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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
##### Keywords:
diagonal arrangement; configuration space; colimit diagram
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