A remark on the homotopy equivalence of SU_n and SL_nC

TL;DR

The paper investigates the topological relationship between SU_n and SL_nC, showing that for large n, no retraction preserving commutativity exists from SL_nC to SU_n, highlighting a limitation in their homotopy equivalence.

Contribution

It proves that for sufficiently large n, there is no commutativity-preserving retraction from SL_nC to SU_n, refining understanding of their topological relationship.

Findings

SU_n is a deformation retract of SL_nC.

No retraction preserving commutativity exists for large n.

Highlights limitations in homotopy equivalence for large n.

Abstract

Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.