Remarks on homotopy equivalence of configuration spaces of a polyhedron

We show that the configuration space $$F_n(M)$$ F n ( M ) of n particles in a compact connected PL manifold M with nonempty boundary $$\partial M$$ ∂ M is homotopy equivalent to the configuration space $$F_n({\mathrm{Int}}\, M)$$ F n ( Int M ) where . 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 .


Introduction
Let X be a topological space and X k its k-fold Cartesian product, k 2. Define the diagonal D of X k as follows: D = {(x 1 , . . . , x k ) ∈ X k : x i = x j for some i = j }.
For a given topological space X , denote by F k (X ) the space X k \ D, the configuration space of k particles in X without collisions. The symmetric group k acts freely on F k (X ) by permuting coordinates of X k . The topology of classical configuration spaces F k (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 [7], in which the case of sphere X = S m is also treated. The homology structure of F k (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 (Q, P) is a pair of compact polyhedra, the subpolyhedron P has a collar in Q, and the homotopy equivalence of the space Q \ P 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 M \ ∂ M and M , where M ⊂ M, M ∼ = M, is given by a deformation retraction of the first one onto the other one inside a collar of the boundary ∂ M, then it descends to a deformation of corresponding configuration spaces.

Configuration spaces of polyhedra and compact manifolds with boundary
Let (Q, P) 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 Q \ P. In particular, we will show that if M is a compact PL manifold with nonempty boundary ∂ M, then the configuration spaces F k (Int M) and F k (M) are 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 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
Proof Let S n be the n-dimensional sphere, S n = R n ∪ {∞}. Decompose S n into two half-spheres, S 0 and S ∞ , where S 0 = {w ∈ R n : |w| 1} and S ∞ = cl (S n \ S 0 ). Consider the subspace R = S n \{0} of S n which is obviously homeomorphic to R n . There is a k -equivariant deformation retraction g t of F k (R) on F k (S ∞ ). To show this, consider in R n ⊂ S n 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, . . . , k, define a map f s : F k (R) → S ∞ as follows: Each coordinate function f s is fixed on points x ∈ F k (S ∞ ) and on the points it acts along the rays in R n originating at 0. In this case, it looks like a monotone PL function h : and x s is in the interior of the disc D 2 , the value f s (x) depends continuously on the parameter ρ. On the other hand, the function ρ is the minimum of finite number of continuous functions (the norms |x i |). So within a small neighborhood U (x) the parameter ρ(x) also changes very little. It follows that f s is continuous at the points 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 : 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 , . . . , x s , . . . , x k ) with distinct coordinates x s to the points y = (y 1 , . . . , y s , . . . , y k ) with different coordinates y s . This is obvious for the points x ∈ F k (S ∞ ) and for the points x = (x 1 , . . . , x s , . . . , x k ) such that all x s lie on different rays of the space R n . On the other hand, if some coordinates x i and x j of x ∈ F k (R) \ F k (S ∞ ) are on the same ray, then |x i | = |x j | and | f i (x)| = | f j (x)|, 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 ∞ ). By the properties of the coordinate functions f s , f retracts the space F k (R) onto the space F k (S ∞ ).
Moreover it is not difficult to see that f is actually a k -equivariant retraction of Each map f s obviously admits an extension to a homotopy g s t via the following formula: For each s, 1 s k, and each x ∈ F k (S ∞ ) the homotopy g s t keeps the coordinate 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 h : P × [0, 2] → Q where h(P × {0}) 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 h(P × [0, 1)). Obviously, Q \U is homeomorphic to Q. It follows that F k (Q \U ) and F k (Q) are homeomorphic in a natural way.

Theorem 2.2 For each k the space F k (Q \ P) deformation retracts onto the subspace F k (Q \U ). Moreover, the configuration space F k (Q) is k -equivariantly homotopy equivalent to the configuration space F k (Q \ P).
Proof Let R 1 , . . . , R m be the connected components of P. Moreover, let C i , i = 1, . . . , m, be the closed collars of R 1 , . . . , R m , respectively, in Q where each C i is identified with R i × [0, 2], i = 1, . . . , m, via the PL embedding h and R i is identified ) with an open collar of P in Q as before. Put C = m i=1 C i . By the above identification, each z ∈ C can be uniquely represented as z = (x, τ ) where x ∈ R j for some j and 0 τ 2. Now we define a deformation retraction of the space F k (Q \ P) onto the space F k (Q \U ) as follows.
It follows that for each s = 1, . . . , k the map f s : F k (Q \ P) → Q \U 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 f = ( f 1 , . . . , f s , . . . , f k ) : F k (Q \ P) → (Q \U ) k is also continuous. Let y = (y 1 , . . . , y k ) be any point of the configuration space F k (Q \ P) and let f i and f j be two coordinate functions of the map f where i = j. If y ∈ F k (Q \U ), we have f i (y) = y i = y j = f j (y). Now assume that some coordinate y s of y is in the set U . If one of the coordinates y i and y j is outside the collar C, it follows immediately that f i (y) = f j (y). Assume that both y i and y j belong to the collar C. The coordinates y i and y j have the following presentation: y i = (x i , τ i ) and y j = (x j , τ j ) where x i , x j ∈ P and ρ(y) τ i , τ j 2. If x i = x j it follows immediately that f i (y) = f j (y). On the other hand, if x i = x j , then τ i = τ j . By the monotonic property of the function h ρ , we get h ρ (τ i ) = h ρ (τ j ) which implies that f i (y) = f j (y). It follows that f maps k-tuples (y 1 , . . . , y k ) with distinct coordinates into k-tuples (z 1 , . . . , z k ) with distinct coordinates. Therefore f is actually a map from the configuration space F k (Q \ P) onto the configuration F k (Q \U ). Moreover, by the properties of the coordinate functions f s , f is a k -equivariant retraction of the space F k (Q \ P) onto the subspace F k (Q \U ).
The map f can be extended to the deformation retraction g t : F k (Q \ P) → F k (Q \ P), t ∈ [0, 1], with g 0 = id F k (Q\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 k -equivariant, we can arrange that the deformation retraction g t of the space F k (Q \ P) onto the space F k (Q \U ) is also k -equivariant. This completes the proof of the theorem.
Let M be a connected, compact and smooth or PL manifold with the nonempty boundary ∂ M. Then ∂ M is collared in M. Moreover we have the following

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 n the subcomplex D n (K ) of the cell complex K n is defined in the following way: D n (K ) = σ 1 × · · · ×σ n 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 complex K with the parameter n. This is the largest cell complex that is contained in the product K n minus its diagonal {(x 1 , . . . , x n ) ∈ |K | n : x i = x j for some i = j }. The symmetric group n acts naturally on D n (K ) by permuting the cells in the product. The polyhedron |D n (K )| has natural n -equivariant embedding in the configuration space F n (|K |) for each n 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 2.
The problem of a cell approximation of the space F n (X ) where X is a polyhedron of dimension 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 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 2 and let n > 2. Show that there is a subdivision K of X such that the manifold F n (X ) admits a 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 2, the question of cell approximation of configuration spaces remains open.