Line Bundle Twists for Unitary Bordism are Ghosts

TL;DR

This paper demonstrates that a specific canonical twist in complex cobordism theory cannot be extended to unitary bordism, revealing fundamental differences in the structure of these theories.

Contribution

It proves that the canonical twist for complex cobordism does not extend to unitary bordism by analyzing the homotopy properties of associated maps.

Findings

The map from $K(bZ,3)$ to $BGL_1(MU)$ loops to a null homotopic map.

The canonical twist $Z$ does not extend to a twist for unitary bordism.

Homotopy analysis shows the non-extendability of the twist.

Abstract

We prove that the canonical twist $\zeta \colon K(\mathbb{Z},3) \rightarrow BGL_1(MSpin^c)$ does not extend to a twist for unitary bordism by showing that every continuous map $f \colon K(\mathbb{Z},3) \rightarrow BGL_1(MU)$ loops to a null homotopic map.