Remarks on homotopy equivalence of configuration spaces of a polyhedron
For a given topological space, \(F_k(X)\) denotes \textit{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 \textit{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 \textit{discrete configuration spaces}.
scientific article; zbMATH DE number 7189616