A note on common complements

We discuss the structure of the set $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$.