Remarks on Topological Rigidity of Real Moment-Angle Manifolds

TL;DR

This paper investigates the topological rigidity of real moment-angle manifolds linked to flag complexes, showing they admit CAT(0) metrics and satisfy the Borel Conjecture, with distinctions from complex cases.

Contribution

It demonstrates that real moment-angle manifolds of dimension five or more are topologically rigid and satisfy the Borel Conjecture, using cubical geometry and surgery theory.

Findings

The universal cover of these manifolds admits a CAT(0) metric.

Their fundamental groups satisfy the Farrell--Jones conjecture.

They satisfy the Borel Conjecture in dimension at least five.

Abstract

We study topological rigidity of real moment-angle manifolds associated to flag simplicial complexes. Using the cubical geometry arising from the Davis construction, we identify the universal cover with the Davis complex and deduce that it admits a CAT(0) metric. As a consequence, its fundamental group satisfies the Farrell--Jones conjecture. Applying surgery theory, we deduce that real moment-angle manifolds of dimension at least five associated to flag complexes satisfy the Borel Conjecture. We also explain why this rigidity phenomenon is specific to the real case and fails for complex and quaternionic moment-angle complexes.