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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.

MSC:
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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