On a theorem of Jaworowski on locally equivariant contractible spaces
Authors:
Sergei M. Ageev and Dusan Repovs
Journal:
Proc. Amer. Math. Soc. 130 (2002), 15391550
MSC (1991):
Primary 54C55, 57S10; Secondary 54C20, 54C60, 54F65, 55M15
Published electronically:
October 5, 2001
MathSciNet review:
1879981
Fulltext PDF Free Access
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Abstract: Ancel's method of fiberwise trivial relations is applied to the problem of characterization of absolute equivariant extensors. We obtain a generalization of Jaworowski's theorem on characterization of equivariant extensors lying in to the case when the space is infinitedimensional, has infinitely many orbit types and the acting compact group is not necessarily a Lie group.
 [Ab]
Herbert
Abels, A universal proper 𝐺space, Math. Z.
159 (1978), no. 2, 143–158. MR 0501039
(58 #18504)
 [AG]
David
F. Addis and John
H. Gresham, A class of infinitedimensional spaces. I. Dimension
theory and Alexandroff’s problem, Fund. Math.
101 (1978), no. 3, 195–205. MR 521122
(80b:54041)
 [Ag]
S.
M. Ageev, An equivariant Dugundji theorem, Uspekhi Mat. Nauk
45 (1990), no. 5(275), 179–180 (Russian);
English transl., Russian Math. Surveys 45 (1990),
no. 5, 219–220. MR 1084993
(92h:54026), http://dx.doi.org/10.1070/RM1990v045n05ABEH002654
 [An]
Fredric
D. Ancel, The role of countable dimensionality
in the theory of celllike relations, Trans.
Amer. Math. Soc. 287 (1985), no. 1, 1–40. MR 766204
(86b:54012), http://dx.doi.org/10.1090/S0002994719850766204X
 [Bo]
Karol
Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44,
Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
(35 #7306)
 [Br]
Glen
E. Bredon, Introduction to compact transformation groups,
Academic Press, New York, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
(54 #1265)
 [Ca]
Robert
Cauty, Un espace métrique linéaire qui n’est
pas un rétracte absolu, Fund. Math. 146
(1994), no. 1, 85–99 (French, with English summary). MR 1305261
(95j:54022)
 [Do]
C.
H. Dowker, Homotopy extension theorems, Proc. London Math.
Soc. (3) 6 (1956), 100–116. MR 0073982
(17,518f)
 [Ha]
William
E. Haver, Locally contractible spaces that are
absolute neighborhood retracts, Proc. Amer.
Math. Soc. 40
(1973), 280–284. MR 0331311
(48 #9645), http://dx.doi.org/10.1090/S0002993919730331311X
 [Hu]
Szetsen
Hu, Theory of retracts, Wayne State University Press, Detroit,
1965. MR
0181977 (31 #6202)
 [Ja]
Jan
W. Jaworowski, Extensions of 𝐺maps and Euclidean
𝐺retracts, Math. Z. 146 (1976), no. 2,
143–148. MR 0394550
(52 #15351)
 [Ma]
Sibe
Mardešić, Approximate polyhedra, resolutions of maps
and shape fibrations, Fund. Math. 114 (1981),
no. 1, 53–78. MR 643305
(83d:55016)
 [Mx]
Charles
N. Maxwell, Homomorphisms of topological transformation groups into
function spaces, Duke Math. J. 33 (1966),
567–574. MR 0203714
(34 #3563)
 [Mu]
Mitutaka
Murayama, On 𝐺ANRs and their 𝐺homotopy
types, Osaka J. Math. 20 (1983), no. 3,
479–512. MR
718960 (85f:57022)
 [Pa]
Richard
S. Palais, The classification of 𝐺spaces, Mem. Amer.
Math. Soc. No. 36, 1960. MR 0177401
(31 #1664)
 [RS]
Dušan
Repovš and Pavel
Vladimirovič Semenov, Continuous selections of multivalued
mappings, Mathematics and its Applications, vol. 455, Kluwer
Academic Publishers, Dordrecht, 1998. MR 1659914
(2000a:54002)
 [To]
H.
Toruńczyk, Concerning locally homotopy negligible sets and
characterization of 𝑙₂manifolds, Fund. Math.
101 (1978), no. 2, 93–110. MR 518344
(80g:57019)
 [Vi]
G. Villalobos, Inversion of the theorem on the existence of slices, Vestnik MSU (1) (2000), 2629 (in Russian).
 [Ab]
 H. Abels, A universal proper space, Math. Z. 159 (1978), 143158. MR 58:18504
 [AG]
 D. F. Addis and J. H. Gresham, A class of infinitedimensional spaces I, Dimension theory and Aleksandroff's problem, Fund. Math. 101 (1978), 195205. MR 80b:54041
 [Ag]
 S. M. Ageev, An equivariant Dugundji theorem, Uspehi Mat. Nauk 45:5 (1990), 116117 (in Russian); Engl. transl. in: Russ. Math. Surv. 45:5 (1990), 219220. MR 92h:54026
 [An]
 F. D. Ancel, The role of countable dimensionality in the theory of celllike relations, Trans. Amer. Math. Soc. 287 (1985), 140. MR 86b:54012
 [Bo]
 K. Borsuk, Theory of Retracts, Monogr. Matem. 44, PWN, Warsaw, 1967. MR 35:7306
 [Br]
 G. E. Bredon, Introduction to Compact Transformation Groups, Pure and Appl. Math. 46, Academic Press, New YorkLondon, 1972. MR 54:1265
 [Ca]
 R. Cauty, Un espace metrique lineaire qui n'est pas un retracte absolu, Fund. Math. 146 (1994), 8599. MR 95j:54022
 [Do]
 C. H. Dowker, Homotopy extension theorems, Proc. London Math. Soc. 6 (1956), 100116. MR 17:518f
 [Ha]
 W. E. Haver, Locally contractible spaces that are absolute neighborhood retracts, Proc. Amer. Math. Soc. 40 (1973), 280284. MR 48:9645
 [Hu]
 S.T. Hu, Theory of Retracts, Wayne State Univ. Press, Detroit, 1965. MR 31:6202
 [Ja]
 J. Jaworowski, Extensions of maps and Euclidean retracts, Math. Z. 146 (1976), 143148. MR 52:15351
 [Ma]
 S. Mardesic, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), 5378. MR 83d:55016
 [Mx]
 C. N. Maxwell, Homeomorphisms of topological transformation groups into function spaces, Duke Math. J. 33 (1966), 567574. MR 34:3563
 [Mu]
 T. Murayama, On 's and their homotopy types, Osaka J. Math. 20 (1983), 479512. MR 85f:57022
 [Pa]
 R. S. Palais, The classification of spaces, Memoirs Amer. Math. Soc. 36 (1960). MR 31:1664
 [RS]
 D. Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Mathematics and Its Appl. 455, Kluwer, Dordrecht, 1998. MR 2000a:54002
 [To]
 H. Torunczyk, Concerning locally homotopy negligible sets and characterization of manifolds, Fund. Math. 101 (1978), 93110. MR 80g:57019
 [Vi]
 G. Villalobos, Inversion of the theorem on the existence of slices, Vestnik MSU (1) (2000), 2629 (in Russian).
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Additional Information
Sergei M. Ageev
Affiliation:
Department of Mathematics, Brest State University, Brest, Belarus 224011
Email:
ageev@highmath.brsu.brest.by
Dusan Repovs
Affiliation:
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001
Email:
dusan.repovs@unilj.si
DOI:
http://dx.doi.org/10.1090/S0002993901061846
PII:
S 00029939(01)061846
Keywords:
Local equivariant contractibility,
absolute equivariant extensor,
fiberwise trivial relation,
countable dimensional orbit space,
selections of multivalued maps
Received by editor(s):
May 17, 2000
Received by editor(s) in revised form:
November 10, 2000
Published electronically:
October 5, 2001
Communicated by:
Alan Dow
Article copyright:
© Copyright 2001 American Mathematical Society
