Directed path spaces via discrete vector fields
Abstract
Let K be an arbitrary semicubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CWcomplex that is homotopy equivalent to the space \(\vec {P}(K)_v^w\) of directed paths between two given vertices v, w of K. In many cases, this construction is minimal: the cells of the constructed CWcomplex are in 1–1 correspondence with the generators of the homology of \(\vec {P}(K)_v^w\).
Keywords
Semicubical set Directed path space Discrete vector field Permutahedron Configuration space1 Introduction
The spaces of directed paths on semicubical sets play an important role in Theoretical Computer Science [4, 5]. In the previous paper [13] the author constructed, for every bipointed semicubical set (K, v, w) satisfying certain mild assumptions, a regular CWcomplex \(W(K)_v^w\) that is homotopy equivalent to the space of directed paths \(\vec {P}(K)_v^w\) on K from v to w. This construction is functorial, and even minimal amongst functorial constructions. The main goal of this paper is to provide a further reduction of this model.
We restrict our attention to semicubical sets that can be embedded into a standard cube, regarded as a semicubical complex. This special case is general enough to encompass most of interesting examples appearing in Concurrency. The main result of this paper is a construction of a discrete gradient field [6] \(\mathcal {W}_K\) on \(W(K)_v^w\). It is used to show that \(\mathcal {P}(K)_v^w\) is homotopy equivalent to an even smaller CWcomplex X(K) whose cells correspond to the critical cells of \(\mathcal {W}_K\). Furthermore, explicit formulas describing the set of critical cells of \(\mathcal {W}_K\) are provided.
This construction allows to calculate the homology groups of \(\mathcal {P}(K)_v^w\), since the differentials in the cellular homology chain complex of X(K) can be recovered using methods from [9, Chapter 11]. We do not examine these differentials in detail. It appears that in many important cases it is not necessary since the differentials vanish for dimensional reasons. This way we reprove here the result of Björner and Welker [2], who calculate the homology of “no \((k+1)\)–equal’ configuration spaces on the real line, as well as a generalization due to Meshulam and Raussen [10].
We pay a special attention to the case when K is a Euclidean cubical complex, i.e., a union of cubes having integral coordinates in the directed Euclidean space \(\vec {\mathbb {R}}^n\). Since state spaces of PVprograms [3] are Euclidean cubical complexes, this case seems important for potential applications in concurrency. Since every finite Euclidean cubical complex can be embedded into a standard cube, our results apply in this case; also, a description of the critical cells of \(\mathcal {W}_K\) is given in this context.
2 Preliminaries
Let us recall some definitions and results obtained in [13].
A dspace [8] is a pair \((X,\vec {P}(X))\), where X is a topological space and \(\vec {P}(X)\subseteq P(X)={{\mathrm{map}}}([0,1],X)\) is a family of paths that contains all constant paths and is closed with respect to concatenation and nondecreasing reparametrizations. Paths that belong to \(\vec {P}(X)\) will be called directed paths or dpaths. For \(x,y\in X\), \(\vec {P}(X)_x^y\) denotes the space of dpaths starting at x and ending at y. Prominent examples of dspaces are the directedncube\(\vec {I}^n=(I^n,\vec {P}(\vec {I}^n))\) and the directed Euclidean space\(\vec {\mathbb {R}}^n=(\mathbb {R}^n,\vec {P}(\vec {\mathbb {R}}^n))\), where \(\vec {P}(\vec {I}^n)\) and \(\vec {P}(\vec {\mathbb {R}}^n)\) are the spaces of all paths having nondecreasing coordinate functions.
A semicubical setK is a sequence of disjoint sets \((K[n])_{n\ge 0}\), equipped with face maps\(d^\varepsilon _i:K[n]\rightarrow K[n1]\), where \(n\ge 0\), \(i\in \{1,\ldots ,n\}\) and \(\varepsilon \in \{0,1\}\), that satisfy the precubical relations, i.e., \(d_i^\varepsilon d_j^\eta = d_{j1}^\eta d_i^\varepsilon \) for \(i<j\). Elements of K[n] will be called cubes or ncubes if one needs to emphasize their dimension; 0cubes and 1cubes will be called vertices and edges, respectively. The set of all cubes of a semicubical set K will be denoted by \({{\mathrm{Cell}}}(K)\) or by K if that does not lead to confusion. This set is partially ordered by inclusion, i.e. \(c\subseteq c'\) if c is the image of \(c'\) under some composition of face maps. Every cube \(c\in K[n]\) has the initial vertex\(d^0(c)=d^0_1\ldots d^0_1(c)\) and the final vertex\(d^1(c)=d^1_1\ldots d^1_1(c)\), where n face maps appear in both compositions.

\(d^0(c_1)=v\),

\(d^1(c_l)=w\),

\(d^1(c_i)=d^0(c_{i+1})\) for \(i\in \{1,\ldots ,l1\}\).
Assume that a semicubical set K is proper, i.e., if \(c\ne c'\) are cubes of K, then \(\{d^0(c),d^1(c)\}\ne \{d^0(c'),d^1(c')\}\). Under this assumption, the following holds:
Theorem 2.1
 (a)There is a homotopy equivalencewhere \({{\mathrm{Ch}}}(K)_v^w\) denotes the geometric realization of the nerve of \({{\mathrm{Ch}}}(K)_v^w\).$$\begin{aligned} \vec {P}(K)_v^w\simeq {{\mathrm{Ch}}}(K)_v^w, \end{aligned}$$
 (b)
\({{\mathrm{Ch}}}(K)_v^w\) carries a natural structure of a regular CWcomplex with closed cells having the form \({{\mathrm{Ch}}}_{\le \mathbf {c}}(K)\) for \(\mathbf {c}\in {{\mathrm{Ch}}}(K)_v^w\), where \({{\mathrm{Ch}}}_{\le \mathbf {c}}(K)\subseteq {{\mathrm{Ch}}}(K)_v^w\) is the subposet of cube chains that are finer than \(\mathbf {c}\), including \(\mathbf {c}\) itself. \(\square \)
In this paper we restrict to the case when K is a semicubical subset of the standard cube. The standardncube\(\square ^n\) is a semicubical set whose kcubes \(\square ^n[k]\) are sequences \((e_1,\ldots ,e_n)\), \(e_i\in \{0,1,*\}\) having exactly k entries equal to \(*\). A face map \(d_i^\varepsilon \) converts the ith occurrence of \(*\) into \(\varepsilon \). It is easy to see that the geometric realization of \(\square ^n\) is dhomeomorphic to the directed cube \(\vec {I}^n\). Furthermore, every semicubical subset of \(\square ^n\) is proper, so the results of [13] can be applied in this situation.
The majority of proofs in this paper is inductive with respect to the dimension of the ambient cube \(\square ^n\). Thus, for convenience, the coordinates will be indexed by an arbitrary finite ordered set A rather than by \(\{1,\ldots ,n\}\). In the case when A is nonempty, \(m\in A\) denotes its maximal element and \(A'=A{\setminus }\{m\}\).
Let \(\#X\) denote the cardinality of a finite set X.
Definition 2.2

\(\square ^A[k]\) is the set of all functions \(c:A\rightarrow \{0,1,*\}\) such that \(\#(c^{1}(*))=k\).
 For \(c\in \square ^A[k]\), if \(c^{1}(*)=\{b_1<b_2<\cdots <b_k\}\), then$$\begin{aligned} d^\varepsilon _i(c)(a)={\left\{ \begin{array}{ll} \varepsilon &{} \quad \hbox {if}\ a=b_i \\ c(a) &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$
We identify \(\square ^A\) with the directed Acube \(\vec {I}^A\); thus, the geometric realization of an Acubical complex is a subspace of \(\vec {I}^A\).
3 Ordered partitions and cube chains in Acomplexes
Definition 3.1

\(c_i(a)=0\) implies \(c_{i1}(a)=0\),

\(c_i(a)=1\) implies \(c_{i+1}(a)=1\),

\(c_i(a)=*\) implies \(c_{i1}(a)=0\) and \(c_{i+1}(a)=1\).

\(c_1(a)\ne 1\) for all \(a\in A\), since \(d^0(c_1)=\mathbf {0}\),

\(c_l(a)\ne 0\) for all \(a\in A\), since \(d^1(c_l)=\mathbf {1}\).
Proposition 3.2
This is a slight reformulation of [13, Proposition 8.1].
Proposition 3.3
 (a)
\(\lambda B_1\cdots B_k\mu \in \mathcal {P}_K\).
 (b)\(\lambda \in \mathcal {P}(K^0_C)\), \(\mu \in \mathcal {P}(K^1_D)\) andfor every \(i\in \{1,\ldots ,k\}\).\(\square \)$$\begin{aligned} c(C\cup B_1\cup \cdots \cup B_{i1},B_i,B_{i+1}\cup \cdots \cup B_k\cup D)\in K \end{aligned}$$
In the remaining part of the paper we will examine the poset \(\mathcal {P}_K\) by means of Discrete Morse Theory.
4 Discrete Morse Theory for CWposets
In this section we recall some basic facts from Discrete Morse Theory for regular CWcomplexes. For detailed expositions of this topic see, for example, [6, 7, 9, Chapter 11].
Definition 4.1
([1]) A poset P is a CWposet if, for every \(a\in P\), \(P_{<a}\) is homeomorphic to a sphere \(S^{d(a)1}\). Elements of a CWposet will be called cells and the integer d(a) will be called the dimension of a cell a. If \(a<b\in P\) and \(d(a)=d(b)1\), then a will be called a facet ofb and we will write \(a\prec b\).
For a CWposet P, P has a natural CWstructure; its closed kcells have the form \(P_{\le a}\), for \(a\in P\) having dimension k. The cell poset C(W) of a regular CWcomplex W is a CWposet and C(W) is homeomorphic to W by a cellpreserving homeomorphism. In particular, the face poset of a convex polytope is a CWposet.
Definition 4.2

A discrete vector field \(\mathcal {V}\)onP is a set of pairwise disjoint pairs (called vectors) (a, b), \(a,b\in P\) such that a is a facet of b.
 A flow of \(\mathcal {V}\) is a sequence of cells of Psuch that, for all \(i\in \{1,\ldots ,k\}\),$$\begin{aligned} (a_1,b_1,a_2,b_2,\ldots ,a_{k},b_k,a_{k+1}) \end{aligned}$$(4.1)

\((a_i,b_i)\in \mathcal {V}\),

\((a_{i+1},b_i)\not \in \mathcal {V}\)

\(a_{i+1}\prec b_i\).


A cycle of \(\mathcal {V}\) is a flow (4.1) such that \(a_1=a_{k+1}\), \(k>0\).

A discrete vector field \(\mathcal {V}\) is a gradient field if it admits no cycle.
The importance of gradient fields follows from the following theorem:
Theorem 4.3
([9, Theorem 11.13]) Assume that P is a CWposet and \(\mathcal {V}\) is a gradient field on P. Then there exists a CWcomplex \(W(\mathcal {V})\) that is homotopy equivalent to P, whose ddimensional cells are in 1–1 correspondence with the ddimensional critical cells of \(\mathcal {V}\).
For convenience, we will use a notion of discrete Morse function which is slightly different from the original one.
Definition 4.4
 (a)
\((a,b)\in \mathcal {V}\) implies that \(h(a)>h(b)\),
 (b)
\((a,b)\not \in \mathcal {V}\) implies that \(h(a)\le h(b)\).
Lemma 4.5
Let \(\mathcal {V}\) be a discrete vector field on a finite CWposet P. If there exists a Morse function associated to \(\mathcal {V}\), then \(\mathcal {V}\) is a gradient field.
Proof
We say that a subposet Q of a poset P is closed if, for \(x\le y\in P\), \(y\in Q\) implies that \(x\in Q\).
Lemma 4.6
Let P be finite CWposet and let \(Q\subseteq P\) is a closed subposet. Let \(\mathcal {V}_Q, \mathcal {V}_{P{\setminus } Q}\) be discrete vector fields on Q and \(P{\setminus } Q\) respectively. Then every cycle of \(\mathcal {V}=\mathcal {V}_Q\cup \mathcal {V}_{P{\setminus } Q}\) is contained either in Q or in \(P{\setminus } Q\). In particular, if both \(\mathcal {V}_Q\) and \(\mathcal {V}_{P{\setminus } Q}\) are gradient fields, then \(\mathcal {V}\) is also a gradient field.
Proof
Let \((a_1,b_1,\ldots ,a_k,b_k,a_1)\) be a cycle. By the assumptions, \(a_i\in Q\) implies \(b_i\in Q\), and \(b_i\in Q\) implies \(a_{i+1}\in Q\), since \(a_{i+1}\prec b_i\). Thus, either all elements of the cycle are in Q or none is. \(\square \)
Now we will give some examples of gradient fields. While the first two examples are not crucial in proving the main results of this paper, they can be helpful in understanding similar constructions performed on permutahedra.
In all following examples, A is a nonempty finite ordered set, m is its maximal element and \(A'=A{\setminus } \{m\}\).
Example 4.7
Example 4.8
 (a)
\(\mathcal {S}^{\square m}_A=\{(f,g):\; (f_{A'},g_{{A'}})\in \mathcal {S}^\square _{{A'}}, \; f(m)=g(m)=0\}.\)
 (cb)
\(\mathcal {S}^{\square r}_A=\{(f,g):\; f_{{A'}}=g_{{A'}},\; f(m)=1,\; g(m)=*\}\)
 (c)
\(\mathcal {S}^{\square }_A=\mathcal {S}^{\square m}_A\cup \mathcal {S}^{\square r}_A\).
Example 4.9
Proposition 4.10
Assume that P, Q are finite CWposets and \(\mathcal {V}\) and \(\mathcal {W}\) are gradient fields on P and Q, respectively. Then \(\mathcal {V}\times \mathcal {W}\) is a gradient field on \(P\times Q\).
Proof
Notice that the standard gradient fields on cubes can be defined alternatively by formulas \(\mathcal {S}^\square _{\{x\}}=\{(1,*)\}\), \(\mathcal {S}^\square _A=\mathcal {S}^\square _{\{m\}}\times \mathcal {S}^\square _{A'}\).
5 Permutahedra
The main goal of this section is to construct “standard” gradient fields on permutahedra. As before, A is a finite ordered set, \(m\in A\) is a maximal element and \(A'=A{\setminus } \{m\}\). We will write \(A=B_1\mathop {\dot{\cup }}\cdots \mathop {\dot{\cup }}B_n\) when \(B_1,\ldots ,B_n\) are pairwise disjoint subsets of A such that \(\bigcup B_i=A\) and the order on every \(B_i\) is inherited from A.
Proposition 5.1
Proof
Proposition 5.2
\(\mathcal {V}_A\) is a gradient field.
Proof
This is obvious for \(A=\emptyset \), so assume otherwise. There is an isomorphism \((\mathcal {P}^m_A,\mathcal {V}^m_A)\cong (\mathcal {P}_{A'},\mathcal {V}_{A'})\); hence, by the inductive hypothesis, \(\mathcal {V}^m_A\) is a gradient field on \(\mathcal {P}^m_A\). By Lemma 4.6, it remains to prove that \(\mathcal {V}^r_A\) is a gradient field on \(\mathcal {P}^r_A\).

\(s>s'\),

\(s=s'\) and \(1\ne t\le t'\),

\(s=s'\) and \(t'=1\).
 \(B_1=\{m\}\). Then \((\mu ,\lambda )\in \mathcal {V}^r_A\), and$$\begin{aligned} h_A(\mu )=(\#C,1)\mathrel {\dot{>}}(\#C,\#B)=h_A(\lambda ). \end{aligned}$$
 \(\{m\}\subsetneq B_1\). Then \((\mu ,\lambda )\not \in \mathcal {V}^r_A\) andsince \(\#B_1>1\).$$\begin{aligned} h_A(\mu )=(\#C,\#B_1)\mathrel {\dot{<}}(\#C,\#B)=h_A(\lambda ), \end{aligned}$$
 \(m\in B_2\). Then \((\mu ,\lambda )\not \in \mathcal {V}^r_A\) and$$\begin{aligned} h_A(\mu )=(\#C+\#B_1,\#B_2)\mathrel {\dot{<}}(\#C,\#B)=h_A(\lambda ). \end{aligned}$$
6 A gradient field on \(\mathcal {P}_K\)
Definition 6.1

\(c(C,m\cup B,D)\not \in K\),

\(c(C,m,B\cup D),c(C\cup m, B,D)\in K\).
Proposition 6.2
For an Acubical complex K, we have \({{\mathrm{Crit}}}_{\mathcal {P}^r_K}(\mathcal {V}^r_K)=\mathcal {R}_K\).
Proof
If \(\lambda =\pi mB\varrho \in \mathcal {R}_{(C,B,D)}\), then \((\pi mB\varrho ,\pi m\cup B \varrho )\in \mathcal {V}^r_A\) and \(c(C,m\cup B,D)\not \in K\); as a consequence, \(\pi m\cup B \varrho \not \in \mathcal {P}^r_K\) and then \(\lambda \in {{\mathrm{Crit}}}_{\mathcal {P}^r_K}(\mathcal {V}^r_K)\). This proves that \(\mathcal {R}_K\subseteq {{\mathrm{Crit}}}_{\mathcal {P}^r_K}(\mathcal {V}^r_K)\).
 On \(\mathcal {P}_K^m\):if \(c(A', m, \emptyset )\in K\); otherwise, \(\mathcal {W}^m_K=\emptyset \). Notice that if \(\lambda m\in \mathcal {P}^m_K\) and \(\mu \in \mathcal {P}_{K^0_{A'}}\), then also \(\mu m\in \mathcal {P}^m_K\), which guarantees that this definition is valid.$$\begin{aligned} \mathcal {W}^m_K=\mathcal {W}_{K^0_{A'}}m=\{(\lambda m,\mu m):\; (\lambda ,\mu )\in \mathcal {W}_{K^0_{A'}}\} \end{aligned}$$

On \({{\mathrm{Reg}}}(\mathcal {V}^r_K)\) we take \(\mathcal {V}^r_K\).
 For \((C,B,D)\in {{\mathrm{Br}}}(K)\), let \(\mathcal {Y}_{(C,B,D)}\) be the discrete vector field on \(\mathcal {R}_{(C,B,D)}\) given byThis is isomorphic to the product discrete vector field \(\mathcal {W}_{K_{C}^0}\times \mathcal {W}_{K_D^1}\) on \(\mathcal {P}_{K_{C}^0}\times \mathcal {P}_{K_{D}^1}\) via the isomorphism$$\begin{aligned}&\mathcal {Y}_{(C,B,D)}=\mathcal {W}_{K_{C}^0}mB\mathcal {W}_{K_D^1}\\&\quad =\{(\pi mB\varrho , \pi mB\varrho '):\; \pi \in \mathcal {P}_{K_C^0}, (\varrho ,\varrho ')\in \mathcal {W}_{K_D^1}\}\\&\quad \cup \{(\pi mB\varrho , \pi 'mB\varrho ):\; (\pi ,\pi ')\in \mathcal {W}_{K_C^0}, \varrho \in {{\mathrm{Crit}}}(\mathcal {W}_{K_D^1})\}. \end{aligned}$$of the underlying posets.$$\begin{aligned} \mathcal {P}_{K_{C}^0} \times \mathcal {P}_{K_{D}^1}\ni (\lambda ,\mu ) \mapsto \lambda mB\mu \in \mathcal {R}_{(C,B,D)} \end{aligned}$$
 On \(\mathcal {R}_K\):$$\begin{aligned} \mathcal {Y}_K=\bigcup \mathcal {Y}_{(C,B,D)}. \end{aligned}$$
Recall (5.5) that \(h_A:\mathcal {P}^r_A\rightarrow H\) is the weak Morse function associated to \(\mathcal {V}^r_A\).
Proposition 6.3
If \((\lambda ,\mu )\in \mathcal {Y}^r_K\), then \(h_A(\lambda )=h_A(\mu )\). As a consequence, \((\lambda ,\mu )\in \mathcal {W}^r_K\) implies that \(h_A(\lambda )\mathrel {\dot{\ge }} h_A(\mu )\).
Proof
We have \(h_A(\lambda )=h_A(\mu )=(\#C,1)\) for \((\lambda ,\mu )\in \mathcal {Y}_{(C,B,D)}\). \(\square \)
Proposition 6.4
\(\mathcal {W}_K\) is a gradient field.
Proof
As a consequence of [13, Theorem 1.2] and Theorem 4.3 we obtain
Corollary 6.5
For an Acubical complex \(K\subseteq \square ^A\), the space \(\vec {P}(K)_{\mathbf {0}}^\mathbf {1}\) is homotopy equivalent to a CWcomplex whose kcells correspond to the kdimensional critical cells of \(\mathcal {W}_K\).
The following inductive formula for critical cells of \(\mathcal {W}_K\) is an immediate consequence of the definition of \(\mathcal {W}_K\):
Proposition 6.6
In the next section we will obtain an explicit formula for the critical cells of \(\mathcal {W}_K\).
7 Explicit formula for the critical cells
Definition 7.1
 (a)
\(A=E_1\mathop {\dot{\cup }}\cdots \mathop {\dot{\cup }}E_q\mathop {\dot{\cup }}F_0\mathop {\dot{\cup }}F_1\mathop {\dot{\cup }}\cdots \mathop {\dot{\cup }}F_q\).
 (b)The critical cellassociated to \(((E_j),(F_j))\) belongs to \(\mathcal {P}_K\).$$\begin{aligned} \sigma ((E_j),(F_j)):=\tau _{F_0}\kappa _{E_1}\tau _{F_1}\kappa _{E_2}\cdots \tau _{F_{q1}}\kappa _{E_q}\tau _{F_q}\in \mathcal {P}_A \end{aligned}$$
 (c)
For every \(j\in \{1,\ldots ,q\}\), either \(F_{j1}=\emptyset \) or \(\max (F_{j1})<\max (E_j)\),
 (d)For every \(j\in \{1,\ldots ,q\}\), \(c(C_j,E_j,D_j)\not \in K\), whereIf (b) is satisfied, this is equivalent to the condition \(\tau _{F_0}\kappa _{E_1}\cdots \tau _{F_{j1}}E_j\tau _{F_j}\cdots \kappa _{E_q}\tau _{F_q}\not \in \mathcal {P}_K\). Let \({{\mathrm{CrSeq}}}(K)\) be the set of all critical sequences in K.$$\begin{aligned} C_j&= F_0\cup E_1\cup F_1 \cup E_2\cup \cdots \cup E_{j1}\cup F_{j1}\nonumber \\ D_j&= F_j\cup E_{j+1}\cup F_{j+1} \cup E_{j+2}\cup \cdots \cup E_{q}\cup F_{q} \end{aligned}$$(7.2)
Proposition 7.2
Proof
 There existsrsuch that\(m\in E_r\). Thenwhere \(C_j, D_j\) are defined as in (7.2). From the inductive hypothesis,$$\begin{aligned} \left( (E_j)_{j=1}^{r1},(F_j)_{j=0}^{r1}\right)&\in {{\mathrm{CrSeq}}}(K^0_{C_j})\\ \left( (E_j)_{j=r+1}^q,(F_j)_{j=r}^q\right)&\in {{\mathrm{CrSeq}}}(K^1_{D_j}) \end{aligned}$$The conditions (b) and (d) imply that \((C_j,E_j{\setminus }\{m\}, D_j)\in {{\mathrm{Br}}}(K)\). Therefore, it follows from 6.6 that$$\begin{aligned} \pi&:= \sigma \left( (E_j)_{j=1}^{r1},(F_j)_{j=0}^{r1}\right) \in {{\mathrm{Crit}}}(K^0_{C_j}) \text { and} \\ \varrho&:= \sigma \left( (E_j)_{j=r+1}^{q},(F_j)_{j=r}^{q}\right) \in {{\mathrm{Crit}}}(K^1_{D_j}). \end{aligned}$$$$\begin{aligned} \lambda =\sigma ((E_j)_{j=1}^q,(F_j)_{j=0}^q)= \pi  \kappa _{E_j} \varrho = \pi  m  E_j{\setminus }\{m\} \varrho \in {{\mathrm{Crit}}}(\mathcal {W}_K). \end{aligned}$$
 \(m\not \in E_r\)for everyr. Then the condition (c) guarantees that \(m\in F_q\). Letit is easy to check that \(\lambda '\in {{\mathrm{CrCell}}}(K^0_{A'})\). As above, it follows from the inductive hypothesis and 6.6 that \(\lambda =\lambda 'm\in {{\mathrm{Crit}}}(\mathcal {W}_K)\).$$\begin{aligned} \lambda '=\sigma ((E_j)_{j=1}^q, (F_0,F_1,\ldots , F_{q1}, F_q{\setminus }\{m\})); \end{aligned}$$
 \(\lambda =\lambda 'm\)for\(\lambda '\in {{\mathrm{Crit}}}(\mathcal {W}_{K^0_{A'}})\). By the inductive hypothesis, \(\lambda '\in {{\mathrm{CrCell}}}(K^0_{A'})\) and then \(\lambda '=\sigma ((E_j)_{j=1}^q,(F_j)_{j=0}^q)\) for \(((E_j),(F_j))\in {{\mathrm{CrSeq}}}(K)\). Clearly,Thus, \(\lambda \in {{\mathrm{CrCell}}}(K)\), since it is the critical cell of the sequence above.$$\begin{aligned} ((E_j)_{j=1}^q, (F_0,\ldots ,F_{q1},F_{q}\cup \{m\}))\in {{\mathrm{CrSeq}}}(K). \end{aligned}$$
 \(\lambda =\pi mB\varrho \) for \((C,B,D)\in {{\mathrm{Br}}}(K)\), \(\pi \in {{\mathrm{Crit}}}(\mathcal {W}_{K^0_{C}})\), \(\varrho \in {{\mathrm{Crit}}}(\mathcal {W}_{K^1_{D}})\). By the inductive hypothesis,for \(((E^\pi _j),(F^\pi _j))\in {{\mathrm{CrSeq}}}(K^0_C)\), \(((E^\varrho _j),(F^\varrho _j))\in {{\mathrm{CrSeq}}}(K^1_D)\). Now \(\lambda \) is associated to a sequence$$\begin{aligned} \pi&=\sigma \left( (E^\pi _j)_{j=1}^q,(F^\pi _j)_{j=0}^q\right) ,\\ \varrho&=\sigma \left( (E^\varrho _j)_{j=1}^r,(F^\varrho _j\right) _{j=0}^r), \end{aligned}$$which is critical in K; the only nontrivial fact to check is that either \(E^\pi _q=\emptyset \) or \(\max (E^\pi _q)<\max (B\cup \{m\})\), which is guaranteed since m is a maximal element of A.$$\begin{aligned} \left( (E_1^\pi ,\ldots ,E_q^\pi ,B\cup \{m\},E_1^\varrho ,\ldots ,E_r^\varrho ),(F_0^\pi ,\ldots ,F_q^\pi ,F_0^\varrho ,\ldots ,F^\varrho _r)\right) , \end{aligned}$$
Theorem 7.3
Let K be an Acubical complex. Then \(\mathcal {P}_K\simeq \vec {P}(K)_\mathbf {0}^\mathbf {1}\) is homotopy equivalent to a CWcomplex \(X_K\) that has exactly \(\# {{\mathrm{CrSeq}}}^d(K)\) cells of dimension d.
8 Euclidean cubical complexes
Euclidean cubical complexes [11] constitute a class of semicubical sets which is especially important for applications in concurrency, since they include state spaces of PVprograms [3, 5, 14]. We recall the definition here and show that every finite Euclidean cubical complex can be embedded into the standard cube. Therefore, Euclidean cubical complexes can be regarded as Acubical complexes.
Definition 8.1
Remark
The geometric realization of a Euclidean complex K regarded as a semicubical set is homeomorphic, in a canonical way, with the union of these cubes regarded as subsets of \(\mathbb {R}^n\).
Remark
Instead of the order (8.1) we can use any order such that \((i,j)<(i,j')\) for \(j<j'\). This leads to a different vector field \(\mathcal {W}_{i_\mathbf {k}(K)}\), with possibly another set of critical cells.
The following observation is elementary but will be used frequently:
Proposition 8.2
 (a)
\(\lambda \in \mathcal {P}_{{\boxplus }^\mathbf {k}}\).
 (b)
For every \(i\in \{1,\ldots ,n\}\) and \(j<j'\in \{1,\ldots ,k_i\}\), if \((i,j)\in B_r\) and \((i,j')\in B_{r'}\), then \(r<r'\). \(\square \)
Proposition 8.3
If \(\lambda =B_1\cdots B_l\in \mathcal {P}_{{\boxplus }^\mathbf {k}}\), then, for every \(r\in \{1,\ldots ,l\}\), \(\bar{B}_r\) is a set.
Proof
This follows from Proposition 8.2. \(\square \)
Let \([\mathbf {k}]\) denote the multiset having characteristic function \(\mathbf {k}\), i.e., such that it contains \(i\in \{1,\ldots ,n\}\) exactly \(k_i\) times. An ordered partition of\([\mathbf {k}]\) is a sequence \(\mu =C_1C_2\cdots C_l\), where the \(C_i\) are multisets with all elements in \(\{1,\ldots ,n\}\), such that \(\sum _{i=1}^l \chi _{C_i}=\mathbf {k}\). An ordered partition \(\mu \) is proper if all multisets \(C_i\) are sets, i.e., \(\chi _{C_i}\le \mathbf {1}\). Let \(\mathcal {R}_\mathbf {k}\) be the poset of ordered partitions of \(\mathbf {k}\), ordered by refinement, and let \(\mathcal {R}_\mathbf {k}^{pr}\subseteq \mathcal {R}_\mathbf {k}\) be the subposet of proper partitions.
Proposition 8.4
If \(\lambda =B_1\cdots B_l\in \mathcal {P}_{{\boxplus }^\mathbf {k}}\), then \({{\bar{\lambda }}}=\bar{B}_1\cdots \bar{B}_l\in \mathcal {R}_\mathbf {k}^{pr}\).
Proof
Proposition 8.5
For every proper ordered partition \(E_1E_2\cdots E_l\) of \([\mathbf {k}]\) there exists a unique ordered partition \(\lambda =B_1\cdots B_l\in \mathcal {P}_{{\boxplus }^\mathbf {k}}\) such that \(E_r=\bar{B}_r\) for all \(r\in \{1,\ldots ,l\}\).
Proof
Proposition 8.6
Proof
Proposition 8.7
Assume that \((({E}_j)_{j=1}^q,({F}_j)_{j=0}^q)\in {{\mathrm{CrSeq}}}(i_\mathbf {k}(K))\). Then, for all j, \(\bar{E}_j\) is a set.
Proof
The associated critical cell \(\sigma (({E}_j),({F}_j))\) belongs to \(\mathcal {P}_{i_\mathbf {k}(K)}\) and has the form \(\lambda (r,s)E'_j\mu \), where \((r,s)=\max (E_j)\), \(E'_j=E_j{\setminus }(r,s)\). Then, by 8.3, \(\bar{E}'_j\) is a set. If \(\max (\bar{E}'_j)=r\), then \((r,s')\in E'_j\) for some \(s'\). But 8.2 implies that \(s<s'\), which contradicts the assumption that \((r,s)=\max (E_j)\). Thus, \(r>\max (\bar{E}'_j)\) and then \(\bar{E}_j=\{r\}\cup \bar{E}'_j\) is a set. \(\square \)
Definition 8.8
 (a)
\(\mathbf {b}^0=\mathbf {0}\)
 (b)
\(\mathbf {a}^{q+1}=\mathbf {k}\),
 (c)
\(\mathbf {b}^j\le \mathbf {a}^{j+1}\) for \(0\le j\le q\),
 (d)
\(\mathbf {0}<\mathbf {b}^j  \mathbf {a}^j \le \mathbf {1}\) for \(0<j\le q\).
 (e)
\(L_{\mathbf {b}^{j},\mathbf {a}^{j+1}}\subseteq K\) for \(0\le j\le q\),
 (f)
\([\mathbf {a}^j,\mathbf {b}^j]\not \in K\) for \(0<j\le q\),
 (g)
\([\mathbf {a}^j,\mathbf {a}^j+\mathbf {e}_{m_j}], [\mathbf {a}^j+\mathbf {e}_{m_j},\mathbf {b}_j]\in K\) for \(0<j\le q\), where \(m_j=\max ({{\mathrm{dir}}}([\mathbf {a}^j,\mathbf {b}^j]))\).
We will prove that there is 1–1 correspondence between critical sequences in \(i_\mathbf {k}(K)\) and critical routes in K.
Both constructions preserve the dimension. Let \({{\mathrm{Rt}}}^d(K)\) be the set of critical routes in K having dimension d. By combining the argument above with Proposition 7.2, we obtain
Proposition 8.9
Here follows the main theorem of this Section.
Theorem 8.10
Let \(\mathbf {k}\in \mathbb {Z}^n_{\ge 0}\) and let \(K\subseteq [\mathbf {0},\mathbf {k}]\) be a Euclidean cubical complex. Then \(\vec {P}(K)_\mathbf {0}^\mathbf {k}\) is homotopy equivalent to a CWcomplex \(X_K\) which has exactly \(\#{{\mathrm{Rt}}}^d(K)\) cells of dimension d.
9 Applications
At first glance, it is not clear to which extent the description of the space of directed paths on an Acubical complex provided in Theorems 7.3 and 8.10 can be used for actual calculations. In this section we describe three cases in which this description is optimal, i.e., the cells of the CWcomplex \(X_K\) correspond to the generators of the homology of \(\vec {P}(K)_\mathbf {0}^\mathbf {1}\).
9.1 “No \((s+1)\)–equal” configuration spaces
Fix \(n>0\) and let \(A=\{1<2<\cdots <n\}\); we will write \(\square ^n\) instead of \(\square ^{\{1<2<\cdots <n\}}\).
Proposition 9.1
 (a)
\(\vec {P}(\square ^n_{(s)})_\mathbf {0}^\mathbf {1}\),
 (b)
\(\mathcal {P}_{\square ^n_{(s)}}\),
 (c)
\({{\mathrm{Conf}}}_{n,s}(\mathbb {R})\).
The space \(\vec {P}(\square ^n_{(s)})_\mathbf {0}^\mathbf {1}\) plays an important role in concurrency, since it is the execution space of a PVprogram consisting of n processes each of which using a resource of capacity s once.
Proposition 9.2
Proof
Condition (b) in Definition 7.1 implies that \(\# E_j\le s+1\), and condition (d) implies that \(\#E_j>s\) for all j. \(\square \)
As a consequence, we obtain
Proposition 9.3
Proof
The first statement follows from 7.3 and 9.2. If \(s>2\), no cells having consecutive dimensions appear, which implies the second statement. \(\square \)
For \(s=2\), a calculation of the homology groups requires checking the incidence numbers of cells having consecutive dimensions. This can be done using methods from, for example, [9, Chapter 11]. We omit these technical calculations here.
9.2 Generalized “no \((s+1)\)–equal” configuration spaces.
The following proposition is an analogue of 9.1, and its proof is similar.
Proposition 9.4
 (a)
\(\vec {P}([\mathbf {0},\mathbf {k}]_{(s)})_\mathbf {0}^\mathbf {k}\),
 (b)
\(\vec {P}({\boxplus ^\mathbf {k}_{(s)}})_\mathbf {0}^\mathbf {1}\),
 (c)
\(\mathcal {P}_{{\boxplus }^\mathbf {k}_{(s)}}\),
 (d)
\({{\mathrm{Conf}}}_{\mathbf {k},s}(\mathbb {R})\).\(\square \)
In terms of PVprograms, the space \(\vec {P}([\mathbf {0},\mathbf {k}]_{(s)})_\mathbf {0}^\mathbf {k}\) is the execution space of a PVprogram with a single resource of capacity s and n processes. The ith process acquires the resource exactly \(k_i\) times.
Proposition 9.5
Fix \(s>0\). A route \(((\mathbf {a}^j)_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) to \(\mathbf {k}\) is a critical route in \([\mathbf {0},\mathbf {k}]_{(s)}\) if and only if \(\dim ([\mathbf {a}^j,\mathbf {b}^j])=s+1\) for \(j\in \{1,\ldots ,q\}\). In particular, the dimension of the critical route equals \(q(s1)\).
Proof
The condition 8.8(e) is satisfied since \(s>0\), and the condition \(\dim ([\mathbf {a}^j,\mathbf {b}^j])=s+1\) is equivalent to the conditions 8.8(f)–(g). \(\square \)
Immediately from Theorem 8.10 follows the analogue of Proposition 9.3.
Proposition 9.6
This recovers results obtained by Meshulam and Raussen in [10, Section 5.3].
9.3 Directed path spaces on K for \([\mathbf {0},\mathbf {k}]_{(n1)}\subseteq K\subseteq [\mathbf {0},\mathbf {k}]\)
This case was considered in [11].
Proposition 9.7
Assume that \(n\ge 2\), \(\mathbf {0}\le \mathbf {k}\in \mathbb {Z}^n\) and that K is a Euclidean cubical complex such that \([\mathbf {0},\mathbf {k}]_{(n1)}\subseteq K\subseteq [\mathbf {0},\mathbf {k}]\). Then a route to \(\mathbf {k}\) given by \(((\mathbf {a}^j)_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) is a critical route in K if and only if \([\mathbf {a}^j,\mathbf {b}^j]\not \in K\) for \(j\in \{1,\ldots ,n\}\). In particular, this implies that \(\mathbf {a}^j=\mathbf {b}^j\mathbf {1}\) for \(j\in \{1,\ldots ,n\}\) and that the dimension of the critical route equals \(q(n2)\).
Proof
The conditions (e) and (g) in Definition 8.8 are trivially satisfied so only the condition (f) remains. \(\square \)
Note that there is 1–1 correspondence between critical routes in K and cube sequences in K defined in [11, Section 1.4]: if \(((\mathbf {a}^j)_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) is a critical route in K, then \([\mathbf {b}^1,\ldots ,\mathbf {b}^q]\) is a cube sequence and, inversely, a cube sequence \([\mathbf {b}^1,\ldots ,\mathbf {b}^q]\) determines the critical route \(((\mathbf {b}^j\mathbf {1})_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) (where \(\mathbf {b}^0=\mathbf {0}\), \(\mathbf {b}^{q+1}=\mathbf {k}+\mathbf {1}\)).
Thus, the main theorem of [11] (Theorem 1.1) follows immediately from Theorem 8.10 if \(n\ne 3\) since there are no critical routes having consecutive dimensions. For \(n=3\), the homology calculation requires, as in the previous cases, some additional calculations we do not present here.
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