The homology of ``\(k\)-equal'' manifolds and related partition lattices (Q1842229)

This paper analyzes the topology of the spaces \(V(n,k)\) consisting of sets of points \((x_1,\dots, x_n)\) in \(\mathbb{R}^n\) (or \(\mathbb{C}^n\)) which satisfy \(x_{x_1} = x_{i_2} = \dots = x_{i_n}\) for some set of \(k\) indices, and \(M(n,k) = \mathbb{R}^n - V(n,k)\) (or \(\mathbb{C}^n - V(n,k)\)). The \(M(n,k)\) are the ``\(k\)-equal'' manifolds of the title. For \(k = 2\), these spaces have been much studied, so the emphasis is on larger \(k\) values. The results generally say that the cohomology is free abelian, and the ranks are determined.