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Remarks on homotopy equivalence of configuration spaces of a polyhedron

  • Leonid PlachtaEmail author
Open Access
Research Article
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Abstract

We show that the configuration space \(F_n(M)\) of n particles in a compact connected PL manifold M with nonempty boundary \(\partial M\) is homotopy equivalent to the configuration space \(F_n({\mathrm{Int}}\, M)\) where Open image in new window . Actually we prove some generalization of this result for polyhedra. Similar results recently have been obtained independently for topological manifolds by Zapata (Collision-free motion planning on manifolds with boundary, 2017. arXiv:1710.00293), using different techniques. We also address the question of whether a compact PL manifold M can be approximated up to homotopy type by discrete configuration spaces defined combinatorially via a simplicial subdivision of M.

Keywords

Configuration space PL manifold Deformation retraction Equivariant map Polyhedron Collar Discrete configuration space 

Mathematics Subject Classification

57Q91 

1 Introduction

Let X be a topological space and \(X^k\) its k-fold Cartesian product, \(k\geqslant 2\). Define the diagonalD of \(X^k\) as follows: Open image in new window .

For a given topological space X, denote by \(F_k(X)\) the space Open image in new window , the configuration space of k particles in X without collisions. The symmetric group\(\Sigma _k\) acts freely on \(F_k(X)\) by permuting coordinates of \(X^k\). The topology of classical configuration spaces \(F_k(\mathbf{R}^n)\) was studied by many authors (see, for example, [5, 8] for the background). A fundamental work on this topic is the monograph by Fadell and Husseini [7], in which the case of sphere \(X=S^m\) is also treated. The homology structure of \(F_k(\mathbf{R}^n)\) was described, for example, in [4]. It is also known that configuration spaces are not homotopy invariant even for closed manifolds (see [11]).

In this paper, we prove that if (QP) is a pair of compact polyhedra, the subpolyhedron P has a collar in Q, and the homotopy equivalence of the space Open image in new window and the polyhedron \(Q'\) is given by a deformation retraction of one onto another inside a collar of the subpolyhedron P, then it extends to a retraction of corresponding configuration spaces. It follows that if M is a compact piecewise linear (PL) manifold and the homotopy equivalence of manifolds Open image in new window and \(M'\), where Open image in new window , Open image in new window , is given by a deformation retraction of the first one onto the other one inside a collar of the boundary \(\partial M\), then it descends to a deformation of corresponding configuration spaces.

2 Configuration spaces of polyhedra and compact manifolds with boundary

Let (QP) be a pair of polyhedra such that P is a compact subpolyhedron of Q that has a PL collar in Q. In this section, we compare the configuration space of the polyhedron Q with the configuration space of the “open” subspace Open image in new window . In particular, we will show that if M is a compact PL manifold with nonempty boundary \(\partial M\), then the configuration spaces \(F_k({\mathrm{Int}}\,M)\) and \(F_k(M)\) are \(\Sigma _k\)-equivariantly homotopy equivalent.

Before proving a general result, we first demonstrate how our approach works in the particular case, when M is a closed unit disk of the Euclidean space. Let \(D^n\) be a closed n-dimensional disc in \(\mathbf{R}^n\). The proof of the following lemma uses the techniques developed by Crowley and Skopenkov in [6].

Lemma 2.1

For each positive integer k the space \(F_k(D^n)\) is a deformation retract of the space \(F_k(\mathbf{R}^n)\). Moreover, there is a \(\Sigma _k\)-equivariant deformation retraction of \(F_k(\mathbf{R}^n)\) onto \(F_k(D^n)\).

Proof

Let \(S^n\) be the n-dimensional sphere, Open image in new window . Decompose \(S^n\) into two half-spheres, \(S_0\) and \(S_\infty \), where Open image in new window and Open image in new window . Consider the subspace Open image in new window of \(S^n\) which is obviously homeomorphic to \(\mathbf{R}^n\). There is a \(\Sigma _k\)-equivariant deformation retraction \(g_t\) of \(F_k(R)\) on \(F_k(S_\infty )\). To show this, consider in Open image in new window a closed disc \(D_2\) of radius 2 centered at 0. The half-sphere \(S_0\) is identified with a closed unit disc \(D_1\).

For each \(s=1, \ldots ,k\), define a map \(f_s:F_k(R)\rightarrow S_\infty \) as follows:
  1. (i)

    \(f_s(x_1, \ldots , x_k)=x_s\) if no \(x_i\), \(i=1,\ldots , k\), is contained in \({\mathrm{Int}}\, D_1\);

     
  2. (ii)
    If Open image in new window , take any j such that Open image in new window . Denote Open image in new window by \(\rho \). We obviously have \(0<\rho <1\). Put
    $$\begin{aligned} f_s(x_1, \dots , x_k)=\frac{x_s}{|x_s|}\,\frac{2-2\rho +|x_s|}{2-\rho }\qquad \text {if}\quad x_s\in D_2; \end{aligned}$$
     
  3. (iii)

    \( f_s(x_1, \ldots , x_k) =x_s\) if \(x_s\) is not in \({\mathrm{Int}} \,D_2\).

     
Each coordinate function \(f_s\) is fixed on points \(x\in F_k(S_\infty )\) and on the points Open image in new window with \(|x_s|\geqslant 2\). For the points \((x_1,\ldots , x_s, \dots , x_k)\in F_k(S_\infty )\) with \(|x_s|\leqslant 2\), it acts along the rays in \(\mathbf{R}^n\) originating at 0. In this case, it looks like a monotone PL function \(h:[0,2] \rightarrow [0,2]\) with \(h(0)=0\), \(h(2)=2\) and \(h(\rho )=1\), which is linear on the intervals \([0,\rho ]\) and \([\rho , 2]\). In particular, if \(|x_s|=\rho \), where \(x_s\) is the s-th coordinate of the point \((x_1, \ldots , x_k)\), we have \(|f_s(x)|=1\). Substituting \(|x_s|=2\) in the formula \(\frac{x_s}{|x_s|}\,\frac{2-2\rho +|x_s|}{2-\rho }\), we get \(f_s(x)=x_s\). Moreover if \(\rho<|x_s|<2\), then \(1<|f_s(x_1, \dots , x_k)|<2\). It follows that each function \(f_s\) is well defined.

It is clear that \(f_s\) is continuous at \(x\in F_k(S_\infty )\). If Open image in new window and \(x_s\) is in the exterior of the disc \(D_2\) or in \(\partial D_2\), the function \(f_s\) depends only on \(x_s\) and we have \(f_s(x)=x_s\). If Open image in new window and \(x_s\) is in the interior of the disc \(D_2\), the value \(f_s(x)\) depends continuously on the parameter \(\rho \). On the other hand, the function \(\rho \) is the minimum of finite number of continuous functions (the norms \(|x_i|\)). So within a small neighborhood U(x) the parameter \(\rho (x)\) also changes very little. It follows that \(f_s\) is continuous at the points Open image in new window with Open image in new window and \(|x_s|=1\), the above remarks and formula (ii) show that \(f_s\) is continuous also at the point x.

Define a map \(f:F_k(R)\rightarrow (S_\infty )^k\) by the formula \(f=(f_1,\ldots , f_s, \ldots , f_k)\). It follows that \(f_s\) is the s-th coordinate function of f and the map f itself is continuous. Actually f maps the points \(x=(x_1,\ldots , x_s,\ldots , x_k)\) with distinct coordinates \(x_s\) to the points \(y=(y_1,\ldots ,y_s,\ldots , y_k)\) with different coordinates \(y_s\). This is obvious for the points \(x\in F_k(S_\infty )\) and for the points \(x=(x_1,\ldots , x_s,\ldots , x_k)\) such that all \(x_s\) lie on different rays of the space \(\mathbf{R}^n\). On the other hand, if some coordinates \(x_i\) and Open image in new window of Open image in new window are on the same ray, then Open image in new window and Open image in new window , according to the monotonic property of each coordinate function \(f_s\). Therefore f maps the configuration space \(F_k(R)\) onto the configuration space \(F_k(S_\infty )\). By the properties of the coordinate functions \(f_s\), f retracts the space \(F_k(R)\) onto the space \(F_k(S_\infty )\). Moreover it is not difficult to see that f is actually a \(\Sigma _k\)-equivariant retraction of \(F_k(R)\) onto \(F_k(S_\infty )\).

Each map \(f_s\) obviously admits an extension to a homotopy \(g^s_t\) via the following formula:For each s, \(1\leqslant s\leqslant k\), and each \(x\in F_k(S_\infty )\) the homotopy \(g^s_t\) keeps the coordinate \(x_s\) of x point-wise fixed. Put \(g_t=(g^1_t,\ldots , g^k_t)\) for each \(t\in [0,1]\). \(g_t\) is obviously a \(\Sigma _k\)-equivariant deformation retraction of \(F_k(R)\) onto \(F_k(S_\infty )\). \(\square \)

Let Q be a polyhedron and P its compact subpolyhedron which has a collar in Q. A closed collar of P in Q is represented by the image of PL embedding Open image in new window where Open image in new window is identified with P. It is a regular neighborhood of P in Q [10]. Denote by U a small open collar of P in Q which is identified with the image Open image in new window . Obviously, Open image in new window is homeomorphic to Q. It follows that Open image in new window and \(F_k(Q)\) are homeomorphic in a natural way.

Theorem 2.2

For each k the space Open image in new window deformation retracts onto the subspace Open image in new window . Moreover, the configuration space \(F_k(Q)\) is \(\Sigma _k\)-equivariantly homotopy equivalent to the configuration space Open image in new window .

Proof

Let \(R_1,\ldots , R_m\) be the connected components of P. Moreover, let \(C_i\), \(i=1,\ldots , m\), be the closed collars of \(R_1,\ldots ,R_m\), respectively, in Q where each \(C_i\) is identified with Open image in new window , \(i=1,\ldots ,m\), via the PL embedding h and \(R_i\) is identified with Open image in new window and Open image in new window if \(i \ne j\). We also identify Open image in new window with an open collar of P in Q as before. Put \(C=\bigcup _{\,i=1}^{\,m} C_i\).

By the above identification, each \(z\in C\) can be uniquely represented as \(z=(x, \tau )\) where Open image in new window for some j and \(0\leqslant \tau \leqslant 2\). Now we define a deformation retraction of the space Open image in new window onto the space Open image in new window as follows.

For each \(\rho \), \(0< \rho <1\), take a monotone PL function \(h_\rho :[0,2] \rightarrow [0,2]\) such that \(h_\rho (0)=0\), \(h_\rho (2)=2\) and \(h_\rho (\rho )=1\). For \(\rho \leqslant \lambda \leqslant 2\), the function \(h_\rho \) can be expressed as follows: \(h(\lambda ) =\frac{2-2\rho +\lambda }{2-\rho }\) (see also the proof of Lemma 2.1). First, for each \(1\leqslant s\leqslant k\) we shall define a map Open image in new window . Let Open image in new window . Let Open image in new window be the set of coordinates of y which belong to the open collar U of P. If \(V=\varnothing \), we put \(f_s(y)=y\). Assume that \(V\ne \varnothing \). We have Open image in new window for each \(1\leqslant s\leqslant r\) where Open image in new window . Take any l such that Open image in new window where s runs from 1 to r. Denote \(\tau _l\) by \(\rho \). We obviously have \(0<\rho <1\). Now, the expression for the coordinate function \(f_s\) is the following:
  • \(f_s(y_1, \ldots , y_k)=y_s\) if no \(y_i\) belongs to U, \(i=1,\ldots , k\);

  • \(f_s(y_1, \ldots , y_k)=y_s\) if some \(y_i\) belongs to U, \(i=1,\ldots , k\), but \(y_s\) does not belong to Open image in new window ;

  • \(f_s(y_1,\ldots y_k) =(x_s, h_\rho (\tau _s ))\), if Open image in new window and Open image in new window , where \(y_s=(x_s, \tau _s)\), \(x_s\in P\), \(0<\tau _s\leqslant 2\).

Note that if the coordinate \(y_s\) of \(y=(y_1, \ldots , y_k)\) belongs to the collar C, i.e., \(y_s=(x_s,\tau _s)\) where \(x_s\in P\) and \(\rho (y)\leqslant \tau _s\leqslant 2\), then the s-th coordinate of \(f_s(y_1, \ldots , y_s, \ldots , y_k)\) can be represented as \((x_s, \lambda _s)\) where \(1\leqslant \lambda _s \leqslant 2\). Moreover if \(\tau _s= 2\), then \(\lambda _s=2\) and if \(\tau _s= \rho \), then \(\lambda _s=1\).

It follows that for each \(s=1, \ldots , k\) the map Open image in new window is well defined in its domain. The continuity of \(f_s\) is performed along the same line as the one of the coordinate functions in the proof of Lemma 2.1. We omit the details.

Therefore the map Open image in new window is also continuous. Let \(y=(y_1, \ldots , y_k)\) be any point of the configuration space Open image in new window and let \(f_i\) and Open image in new window be two coordinate functions of the map f where \(i\ne j\). If Open image in new window , we have Open image in new window . Now assume that some coordinate \(y_s\) of y is in the set U. If one of the coordinates \(y_i\) and Open image in new window is outside the collar C, it follows immediately that Open image in new window . Assume that both \(y_i\) and Open image in new window belong to the collar C. The coordinates \(y_i\) and Open image in new window have the following presentation: \(y_i=(x_i, \tau _i )\) and Open image in new window where Open image in new window and Open image in new window . If Open image in new window it follows immediately that Open image in new window . On the other hand, if Open image in new window , then Open image in new window . By the monotonic property of the function \(h_\rho \), we get Open image in new window which implies that Open image in new window . It follows that f maps k-tuples \((y_1,\ldots ,y_k)\) with distinct coordinates into k-tuples \((z_1,\ldots , z_k)\) with distinct coordinates. Therefore f is actually a map from the configuration space Open image in new window onto the configuration Open image in new window . Moreover, by the properties of the coordinate functions \(f_s\), f is a \(\Sigma _k\)-equivariant retraction of the space Open image in new window onto the subspace Open image in new window .

The map f can be extended to the deformation retraction Open image in new window , \(t\in [0,1]\), with \(g'_0={\mathrm{id}}_{F_k(Q\setminus P)}\) and \(g'_1= f\). The deformation retraction \(g'_t\) is defined in the same way as the homotopy \(g_t\) in the proof of Lemma 2.1. We omit the details. Since the map f is \(\Sigma _k\)-equivariant, we can arrange that the deformation retraction \(g'_t\) of the space Open image in new window onto the space Open image in new window is also \(\Sigma _k\)-equivariant. This completes the proof of the theorem. \(\square \)

Let M be a connected, compact and smooth or PL manifold with the nonempty boundary \(\partial M\). Then \(\partial M\) is collared in M. Moreover we have the following

Corollary 2.3

For each \(k\geqslant 1\) the configuration space \(F_k(M)\) is \(\Sigma _k\)-equivariantly homotopy equivalent to the configuration space \(F_k({\mathrm{Int}}\, M)\).

3 Discrete configuration spaces of complexes

Let K be a finite simplicial complex. Denote by |K| the underlying topological space of K which is a polyhedron. For each \(k\leqslant n\) the subcomplex \(D_n(K)\) of the cell complex \(K^n\) is defined in the following way: Open image in new window where the sum is over all n pairwise disjoint closed cells in K (see [2, 3]). The subcomplex \(D_n(K)\) is called the discrete configuration space of the complexK with the parameter n. This is the largest cell complex that is contained in the product \(K^n\) minus its diagonal Open image in new window . The symmetric group \(\Sigma _ n\) acts naturally on \(D_n(K)\) by permuting the cells in the product. The polyhedron \(|D_n(K)|\) has natural \(\Sigma _n\)-equivariant embedding in the configuration space \(F_n(|K|)\) for each \(n\geqslant 2\).

A graph G can be considered as a 1-complex. Abrams [1] proved that for each graph G there is a subdivision \(G'\) of G such that the discrete configuration space \(D_n(G')\) is homotopy equivalent to the usual configuration space \(F_n(G)\), \(n\geqslant 2\).

The problem of a cell approximation of the space \(F_n(X)\) where X is a polyhedron of dimension \(\geqslant 2\) was considered and studied in [3]. For \(n=2\), Hu [9] showed that the configuration spaces \(D_2(K)\) and \(F_2(K)\) are homotopy equivalent. Moreover he showed that for any finite simplicial complex K there is a \(\Sigma _ 2\)-equivariant deformation retraction of \(F_2(K)\) onto \(|D_2(K)|\). In general, the problem can be formulated as follows

Problem

Let X be a compact connected PL manifold of dimension \(k\geqslant 2\) and let \(n>2\). Show that there is a subdivision K of X such that the manifold \(F_n(X)\) admits a \(\Sigma _n\)-equivariant deformation retraction onto the polyhedron \(|D_n(K)|\) or give a counterexample.

To the best of our knowledge, for PL manifolds of dimension \(k\geqslant 2\), the question of cell approximation of configuration spaces remains open.

Notes

Acknowledgements

The author thanks the referee for indicating some mistakes and inaccuracies in definitions in the earlier version of the paper, and for useful remarks and valuable comments.

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Authors and Affiliations

  1. 1.Faculty of Applied MathematicsAGH University of Science and TechnologyCracowPoland

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