© 1997 Copyright by E. Santillan. All rights reserved.

Question

Peano Continua which fix the n+1-dimensional triodes E. Santillan

Let I be the closed unit interval of the real line and let T be a (simple) triode with vertex point t. (T is the union of three arcs which are pairwise disjoint, except for the common point t. Both sets I and T are endowed with the standard topology).

Definition The topological space X is said to fix the n+1-dimensional triodes (n is a natural number), if for every mapping f : T ×Iⁿ --> X ×Iⁿ which is a homeomorphism from Y ×Iⁿ onto its image, there exists a point x in X such that f(t ×Iⁿ) 'subset or equal' {x} ×Iⁿ.

If a topological space X fixes the n+1-dimensional triodes, it will contains no copies of the real plane R². I think that the converse implication holds when X is a Peano continuum, but I have not be able to show this fact.

Open Problem Will each Peano continuum X, which contains no copies of the real plane R², fix the n+1-dimensional triodes?

Nowadays, I am working on cancellation laws of topological products, and I got this open problem when I was showing the following theorem (I am preparing a paper with this theorem).

The spaces S¹, R, and R⁰⁺ = [0, \infty) 'subset or equal' R are respectively the closed unit circle, the real line and the semi-closed interval of the real line. (They are all endowed with the standard topology).

Theorem Let H be a metrisable connected n-manifold with boundary (n is a natural number) such that the following four products are not homeomorphic by pairs:

1. S¹ ×H,

2. I ×H,

3. R⁰⁺ ×H,

4. R ×H.

Then the cancellation law: