On the Gray index conjecture for phantom maps

TL;DR

This paper investigates the Gray index of phantom maps, disproves a conjecture about their properties by providing a counterexample, and explores implications for the structure of certain homotopy groups.

Contribution

It constructs a counterexample to the Gray index conjecture for phantom maps and establishes conditions under which the conjecture holds.

Findings

Counterexample to the Gray index conjecture for phantom maps.

The conjecture holds when target spaces are simply connected finite complexes.

Demonstrates non-triviality of ^{} for some finite type space.

Abstract

We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that $\SNT^{\infty}(X)$ can be non-trivial for some space $X$ of finite type.