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Homotopy type of Euclidean configuration spaces. (English) Zbl 0974.55003

Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 161-164 (2001).
Let \(F({\mathbb R}^n, k)\) denote the configuration space of pairwise-disjoint \(k\)-tuples of points in \({\mathbb R}^n\). In this short note the author describes a cellular structure for \(F({\mathbb R}^n, k)\) when \(n \geq 3\). From results in [F. R. Cohen, T. J. Lada and J. P. May, The homology of iterated loop spaces, Lect. Notes Math. 533 (1976; Zbl 0334.55009)], the integral (co)homology of \(F({\mathbb R}^n, k)\) is well-understood. This allows an identification of the location of the cells of \(F({\mathbb R}^n, k)\) in a minimal cell decomposition. Somewhat more detail is provided by the main result here, in which the attaching maps are identified as higher order Whitehead products.
For the entire collection see [Zbl 0961.00020].

MSC:

55P15 Classification of homotopy type
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55Q15 Whitehead products and generalizations

Citations:

Zbl 0334.55009