J. Math.**67**(1976), 155-160. MR**0431184 (55:4186)****[8]**S. Deo and A. S. Shukla,*Cohomological dimension and the sum theorems*(submitted).**[9]**S. Deo and H. S. Tripathi,*Compact Lie group actions on finitistic spaces*, Topology (to appear). MR**670743 (83k:54042)****[10]**L. Gillman and M. Jerison,*Rings of continuous functions*, Springer-Verlag, New York and Berlin, 1960. MR**0116199 (22:6994)****[11]**Wu-Yi Hsiang,*Cohomological theory of topological transformation groups*, Springer-Verlag, New York and Berlin, 1975. MR**0423384 (54:11363)****[12]**K. Nagami,*Dimension theory*, Academic Press, New York, 1972. MR**0271918 (42:6799)****[13]**A. Okuyama,*On cohomological dimension of paracompact Hausdorff spaces*, Proc. Japan Acad.**38**(1962), 489-494. MR**0150770 (27:757)****[14]**R. Oliver,*A proof of Conner conjecture*, Ann. of Math. (2)**103**(1976), 637-644. MR**0415650 (54:3730)****[15]**R. G. Swan,*A new method in fixed point theory*, Comment. Math. Helv.**34**(1960), 1-16. MR**0115176 (22:5978)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0671233-3

Keywords:
Sheaves,
family of supports,
sheaf theoretic cohomological dimension,
real-compactness,
finitistic spaces

Article copyright:
© Copyright 1982
American Mathematical Society