On the Conjecture of Wood and projective homogeneity

In 2005 Kawamura and Rambla, independently, constructed a metric counterexample to Wood's Conjecture from 1982. We exhibit a new nonmetric counterexample of a space Lˆ, such that C0(Lˆ,C) is almost transitive, and show that it is distinct from a nonmetric space whose existence follows from the...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2018-05, Vol.461 (2), p.1733-1747
Main Authors: Boroński, J.P., Smith, M.
Format: Article
Language:English
Subjects:
Almost transitive
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Summary:In 2005 Kawamura and Rambla, independently, constructed a metric counterexample to Wood's Conjecture from 1982. We exhibit a new nonmetric counterexample of a space Lˆ, such that C0(Lˆ,C) is almost transitive, and show that it is distinct from a nonmetric space whose existence follows from the work of Greim and Rajagopalan in 1997. Up to our knowledge, this is only the third known counterexample to Wood's Conjecture. We also show that, contrary to what was expected, if a one-point compactification of a space X is R.H. Bing's pseudo-circle then C0(X,C) is not almost transitive, for a generic choice of points. Finally, we point out close relation of these results on Wood's conjecture to a work of Irwin and Solecki on projective Fraïssé limits and projective homogeneity of the pseudo-arc and, addressing their conjecture, we show that the pseudo-circle is not approximately projectively homogeneous.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2017.12.051