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Remarks on homotopy equivalence of configuration spaces of a polyhedron. (English) Zbl 1437.55022

For a given topological space, \(F_k(X)\) denotes the configuration space of \(k\) particles in \(X\) without collisions, i.e., \(F_k(X)=\{(x_1\ldots,x_k)\in X^k:~x_i\neq x_j \text{ for } i\neq j\}\). The symmetric group \(\Sigma_k\) acts freely on \(F_k(X)\) by permuting coordinates of \(X^k=X\times\cdots\times X\).
The main result in the present article, Theorem 2.2, is to prove the boundary homotopy invariance of the configuration spaces for polyhedra. Explicitly, the author proves that the configuration space \(F_k(Q)\) is \(\Sigma_k\)-equivariantly homotopy equivalent to the configuration space \(F_k(Q\setminus P)\), for any pair of polyhedra \((Q,P)\) where \(P\) is a compact subpolyhedron of \(Q\) that has a PL collar in \(Q\). Furthermore, in the last part of the paper, the author present a question of cell approximation, up to homotopy, of the configuration spaces for PL manifolds via discrete configuration spaces.

MSC:

55R80 Discriminantal varieties and configuration spaces in algebraic topology
57Q91 Equivariant PL-topology
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