1 Introduction
The spaces of directed paths on semi-cubical sets play an important role in Theoretical Computer Science [
4,
5]. In the previous paper [
13] the author constructed, for every bi-pointed semi-cubical set (
K,
v,
w) satisfying certain mild assumptions, a regular CW-complex
\(W(K)_v^w\) 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 semi-cubical sets that can be embedded into a standard cube, regarded as a semi-cubical 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 CW-complex
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 PV-programs [
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 d-space [
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 non-decreasing reparametrizations. Paths that belong to
\(\vec {P}(X)\) will be called
directed paths or
d-paths. For
\(x,y\in X\),
\(\vec {P}(X)_x^y\) denotes the space of d-paths starting at
x and ending at
y. Prominent examples of d-spaces are
the directedn-cube\(\vec {I}^n=(I^n,\vec {P}(\vec {I}^n))\) 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 non-decreasing coordinate functions.
A semi-cubical setK is a sequence of disjoint sets \((K[n])_{n\ge 0}\), equipped with face maps\(d^\varepsilon _i:K[n]\rightarrow K[n-1]\), where \(n\ge 0\), \(i\in \{1,\ldots ,n\}\) and \(\varepsilon \in \{0,1\}\), that satisfy the pre-cubical relations, i.e., \(d_i^\varepsilon d_j^\eta = d_{j-1}^\eta d_i^\varepsilon \) for \(i<j\). Elements of K[n] will be called cubes or n--cubes if one needs to emphasize their dimension; 0-cubes and 1-cubes will be called vertices and edges, respectively. The set of all cubes of a semi-cubical 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.
The geometric realization of a semi-cubical set
K is the d-space
$$\begin{aligned} |K|=\coprod _{n\ge 0} K[n]\times \vec {I}^n/(d^\varepsilon _i(c),x)\sim (c,\delta ^\varepsilon _i(x)), \end{aligned}$$
(2.1)
where
\(\delta ^\varepsilon _i(s_1,\ldots ,s_{n-1})=(s_1,\ldots ,s_{i-1},\varepsilon , s_i,\ldots ,s_{n-1})\). A path
\(\alpha \in P(|K|)\) is directed if there exist numbers
\(0=t_0<t_1<\cdots <t_l=1\), cubes
\(c_i\in K[n_i]\) and directed paths
\(\beta _i\) in
\(\vec {I}^{n_i}\) such that
\(\alpha (t)=(c_i,\beta _i(t))\) for
\(t\in [t_{i-1},t_i]\).
For a semi-cubical set
K and a pair of its vertices
\(v,w\in K[0]\),
a cube chain inKfromv to
w in
K is a sequence of cubes
\(\mathbf {c}=(c_1,\ldots ,c_l)\),
\(c_i\in K[n_i]\),
\(n_i>0\), that satisfies the following conditions:
The set of all cube chains in
K from
v to
w is denoted by
\({{\mathrm{Ch}}}(K)_v^w\). There is a natural partial order on
\({{\mathrm{Ch}}}(K)_v^w\) given by the refinement of cube chains, see [
13, Definition 1.1] for details.
Assume that a semi-cubical 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:
In this paper we restrict to the case when
K is a semi-cubical subset of the standard cube.
The standardn-cube\(\square ^n\) is a semi-cubical set whose
k-cubes
\(\square ^n[k]\) 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
i-th occurrence of
\(*\) into
\(\varepsilon \). It is easy to see that the geometric realization of
\(\square ^n\) is d-homeomorphic to the directed cube
\(\vec {I}^n\). Furthermore, every semi-cubical 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 non-empty, \(m\in A\) denotes its maximal element and \(A'=A{\setminus }\{m\}\).
Let \(\#X\) denote the cardinality of a finite set X.
We identify \(|\square ^A|\) with the directed A-cube \(\vec {I}^A\); thus, the geometric realization of an A-cubical complex is a subspace of \(\vec {I}^A\).
Let us introduce a notation for cubes of
\(\square ^A\). For subsets
\(B_1,B_*,B_0\subseteq A\) such that
\(A=B_1\mathop {\dot{\cup }}B_* \mathop {\dot{\cup }}B_0\) is a partition of
A, let
\(c(B_1,B_*,B_0)\) be the cube of
\(\square ^A\) satisfying
$$\begin{aligned} c(B_1,B_*,B_0)(a)={\left\{ \begin{array}{ll} 0 &{}\quad \hbox {for}\ a\in B_0\\ * &{}\quad \hbox {for}\ a\in B_*\\ 1 &{}\quad \text { for } a\in B_1.\\ \end{array}\right. } \end{aligned}$$
(2.2)
The dimension of
\(c(B_1,B_*,B_0)\) equals
\(\#B_*\), and
\(c(B_0,B_*,B_1)\subseteq c(B'_0,B'_*,B'_1)\) if and only if
\(B'_0\subseteq B_0\),
\(B'_1\subseteq B_1\) and
\(B_*\subseteq B'_*\).
For a subset
\(B\subseteq A\) and
\(\varepsilon \in \{0,1\}\), let
\(K|_B^\varepsilon \subseteq \square ^B\) be the set of functions
\(c:B\rightarrow \{0,1,*\}\) such that the function
$$\begin{aligned} A\ni a \mapsto {\left\{ \begin{array}{ll} c(a) &{}\quad \hbox {for}\ a\in B\\ \varepsilon &{}\quad \hbox {for}\ a\not \in B \end{array}\right. } \end{aligned}$$
(2.3)
belongs to
K. Clearly,
\(K|_B^\varepsilon \) is a
B-cubical complex. After passing to geometric realizations, the restriction of
K corresponds to the intersection with a particular face of the directed
A-cube, i.e., there is a homeomorphism
$$\begin{aligned} |K|^\varepsilon _B|\cong |K| \cap \{(t_a)_{a\in A}:\; \forall _{a\in A{\setminus } B}\; t_a=\varepsilon \}. \end{aligned}$$
(2.4)
3 Ordered partitions and cube chains in A-complexes
We will define an isomorphism between
\(\mathcal {P}_A\) and the poset
\({{\mathrm{Ch}}}(\square ^A)_{\mathbf {0}}^\mathbf {1}\), where
\(\mathbf {0},\mathbf {1}\in \square ^A[0]\) stand for the constant functions having values 0 and 1 respectively. Pick a cube chain
\(\mathbf {c}=(c_1,\ldots ,c_l)\in {{\mathrm{Ch}}}(\square ^A)_\mathbf {0}^\mathbf {1}\), and let
\(B^\mathbf {c}_i=c_i^{-1}(*)\). For every cube
\(c\in \square ^A\) and
\(a\in A\) we have
$$\begin{aligned} d^\varepsilon (c)(a)={\left\{ \begin{array}{ll} c(a) &{}\quad \hbox {for}\ c(a)\ne *\\ \varepsilon &{}\quad \text {for c(a)=*.} \end{array}\right. } \end{aligned}$$
Thus, from the condition
\(d^1(c_i)=d^0(c_{i+1})\) follows that
-
\(c_i(a)=0\) implies \(c_{i-1}(a)=0\),
-
\(c_i(a)=1\) implies \(c_{i+1}(a)=1\),
-
\(c_i(a)=*\) implies \(c_{i-1}(a)=0\) and \(c_{i+1}(a)=1\).
Moreover,
-
\(c_1(a)\ne 1\) for all \(a\in A\), since \(d^0(c_1)=\mathbf {0}\),
-
\(c_l(a)\ne 0\) for all \(a\in A\), since \(d^1(c_l)=\mathbf {1}\).
Thus, for every
\(a\in A\), the value
\(*\) appears exactly once in the sequence
\(\left( c_i(a)\right) _{i=1}^l\); all preceding elements are 0 and all succeeding ones are 1. As a consequence,
\(\mathbf {c}\) determines the ordered partition
\(\lambda ^\mathbf {c}:=B^\mathbf {c}_1|B^\mathbf {c}_2| \cdots | B^\mathbf {c}_l\) of
A.
On the other hand, an ordered partition
\(\lambda =B_1|\cdots | B_l\) determines a cube chain
\(\mathbf {c}^\lambda =(c^\lambda _1,\ldots ,c^\lambda _l)\), where
$$\begin{aligned} c^\lambda _i = c(B_1\cup \cdots \cup B_{i-1},B_i,B_{i+1}\cup \cdots \cup B_l). \end{aligned}$$
(3.1)
It is easy to check that these operations are mutually inverse and that finer cube chains correspond to finer partitions. As a consequence, we obtain
This is a slight reformulation of [
13, Proposition 8.1].
Now let
K be an
A-cubical complex. Define
$$\begin{aligned} \mathcal {P}_K:=\{\lambda \in \mathcal {P}_A:\; \mathbf {c}^\lambda \in {{\mathrm{Ch}}}(K)_\mathbf {0}^\mathbf {1}\}=\{\lambda \in \mathcal {P}_A:\; \forall _{i\in \{1,\ldots ,l(\lambda )\}}\; c^\lambda _i\in K\}. \end{aligned}$$
(3.2)
This is the image of
\({{\mathrm{Ch}}}(K)_\mathbf {0}^\mathbf {1}\) under the isomorphism in Proposition
3.2. As a consequence, there is a sequence of homotopy equivalences
$$\begin{aligned} \vec {P}(|K|)_\mathbf {0}^\mathbf {1}\simeq |{{\mathrm{Ch}}}(K)_\mathbf {0}^\mathbf {1}|\simeq |\mathcal {P}_K|. \end{aligned}$$
(3.3)
The following criterion will be used later; it follows immediately from the definitions.
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 CW-posets
In this section we recall some basic facts from Discrete Morse Theory for regular CW-complexes. For detailed expositions of this topic see, for example, [
6,
7,
9, Chapter 11].
For a CW-poset P, |P| has a natural CW-structure; its closed k-cells have the form \(|P_{\le a}|\), for \(a\in P\) having dimension k. The cell poset C(W) of a regular CW-complex W is a CW-poset and |C(W)| is homeomorphic to W by a cell-preserving homeomorphism. In particular, the face poset of a convex polytope is a CW-poset.
For a discrete vector field
\(\mathcal {V}\) on a CW-poset
P, let
$$\begin{aligned} {{\mathrm{Reg}}}(\mathcal {V})=\bigcup _{(a,b)\in \mathcal {V}} \{a,b\} \end{aligned}$$
(4.2)
be the set of
regular cells of
\(\mathcal {V}\), and let
\({{\mathrm{Crit}}}(\mathcal {V})=P{\setminus }{{\mathrm{Reg}}}(\mathcal {V})\) be the set of
critical cells of
\(\mathcal {V}\). If
\(P\subseteq Q\) and
Q is a CW-poset, then
\(\mathcal {V}\) can be regarded as a discrete vector field on
Q. In such a case, we will write
\({{\mathrm{Crit}}}_{P}(\mathcal {V})\) or
\({{\mathrm{Crit}}}_Q(\mathcal {V})\) for the set of critical cells to emphasize which underlying poset we have in mind.
The importance of gradient fields follows from the following theorem:
For convenience, we will use a notion of discrete Morse function which is slightly different from the original one.
We say that a subposet Q of a poset P is closed if, for \(x\le y\in P\), \(y\in Q\) implies that \(x\in Q\).
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 non-empty finite ordered set, m is its maximal element and \(A'=A{\setminus } \{m\}\).
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.
Recall that
\(\mathcal {P}_A\) denotes the poset of ordered partitions of
A.
\(\mathcal {P}_A\) is a CW-poset whose geometrical realization is a permutahedron on letters in
A [
12, p. 18]. For a cell
\(\lambda =B_1|B_2|\cdots |B_l\in \mathcal {P}_A\), we have
$$\begin{aligned} \dim (\lambda )=(\# B_1-1)+(\# B_2-1)+\cdots +(\# B_l-1). \end{aligned}$$
Denote
$$\begin{aligned} \mathcal {P}_A^m := \mathcal {P}_{A'}|m = \{\lambda |m:\; \lambda \in \mathcal {P}_{A'}\},\quad \mathcal {P}^r_A=\mathcal {P}^A{\setminus } \mathcal {P}_A^m. \end{aligned}$$
(5.1)
Clearly,
\(\mathcal {P}_A^m\) is a closed subposet of
\(\mathcal {P}_A\), which is isomorphic to
\(\mathcal {P}_{A'}\).
Define discrete vector fields
\(\mathcal {V}^{r}_{A}\) on
\(\mathcal {P}_A^r\),
\(\mathcal {V}^{m}_{A}\) on
\(\mathcal {P}_A^m\) and
\(\mathcal {V}_{A}=\mathcal {V}^{r}_{A}\cup \mathcal {V}^{m}_{A}\) on
\(\mathcal {P}_A\) inductively in the following way. We put
\(\mathcal {V}^{r}_\emptyset =\mathcal {V}^{m}_\emptyset =\mathcal {V}_\emptyset =\emptyset \) and, for a
\(A\ne \emptyset \),
$$\begin{aligned} \mathcal {V}^m_A&=\mathcal {V}_{A'}|m=\{(\pi |m, \varrho |m):\; (\pi ,\varrho )\in \mathcal {V}_{A'}\} \end{aligned}$$
(5.2)
$$\begin{aligned} \mathcal {V}^r_A&=\{ (\pi |m|B|\varrho , \pi |m\cup B|\varrho ):\; \pi \in \mathcal {P}_C,\; \varrho \in \mathcal {P}_D,\; B\ne \emptyset ,\; A'=C\mathop {\dot{\cup }}B\mathop {\dot{\cup }}D \}. \end{aligned}$$
(5.3)
The following picture illustrates the gradient field
\(\mathcal {V}_{A}\) for
\(A=\{1,2,3\}\).
6 A gradient field on \(\mathcal {P}_K\)
In this Section, we construct a gradient field on
\(\mathcal {P}_K\), for any finite ordered set
A and an
A-cubical complex
\(K\subseteq \square ^A\). The starting point is the restriction of
\(\mathcal {V}_K\) to
\(\mathcal {P}_A\), i.e.,
$$\begin{aligned} \mathcal {V}_K=\mathcal {V}_A|_{\mathcal {P}_K}=\{(\lambda ,\mu )\in \mathcal {V}_A:\; \lambda ,\mu \in \mathcal {P}_K\}. \end{aligned}$$
In general, the discrete vector field
\(\mathcal {V}_K\) has critical cells that are not critical cells of
\(\mathcal {V}_A\): if
\((\lambda ,\mu )\in \mathcal {V}_A\),
\(\lambda \in \mathcal {P}_K\) and
\(\mu \not \in \mathcal {P}_K\), then
\(\lambda \in {{\mathrm{Crit}}}(\mathcal {V}_K)\). We will add some vectors to
\(\mathcal {V}_K\) to reduce the number of critical cells.
Denote
\(\mathcal {P}^r_K=\mathcal {P}_A^r\cap \mathcal {P}_K\),
\(\mathcal {P}^m_K=\mathcal {P}_A^m\cap \mathcal {P}_K\). Note that
\(\mathcal {P}^m_K\) is a closed subposet of
\(\mathcal {P}_K\), which is empty if
\(c(A',\emptyset ,m)\not \in K\) and otherwise there is an isomorphism of CW-posets
$$\begin{aligned} \mathcal {P}_{K|^0_{A'}}\ni \lambda \mapsto \lambda |m \in \mathcal {P}_K. \end{aligned}$$
(6.1)
For
\((C,B,D)\in {{\mathrm{Br}}}(K)\) let
$$\begin{aligned} \mathcal {R}_{(C,B,D)}=\{\pi |m|B|\varrho :\; \pi \in \mathcal {P}_{K|_C^0},\; \varrho \in \mathcal {P}_{K|_D^1}\}\subseteq \mathcal {P}^r_K \end{aligned}$$
(6.2)
and let
$$\begin{aligned} \mathcal {R}_K=\bigcup _{(C,B,D)\in {{\mathrm{Br}}}(K)} \mathcal {R}_{(C,B,D)} \subseteq \mathcal {P}^r_K. \end{aligned}$$
(6.3)
Clearly, the posets
\(\mathcal {R}_{(C,B,D)}\) are pairwise disjoint, and there is an isomorphism of posets
$$\begin{aligned} \mathcal {R}_{(C,B,D)}\cong \mathcal {P}_{K|_C^0}\times \mathcal {P}_{K|_D^1}, \end{aligned}$$
(6.4)
which shifts the dimensions of elements by
\(\#B-1\).
As a consequence, there is a decomposition
$$\begin{aligned} \mathcal {P}_K=\mathcal {P}_K^m \mathop {\dot{\cup }}{{\mathrm{Reg}}}(\mathcal {V}^r_K)\mathop {\dot{\cup }}\mathcal {R}_K. \end{aligned}$$
(6.5)
We will define inductively discrete vector fields on the components of this decomposition; they are empty if
\(A=\emptyset \), and otherwise they are inductively defined by the following formulas:
Finally, we define
\(\mathcal {W}^r_K=\mathcal {V}^r_K \cup \mathcal {Y}_K\) and
$$\begin{aligned} \mathcal {W}_K=\mathcal {W}^m_K\cup \mathcal {W}^r_K=\mathcal {W}^m_K \cup \mathcal {V}^r_K \cup \mathcal {Y}_K. \end{aligned}$$
(6.6)
An easy inductive argument shows that
\(\mathcal {V}_K\subseteq \mathcal {W}_K\).
Recall (
5.5) that
\(h_A:\mathcal {P}^r_A\rightarrow H\) is the weak Morse function associated to
\(\mathcal {V}^r_A\).
As a consequence of [
13, Theorem 1.2] and Theorem
4.3 we obtain
The following inductive formula for critical cells of \(\mathcal {W}_K\) is an immediate consequence of the definition of \(\mathcal {W}_K\):
In the next section we will obtain an explicit formula for the critical cells of \(\mathcal {W}_K\).
8 Euclidean cubical complexes
Euclidean cubical complexes [
11] constitute a class of semi-cubical sets which is especially important for applications in concurrency, since they include state spaces of PV-programs [
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
A-cubical complexes.
We will define an embedding of the hyperrectangle
\([\mathbf {0},\mathbf {k}]\subseteq \vec {\mathbb {R}}^n\),
\(\mathbf {k}=(k_1,\ldots ,k_n)\in \mathbb {Z}^n\), in the standard cube. For
\(n=1\),
\(k\in \mathbb {Z}_{\ge 0}\) and an elementary cube
\([a,b]\subseteq [0,k]\), define
$$\begin{aligned} i_k([a,a])(j)={\left\{ \begin{array}{ll} 0 &{}\quad \hbox {for}\ a<j\\ 1 &{}\quad \hbox {for}\ a\ge j \end{array}\right. } \in \square ^n[0] \qquad i_k([a-1,a])(j)={\left\{ \begin{array}{ll} 0 &{}\quad \hbox {for}\ a<j\\ 1 &{}\quad \hbox {for}\ a>j\\ * &{}\quad \hbox {for}\ a=j \end{array}\right. } \in \square ^n[1]. \end{aligned}$$
This defines the semi-cubical embedding
\(i_k:[0,k]\rightarrow \square ^k=\square ^{\{1<\cdots <k\}}\). For an arbitrary
n, we will construct the embedding which is the product of the
\(i_k\). Define the ordered set
$$\begin{aligned}&A_\mathbf {k}=\{(1,1)<(1,2)<\cdots<(1,{k_1})\nonumber \\&<(2,1)<\cdots<(2,{k_2})<\cdots<(n,1)<\cdots <(n,{k_n})\}. \end{aligned}$$
(8.1)
For an elementary cube
\([\mathbf {a},\mathbf {b}]\subseteq [\mathbf {0},\mathbf {k}]\) having dimension
d, define
\(i_\mathbf {k}([\mathbf {a},\mathbf {b}])\in \square ^{A_\mathbf {k}}[d]\) by
$$\begin{aligned} i_\mathbf {k}([\mathbf {a},\mathbf {b}])(i,j)={\left\{ \begin{array}{ll} 0 &{}\quad \text { for } b_i< j,\\ 1 &{}\quad \text { for } j\le a_i,\\ * &{}\quad \text { for } a_i<j=b_i. \end{array}\right. } \end{aligned}$$
(8.2)
This definition makes sense since
\(b_i-a_i\in \{0,1\}\) for all
i, and defines the injective semi-cubical map
\(i_\mathbf {k}:[\mathbf {0},\mathbf {k}]\rightarrow \square ^{A_\mathbf {k}}\); it is elementary to check that this map commutes with the face maps. Denote
$$\begin{aligned} \boxplus ^\mathbf {k}:=i_\mathbf {k}([\mathbf {0},\mathbf {k}])\subseteq \square ^{A_\mathbf {k}}. \end{aligned}$$
(8.3)
Let
K be a finite Euclidean cubical complex. Without loss of generality, we can assume that
K is contained in the hyperrectangle
\([\mathbf {0},\mathbf {k}]\),
\(\mathbf {k}=(k_1,\ldots ,k_n)\in \mathbb {Z}^n\), since
K can be shifted if necessary. We have
$$\begin{aligned} i_\mathbf {k}(K)\subseteq \boxplus ^\mathbf {k}\subseteq \square ^{A_\mathbf {k}} \end{aligned}$$
(8.4)
Thus,
K can be regarded as an
\(A_\mathbf {k}\)-cubical complex.
The following observation is elementary but will be used frequently:
For a subset
\({B}\subseteq A_\mathbf {k}\), let
\({\bar{B}}\) be the multiset that contains only elements from
\(\{1,\ldots ,n\}\), and every
\(i\in \{1,\ldots ,n\}\) is contained in
\(\bar{B}\) with multiplicity
\(\#\{j\in \{1,\ldots ,k_i\}:\; (i,j)\in {B} \}\).
\(\bar{B}\) is the image of
B under the projection
\(A_\mathbf {k}\ni (i,j)\mapsto i\in \{1,\ldots ,n\}\), with multiplicities preserved. In terms of characteristic functions, we have
$$\begin{aligned} \chi _{\bar{B}}(i)=\sum _{j=1}^{k_i} \chi _{{B}}(i,j). \end{aligned}$$
(8.5)
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 \(\mu =C_1|C_2|\cdots |C_l\), 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.
As a consequence, the formula
$$\begin{aligned} U_\mathbf {k}: \mathcal {P}_{{\boxplus }^\mathbf {k}}\ni \lambda =B_1|B_2|\cdots |B_l\mapsto {\bar{\lambda }}=\bar{B}_1|\bar{B}_2|\cdots |\bar{B}_l \in \mathcal {R}^{pr}_\mathbf {k}. \end{aligned}$$
(8.7)
defines an isomorphism of posets. For a Euclidean cubical complex
\(K\subseteq [\mathbf {0},\mathbf {k}]\) let
$$\begin{aligned} \mathcal {R}_K=U_\mathbf {k}(\mathcal {P}_{i_\mathbf {k}(K)}). \end{aligned}$$
(8.8)
For
\(\mathbf {a}\le \mathbf {b}\in \mathbb {Z}^n\),
the minimal line from\(\mathbf {a}\)to\(\mathbf {b}\) is a Euclidean cubical complex
\(L_{\mathbf {a},\mathbf {b}}\) such that
$$\begin{aligned} |L_{\mathbf {a},\mathbf {b}}|=\bigcup _{i=1}^n \{a_1\}\times \cdots \times \{a_{i-1}\} \times [a_i,b_i] \times \{b_{i+1}\} \times \cdots \times \{b_n\}\subseteq \mathbb {R}^n.\qquad \end{aligned}$$
(8.10)
We will prove that there is 1–1 correspondence between critical sequences in \(i_\mathbf {k}(K)\) and critical routes in K.
For
\((({E}_j)_{j=1}^q,({F}_j)_{j=0}^q)\in {{\mathrm{CrSeq}}}(i_\mathbf {k}(K))\) define
$$\begin{aligned} \mathbf {a}^j&=\chi _{{\bar{F}}_0}+\chi _{{\bar{E}}_1}+\chi _{{\bar{F}}_1}+\cdots +\chi _{{\bar{E}}_{j-1}}+\chi _{{\bar{F}}_{j-1}}\nonumber \\ \mathbf {b}^j&=\chi _{{\bar{F}}_0}+\chi _{{\bar{E}}_1}+\chi _{\bar{F}_1}+\cdots +\chi _{{\bar{E}}_{j-1}}+\chi _{{\bar{F}}_{j-1}}+\chi _{{\bar{E}}_j}. \end{aligned}$$
(8.11)
We will check that
\(((\mathbf {a}^j)_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) satisfies the conditions (a)–(g) of Definition
8.8. Points (a) and (c) and obvious, and (b) follows from
7.1. (a), since
\(\chi _{\bar{A}_\mathbf {k}}=\mathbf {k}\). Points (e) and (g) follow from
7.1(b) and the definitions of
\(\tau _{F_j}\) and
\(\kappa _{E_j}\) (
7.1), respectively. Finally, point (d) follows from
8.7 and (f) follows from
7.1(d). Thus,
\(((\mathbf {a}^j),(\mathbf {b}^j))\) is a critical route in
K.
On the other hand, if
\(((\mathbf {a}^j)_{j=1}^{q+1}, (\mathbf {b}^j)_{j=0}^{q})\) is a critical route in
K, then we define sequences of subsets of
\(A_\mathbf {k}\):
$$\begin{aligned} E_j&= \left\{ (i, b^j_i):\; i\in {{\mathrm{dir}}}([\mathbf {a}^j,\mathbf {b}^j])\right\} =\left\{ (i,r):\; i\in \{1,\ldots ,n\},\; a^{j}_i < r \le b^j_i\right\} \end{aligned}$$
(8.12)
$$\begin{aligned} F_j&= \left\{ (i,r):\; i\in \{1,\ldots ,n\},\; b^{j-1}_i < r \le a^j_i\right\} . \end{aligned}$$
(8.13)
An argument similar to the one above shows that
\(((E_j)_{j=1}^q, (F_j)_{j=0}^q)\) is a critical sequence in the
\(A_\mathbf {k}\)-cubical complex
\(i_\mathbf {k}(K)\).
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
Here follows the main theorem of this Section.
The following picture illustrates all critical routes in an example of a Euclidean cubical complex.
Each of these routes represents one of three components in the path space, each of which is contractible.
9 Applications
At first glance, it is not clear to which extent the description of the space of directed paths on an
A-cubical 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 CW-complex
\(X_K\) correspond to the generators of the homology of
\(\vec {P}(|K|)_\mathbf {0}^\mathbf {1}\).
9.1 “No \((s+1)\)–equal” configuration spaces
For integers
\(0<s<n\), the “no
\((s+1)\)-equal” configuration space is
$$\begin{aligned} {{\mathrm{Conf}}}_{n,s}(\mathbb {R})=\{(t_i)_{i=1}^n\in \mathbb {R}^n:\; \forall _{t\in \mathbb {R}}\; \#\{i:\; t_i=t\}\le s\}. \end{aligned}$$
(9.1)
Homology groups of these spaces were calculated by Björner and Welker [
2]. We will reprove their result here.
Theq-skeleton of a semi-cubical complex
K is the cubical complex
\(K_{(q)}\) such that
$$\begin{aligned} K_{(q)}[d]={\left\{ \begin{array}{ll} K[d] &{}\quad \hbox {for}\ d\le q\\ \emptyset &{}\quad \text {for d>q.} \end{array}\right. } \end{aligned}$$
(9.2)
and the face maps of
\(K_{(q)}\) are inherited from
K.
Fix \(n>0\) and let \(A=\{1<2<\cdots <n\}\); we will write \(\square ^n\) instead of \(\square ^{\{1<2<\cdots <n\}}\).
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 PV-program consisting of n processes each of which using a resource of capacity s once.
As a consequence, we obtain
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.
For
\(\mathbf {k}=(k_1,\ldots ,k_n)\in \mathbb {Z}_+^n\) define the generalized “no
\((s+1)\)–equal” configuration space:
$$\begin{aligned} {{\mathrm{Conf}}}_{\mathbf {k},s}(\mathbb {R})=\left\{ (t_i^j)_{i=1,\ldots ,n}^{j=1,\ldots ,k_i}:\; \forall _i\; t_i^1<\cdots <t_i^{k_i} \text { and } \forall _{t\in \mathbb {R}}\; \#\{(i,j):\; t^j_i=t\}\le s\right\} .\nonumber \\ \end{aligned}$$
(9.3)
This is a generalization of “no
\((s+1)\)–equal” configuration spaces, since
\({{\mathrm{Conf}}}_{n,s}(\mathbb {R})={{\mathrm{Conf}}}_{(1,\ldots ,1),s}(\mathbb {R})\).
The following proposition is an analogue of
9.1, and its proof is similar.
In terms of PV-programs, the space \(\vec {P}([\mathbf {0},\mathbf {k}]_{(s)})_\mathbf {0}^\mathbf {k}\) is the execution space of a PV-program with a single resource of capacity s and n processes. The i-th process acquires the resource exactly \(k_i\) times.
Immediately from Theorem
8.10 follows the analogue of Proposition
9.3.
This recovers results obtained by Meshulam and Raussen in [
10, Section 5.3].
9.3 Directed path spaces on K for \([\mathbf {0},\mathbf {k}]_{(n-1)}\subseteq K\subseteq [\mathbf {0},\mathbf {k}]\)
This case was considered in [
11].
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.