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Models for configuration space in a simplicial complex. (English) Zbl 1411.55009

For a topological space \(Z\) and a positive integer \(n\geq 1\), let \(\mathrm{Conf}(n,Z)\) denote the ordered configuration space of distinct \(n\) points in \(Z\) defined by \(\mathrm{Conf}(n,Z)=\{(z_1,\dots ,z_n)\in Z^n:z_i\not= z_j\text{ if }i\not= j\}\). For an abstract simplicial complex \(X\), let \(|X|\) denote the geometric realization of \(X\). Now we shall consider the combinatorial models for the configuration space \(\mathrm{Conf}(n,|X|)\). Let \(X\) be an abstract simplicial complex with partial ordering of the vertices in \(X\) such that this ordering gives the total order on every face in \(X\). In this situation a conf matrix means a matrix \((v_{i,j})\) of vertices in \(X\) such that
(a)
each row is weakly increasing in the vertex ordering,
(b)
for every row, the vertices appearing within that row form a face of \(X\), and
(c)
no row is repeated.
A conf matrix is called minimal if deleting a column results in duplicate row, no matter which column is deleted. Then we denote by \(C(n,X)\) the abstract simplicial complex whose vertices are minimal conf matrices for \(X\) with a \(n\) row, and where a collection of matrices forms a face if their columns can be assembled into a single conf matrix.
Then in this paper the author proves that there is a homotopy equivalence \(|C(n,X)|\simeq \mathrm{Conf}(n,|X|)\), and he also gives the combinatorial model of the local configuration space. By way of application, he studies the nodal curve \(y^2z=x^3+x^2z\), for obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its two- and three strand local braid groups near the single point.

MSC:

55R80 Discriminantal varieties and configuration spaces in algebraic topology
55U05 Abstract complexes in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology

Software:

GAP; SageMath
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References:

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