Topological semigroups and universal spaces related to
extension dimension

A. Chigogidze; A. Karasev; M. Zarichnyi

It is proved that there is no structure of left (right) cancellative semigroup on $[L]$-dimensional universal space for the class of compact metrizable spaces of extensional dimension $\le[L]$. Besides, we show that the homeomorphism group of every locally compact separable metric space whose every nonempty open subset is universal for the class of compact metric $[L]$-dimensional spaces is almost 0-dimensional and, therefore, at most one-dimensional.

1. A. Chigogidze, Cohomological dimension of Tychonov spaces. Topology Appl. 79 (1997), no. 3, 197–228.

2. A. Chigogidze, V. Valov, Universal metric spaces and extension dimension. Geometric topology: Dubrovnik 1998. Topology Appl. 79 (1997), no. 3, 197–228.

3. A. Dranishnikov, J. Dydak, Extension dimension and extension types. Tr. Mat. Inst. Steklova 212 (1996), Otobrazh. i Razmer., 61–94; translation in Proc. Steklov Inst. Math. 212 (1996), no. 1, 55–88.

4. A. Chigogidze, M. Zarichnyi, On absolute extensors modulo a complex. Topology Appl. 86 (1998), no. 2, 169–178.

5. A. N. Dranishnikov, The Eilenberg-Borsuk theorem for mappings in an arbitrary complex. (Russian) Mat. Sb. 185 (1994), no. 4, 81–90.

6. A. Chigogidze, K. Kawamura, E. D. Tymchatyn, Menger manifolds. Continua (Cincinnati, OH, 1994), 37–88, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995.

7. L. G. Oversteegen, E. D. Tymchatyn, On the dimension of certain totally disconnected spaces. Proc. Amer. Math. Soc. 122 (1994), no. 3, 885–891.

8. B. L. Brechner, On the dimensions of certain spaces of homeomorphisms. Trans. Amer. Math. Soc. 121 (1966) 516–548.

9. M. Zarichnyi, On topological inverse semigroups homeomorphic to manifolds, In: Algebra and Topology (M. Komarnytskyi, Ed.), Kyiv (1993), 50–53.

	Topological semigroups and universal spaces related to extension dimension
Author	A. Chigogidze, A. Karasev, M. Zarichnyi chigogid@math.usask.ca, topos@franko.lviv.ua Department of Mathematics and Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada, Department of Mechanics and Mathematics, Lviv National University, Universitetska 1, 79000, Lviv, Ukraine
Abstract	It is proved that there is no structure of left (right) cancellative semigroup on $[L]$-dimensional universal space for the class of compact metrizable spaces of extensional dimension $\le[L]$. Besides, we show that the homeomorphism group of every locally compact separable metric space whose every nonempty open subset is universal for the class of compact metric $[L]$-dimensional spaces is almost 0-dimensional and, therefore, at most one-dimensional.
Keywords	right cancellative semigroup, universal spaces, compact metrizable spaces, extensional dimension, homeomorphism groups, locally compact separable metric spaces
DOI	doi:10.30970/ms.16.2.195-198
Reference	1. A. Chigogidze, Cohomological dimension of Tychonov spaces. Topology Appl. 79 (1997), no. 3, 197–228. 2. A. Chigogidze, V. Valov, Universal metric spaces and extension dimension. Geometric topology: Dubrovnik 1998. Topology Appl. 79 (1997), no. 3, 197–228. 3. A. Dranishnikov, J. Dydak, Extension dimension and extension types. Tr. Mat. Inst. Steklova 212 (1996), Otobrazh. i Razmer., 61–94; translation in Proc. Steklov Inst. Math. 212 (1996), no. 1, 55–88. 4. A. Chigogidze, M. Zarichnyi, On absolute extensors modulo a complex. Topology Appl. 86 (1998), no. 2, 169–178. 5. A. N. Dranishnikov, The Eilenberg-Borsuk theorem for mappings in an arbitrary complex. (Russian) Mat. Sb. 185 (1994), no. 4, 81–90. 6. A. Chigogidze, K. Kawamura, E. D. Tymchatyn, Menger manifolds. Continua (Cincinnati, OH, 1994), 37–88, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995. 7. L. G. Oversteegen, E. D. Tymchatyn, On the dimension of certain totally disconnected spaces. Proc. Amer. Math. Soc. 122 (1994), no. 3, 885–891. 8. B. L. Brechner, On the dimensions of certain spaces of homeomorphisms. Trans. Amer. Math. Soc. 121 (1966) 516–548. 9. M. Zarichnyi, On topological inverse semigroups homeomorphic to manifolds, In: Algebra and Topology (M. Komarnytskyi, Ed.), Kyiv (1993), 50–53.
Pages	195-198
Volume	16
Issue	2
Year	2001
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Topological semigroups and universal spaces related to extension dimension