Complexity of atriodic continua

TL;DR

This dissertation explores the complexity relationships between atriodic n-od-like continua and their subcontinua, demonstrating how certain inverse limits can be factored and establishing non-equivalence in their complexity classifications.

Contribution

It provides new insights into the structure of atriodic n-od-like continua and shows how inverse limits can be factored through simpler continua, revealing limitations in their complexity.

Findings

Inverse limits of simple-n-od graphs can be factored through simple-(n-1)-od graphs.

Certain inverse limits with all proper subcontinua as arcs are not simple-(n-1)-od-like.

The results clarify the complexity hierarchy among atriodic continua.

Abstract

This dissertation investigates the relative complexity between a continuum and its proper subcontinua, in particular, providing examples of atriodic n-od-like continua. Let X be a continuum and n be an integer greater than or equal to three. If X is homeomorphic to an inverse limit of simple-n-od graphs with simplicial bonding maps and is simple-(n-1)-od-like, it is shown that the bonding maps can be simplicially factored through a simple-(n-1)-od. This implies, in particular, that X is homeomorphic to an inverse limit of simple-(n-1)-od graphs with simplicial bonding maps. This factoring is subsequently used to show that a specific inverse limit of simple-n-ods with simplicial bonding maps, having the property of every proper nondegenerate subcontinuum being an arc, is not simple-(n-1)-od-like.