A note on common complements

TL;DR

This paper investigates the structure of pairs of closed subspaces with common complements in Hilbert spaces, revealing a fiber bundle structure and determining its homotopy type.

Contribution

It introduces a real analytic fiber bundle structure on the set of such pairs and characterizes its homotopy type, extending previous work on the geometric structure.

Findings

$ ext{Delta}$ is the base space of a real analytic fiber bundle.

The homotopy type of $ ext{Delta}$ is explicitly determined.

The structure relates to geometric objects on the Grassmann manifold.

Abstract

We discuss the structure of the set $\Delta$ consisting of pairs of closed subspaces that have a common complement in a Hilbert space previously studied by Lauzon and Treil (J. Funct. Anal. 212: 500--512, 2004). We prove that $\Delta$ is the base space of a real analytic fiber bundle constructed in terms of geometric objects associated to the Grassmann manifold. As a consequence we determine the homotopy type of $\Delta$.