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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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