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LECs, Local Mixers, Topological Groups and Special Products

Part of: Maps and general types of spaces defined by maps Topological and differentiable algebraic systems

Published online by Cambridge University Press:  09 April 2009

Carlos R. Borges
Carlos R. Borges
Affiliation:
Department of MathematicsUniversity of CaliforniaDavis, California 95616, U.S.A.
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Abstract

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We prove that every (locally) contractible topological group is (L)EC and apply these results to homeomorphism groups, free topological groups, reduced products and symmetric products. Our main results are: The free topological group of a θ-contractible space is equiconnected. A paracompact and weakly locally contractible space is locally equiconnected if and only if it has a local mixer. There exist compact metric contractible spaces X whose reduced (symmetric) products are not retracts of the Graev free topological groups F(X) (A(X)) (thus correcting results we published ibidem).

Keywords

θ-contractiblefree topological groupreduced productsymmetric product(L)EC(local) mixershomeomorphism groups.

MSC classification

Secondary: 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) 22A05: Structure of general topological groups 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 252 - 258
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Borges, C. R., ‘A study of absolute extensor spaces’, Pacific J. Math. 31 (1969), 609–617.CrossRefGoogle Scholar
[2]Borges, C. R., ‘Absolute extensor spaces: A correction and an answer’, Pacific J. Math. 50 (1974), 29–30.CrossRefGoogle Scholar
[3]Borges, C. R., ‘(Local) mixers and (L)EC-spaces’, Math. Japon. 30 (1985), 85–88.Google Scholar
[4]Borges, C. R., ‘Free topological groups’, J. Austral. Math. Soc. Ser. A 23 (1977), 360–365.CrossRefGoogle Scholar
[5]Borges, C. R., ‘Free groups, symmetric and reduced products’, J. Austral. Math. Soc. Ser. A 28 (1979), 174–178.CrossRefGoogle Scholar
[6]Dugundji, J., ‘Locally equiconnected spaces and absolute neighborhood retracts’, Fund. Math. 57 (1965), 187–193.CrossRefGoogle Scholar
[7]Dugundji, J., Topology (Allyn and Bacon, Boston, Mass., 1966).Google Scholar
[8]Dyer, E. and Eilenberg, S., ‘An adjunction theorem for locally equiconnected spaces’, Pacific J. Math. 41 (1972), 669–685.CrossRefGoogle Scholar
[9]Edwards, R. and Kirby, R., ‘Deformations of spaces and embeddings’, Ann. of Math. (2) 93 (1971), 63–88.CrossRefGoogle Scholar
[10]Graev, M. I., ‘Free topological groups’, Amer. Math. Soc. Transl. (Ser. 1) 8 (1962), 305–364.Google Scholar
[11]James, I. M., ‘Reduced product spaces’, Ann. of Math. (2) 62 (1955), 179–197.CrossRefGoogle Scholar
[12]Katz, E., Morris, S. A. and Nickolas, P., ‘Free abelian topological groups and adjunction spaces’ to appear.Google Scholar
[13]Sakai, K., ‘A characterization of local equiconnectedness’, Pacific J. Math. 111 (1984), 231–241.CrossRefGoogle Scholar
[14]Willard, S., General Topology (Addison-Wesley, Reading, Mass., 1970).Google Scholar