On the conjecture of Wood and projective homogeneity

TL;DR

This paper presents new counterexamples to Wood's Conjecture, explores the conditions for almost transitivity of certain function spaces, and investigates the projective homogeneity of the pseudo-circle, linking these topics to recent work on projective Fra"issé limits.

Contribution

It introduces a new nonmetric counterexample to Wood's Conjecture, analyzes the almost transitivity of $C_0( ext{space})$ spaces, and examines the projective homogeneity of the pseudo-circle.

Findings

A new nonmetric counterexample to Wood's Conjecture is constructed.

Almost transitivity of $C_0( ext{space})$ spaces depends on the space being the one-point compactification of a pseudo-circle.

The pseudo-circle is shown not to be approximately projectively homogeneous.

Abstract

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 $\hat L$, such that $C_0(\hat L,\mathbb{C})$ 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 $C_0(X,\mathbb{C})$ 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 Fra\"iss\'e limits and projective homogeneity of the pseudo-arc and, addressing their conjecture, we show that the pseudo-circle is not approximately projectively homogeneous.