Summary: We study the set \(S=\{(a,b)\in A\times A:aba=a,bab=b\}\) which pairs the relatively regular elements of a Banach algebra \(A\) with their pseudoinverses, and prove that it is an analytic submanifold of \(A \times A\). If \(A\) is a \(C^*\)-algebra, and inside \(S\) lies a copy of the set \({\mathcal I}\) of partial isometries, we prove that this set is a \(C^\infty\) submanifold of \(S\) (as well as a submanifold of \(A)\). These manifolds carry actions from, respectively, \(G_A \times G_A\) and \(U_A\times U_A\), where \(G_A\) is the group of invertibles of \(A\) and \(U_A\) is the subgroup of unitary elements. These actions define homogeneous reductive structures for \(S\) and \({\mathcal I}\) (in the differential geometric sense). Certain topological and homotopical properties of these sets are derived. In particular, it is shown that if \(A\) is a von Neumann algebra and \(p\) is a purely infinite projection of \(A\), then the connected component \({\mathcal I}_p\) of \(p\) in \({\mathcal I}\) is simply connected. If \(1-p\) is also purely infinite, then \({\mathcal I}_p\) is contractible.