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Homotopy groups of diagonal complements. (English) Zbl 1355.55012
For a space \(X\) and integers \(1\leq d<n\), let \(\Delta_{d+1}(X,n)\) denote the union of the \((d+1)\)-th diagonal arrangement in \(X^n\) given by \[ \Delta_{d+1}(X,n)=\{(x_1,\cdots x_n)\in X^n:x_{i_0}=\cdots =x_{i_d}\text{ for some }\{x_{i_k}\}_{k=0}^d\in I_d\}, \] where \(\mathbb{N}\) denotes the set of positive integers and \(I_d\subset \mathbb{N}\) denotes the subset of \((d+1)\)-points in \(\mathbb{N}\). Let \(\Delta^d(X,n)\subset X^n\) denote the complement given by \(\Delta^d(X,n)=X^n\setminus \Delta_{d+1}(X)\). The symmetric group \(S_n\) of \(n\)-letters acts on \(X^n\) by coordinate permutation and let \(\mathrm{SP}^d(X)\) be the \(n\)-th symmetric product of \(X\) given by the orbit space \(\mathrm{SP}^n(X)=X^n/S_n\). Since \(\Delta^d(X,n)\) is an \(S_n\)-invariant subspace of \(X^n\), let \(B^d(X,n)\subset \mathrm{SP}^n(X)\) denote the subspace defined by \(B^d(X,n)=\Delta^d(X,n)/S_n\). Note that there are natural filtrations \[ \begin{cases} F(X,n)&=\Delta^1(X,n)\subset \Delta^2(X,n)\subset \cdots \subset \Delta^n(X,n)=X^n \\B(X,n)&=B^1(X,n)\subset B^2(X.n)\subset \cdots \subset B^n(X,n) =\mathrm{SP}^n(X)\end{cases} \] where \(F(X,n)\) (resp. \(B(X,n)\)) denotes the ordered (resp. unordered) configuration space of distinct \(n\)-points in \(X\).
In this paper, the authors study the inclusion maps \(i_d:\Delta^d(X,n)\to X^n\) and \(j_d:B^d(X,n)\to \mathrm{SP}^n(X)\) and show that these induce an isomorphism on the homotopy groups \(\pi_k(\;)\) for any \(k\leq 2d-2\) if \(X\) is a connected finite simplicial complex that is not a point. They also obtain a similar result for the fundamental group \(\pi_1(B^d(X,n))\).

MSC:
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55Q52 Homotopy groups of special spaces
55P10 Homotopy equivalences in algebraic topology
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