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The homology of “\(k\)-equal” manifolds and related partition lattices. (English) Zbl 0845.57020
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.

57R19 Algebraic topology on manifolds and differential topology
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
06A11 Algebraic aspects of posets
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