Noncommutative Borsuk-Ulam-type conjectures

TL;DR

This paper formulates noncommutative analogues of the Borsuk-Ulam conjecture within the framework of free actions of compact quantum groups on unital C*-algebras, extending classical topological results to quantum settings.

Contribution

It proposes two new conjectures in noncommutative topology related to equivariant homomorphisms and proves the second conjecture for the case of the quantum group $SU_q(2)$.

Findings

The first conjecture generalizes the classical Borsuk-Ulam theorem to quantum group actions.

The second conjecture is verified for the quantum group $SU_q(2)$.

The results relate to the non-trivializability of quantum instanton fibrations.

Abstract

Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if $H$ is the C*-algebra of a compact quantum group coacting freely on a unital C*-algebra $A$, then there is no equivariant $*$-homomorphism from $A$ to the join C*-algebra $A*H$. For $A$ being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of funtions on $\mathbb{Z}/2\mathbb{Z}$, we recover the celebrated Borsuk-Ulam theorem. The second conjecture states that there is no equivariant $*$-homomorphism from $H$ to the join C*-algebra $A*H$. We show how to prove the conjecture in the special case $A=C(SU_q(2))=H$, which is tantamount to showing the non-trivializability of Pflaum's quantum instanton fibration built from $SU_q(2)$.