Braid groups in complex Grassmannians
The authors study the ordered configuration spaces \(\mathcal F_h(k,n)\) of \(h\) distinct points in the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space. The subspace \(\mathcal F_h^i(k,n)\) comprises those \(h\)-tuples of points in \(\mathrm{Gr}(k,n)\) which generate a subspace of dimension \(i\). The first main result of the paper under review states that the spaces \(\mathcal F_h^i(k,n)\) are simply connected or empty. As a consequence the unordered version of such a space has as its fundamental group the symmetric group which permutes the \(h\) points. The second main result describes some second homotopy groups subject to conditions on \(i\), \(h\), \(k\) and \(n\).
scientific article; zbMATH DE number 6334515