Extension theory of separable metrizable spaces with applications to dimension theory
HTML articles powered by AMS MathViewer
- by Alexander Dranishnikov and Jerzy Dydak PDF
- Trans. Amer. Math. Soc. 353 (2001), 133-156 Request permission
Abstract:
The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results:
Generalized Eilenberg-Borsuk Theorem. Let $L$ be a countable CW complex. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$ for some CW complex $K$, then for any map $f:A\to K$, $A$ closed in $X$, there is an extension $f’:U\to K$ of $f$ over an open set $U$ such that $L\in AE(X-U)$.
Theorem. Let $K,L$ be countable CW complexes. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$, then there is a subset $Y$ of $X$ such that $K\in AE(Y)$ and $L\in AE(X-Y)$.
Theorem. Suppose $G_{i},\ldots ,G_{n}$ are countable, non-trivial, abelian groups and $k>0$. For any separable metrizable space $X$ of finite dimension $\dim \! X>0$, there is a closed subset $Y$ of $X$ with $\dim _{G_{i}}\! Y=\max (\dim _{G_{i}}\! X-k,1)$ for $i=1,\ldots ,n$.
Theorem. Suppose $W$ is a separable metrizable space of finite dimension and $Y$ is a compactum of finite dimension. Then, for any $k$, $0<k<\dim \! W-\dim \! Y$, there is a closed subset $T$ of $W$ such that $\dim \! T=\dim \! W-k$ and $\dim (T\times Y)=\dim (W\times Y)-k$.
Theorem. Suppose $X$ is a metrizable space of finite dimension and $Y$ is a compactum of finite dimension. If $K\in AE(X)$ and $L\in AE(Y)$ are connected CW complexes, then $K\wedge L\in AE(X\times Y).$
References
- P.S.Alexandroff, Dimensionstheorie, Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161–238.
- K.Borsuk, Un theoreme sur la prolongements des transformations, Fund. Math. 29 (1937), 161–166.
- Robert Cauty, Sur les sous-espaces des complexes simpliciaux, Bull. Soc. Math. France 100 (1972), 129–155 (French). MR 326680
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Jerzy Dydak and Jerzy Mogilski, Universal cell-like maps, Proc. Amer. Math. Soc. 122 (1994), no. 3, 943–948. MR 1242080, DOI 10.1090/S0002-9939-1994-1242080-7
- A. N. Dranishnikov, Homological dimension theory, Uspekhi Mat. Nauk 43 (1988), no. 4(262), 11–55, 255 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 4, 11–63. MR 969565, DOI 10.1070/RM1988v043n04ABEH001900
- A. N. Dranishnikov, On the mapping intersection problem, Pacific J. Math. 173 (1996), no. 2, 403–412. MR 1394397
- Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
- A. N. Dranishnikov, The Eilenberg-Borsuk theorem for mappings in an arbitrary complex, Mat. Sb. 185 (1994), no. 4, 81–90 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 81 (1995), no. 2, 467–475. MR 1272187, DOI 10.1070/SM1995v081n02ABEH003546
- A. N. Dranishnikov, On intersection of compacta in Euclidean space. II, Proc. Amer. Math. Soc. 113 (1991), no. 4, 1149–1154. MR 1060721, DOI 10.1090/S0002-9939-1991-1060721-1
- A. N. Dranishnikov, The cohomological dimension is not preserved under the Stone-Čech compactification, C. R. Acad. Bulgare Sci. 41 (1988), no. 12, 9–10 (Russian). MR 985881
- A.N.Dranishnikov, On the dimensions of the product, the union and the intersection of two spaces, preprint, 1995.
- Aleksander N. Dranišnikov and Dušan Repovš, The Urysohn-Menger sum formula: an extension of the Dydak-Walsh theorem to dimension one, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 273–282. MR 1346635
- A. N. Dranišnikov, D. Repovš, and E. V. Ščepin, On the failure of the Urysohn-Menger sum formula for cohomological dimension, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1267–1270. MR 1205488, DOI 10.1090/S0002-9939-1994-1205488-1
- A. N. Dranishnikov, D. Repovš, and E. V. Ščepin, Dimension of products with continua, Topology Proc. 18 (1993), 57–73. MR 1305123
- Jerzy Dydak and Jack Segal, Shape theory, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. An introduction. MR 520227
- Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
- Jerzy Dydak and John J. Walsh, Spaces without cohomological dimension preserving compactifications, Proc. Amer. Math. Soc. 113 (1991), no. 4, 1155–1162. MR 1081695, DOI 10.1090/S0002-9939-1991-1081695-3
- J.Dydak and J.J.Walsh, Aspects of cohomological dimension for principal ideal domains, preprint.
- Jerzy Dydak, Cohomological dimension and metrizable spaces, Trans. Amer. Math. Soc. 337 (1993), no. 1, 219–234. MR 1153013, DOI 10.1090/S0002-9947-1993-1153013-X
- Jerzy Dydak, Cohomological dimension and metrizable spaces. II, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1647–1661. MR 1333390, DOI 10.1090/S0002-9947-96-01536-X
- Jerzy Dydak, Union theorem for cohomological dimension: a simple counterexample, Proc. Amer. Math. Soc. 121 (1994), no. 1, 295–297. MR 1205489, DOI 10.1090/S0002-9939-1994-1205489-3
- S.Eilenberg, Un theoreme de la dualite, Fund. Math. 26 (1936), 280–282.
- Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. MR 0466081
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- V. I. Kuz′minov, Homological dimension theory, Uspehi Mat. Nauk 23 (1968), no. 5 (143), 3–49 (Russian). MR 0240813
- Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
- William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
- Wojciech Olszewski, Completion theorem for cohomological dimensions, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2261–2264. MR 1307554, DOI 10.1090/S0002-9939-1995-1307554-X
- Wojciech Olszewski, Universal separable metrizable spaces of given cohomological dimension, Topology Appl. 61 (1995), no. 3, 293–299. MR 1317083, DOI 10.1016/0166-8641(94)00036-3
- Leonard R. Rubin, Characterizing cohomological dimension: the cohomological dimension of $A\cup B$, Topology Appl. 40 (1991), no. 3, 233–263. MR 1124840, DOI 10.1016/0166-8641(91)90108-X
- Daciberg Lima Gonçalves, Braid groups and Wecken pairs, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 89–97. MR 956480, DOI 10.1090/conm/072/956480
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Dennis Sullivan, Geometric topology. Part I, Massachusetts Institute of Technology, Cambridge, Mass., 1971. Localization, periodicity, and Galois symmetry; Revised version. MR 0494074
- John J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Shape theory and geometric topology (Dubrovnik, 1981) Lecture Notes in Math., vol. 870, Springer, Berlin-New York, 1981, pp. 105–118. MR 643526
- Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508
- M.Zarichnyi, Universal spaces for cohomological dimension (to appear).
Additional Information
- Alexander Dranishnikov
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 212177
- Email: dranish@math.ufl.edu
- Jerzy Dydak
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
- Email: dydak@math.utk.edu
- Received by editor(s): July 14, 1995
- Received by editor(s) in revised form: February 5, 1999
- Published electronically: August 3, 2000
- Additional Notes: The first and second authors were supported in part by grants DMS-9696238 and DMS-9704372, respectively, from the National Science Foundation.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 133-156
- MSC (1991): Primary 55M10, 54F45
- DOI: https://doi.org/10.1090/S0002-9947-00-02536-8
- MathSciNet review: 1694287