Directed Path Spaces via Discrete Vector Fields

DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS arXiv:1708.02055v1 [math.AT] 7 Aug 2017 KRZYSZTOF ZIEMIAŃSKI Abstract. Let K be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space ~ (K)w P v of directed paths between two given vertices v, w of K. In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1–1 correspondence with the generators of ~ (K)w the homology of P v. 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)w v that is homotopy w ~ equivalent to the space of directed paths P (K)v 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 w gradient field [6] WK on W (K)w v . This shows that P(K)v is homotopy equivalent to an even smaller CW-complex X(K) whose cells correspond to the critical cells of WK . Furthermore, explicit formulas describing the set of critical cells of WK are provided. This construction allows to calculate the homology groups of P(K)w v , 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 by dimensional reasons. This way we reprove here the result of Bjorner and Welker [2], who calculate the homology of ”not (k + 1)–equal” configuration spaces on the real line, as well as its 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 sum of cubes ~ n . Since state spaces of PV-programs [3] having integral coordinates in the directed Euclidean space R 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 critical cells of WK 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, P~ (X)), where X is a topological space and P~ (X) ⊆ P (X) = 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 P~ (X) will be called directed paths or d-paths. For x, y ∈ X, P~ (X)yx denotes the space of d-paths starting at x and ending at y. Prominent examples of ~ n = (Rn , P~ (R ~ n )), d-spaces are the directed n-cube I~n = (I n , P~ (I~n )) and the directed Euclidean space R n n ~ ~ ~ ~ where P (I ) and P (R ) are the spaces of all paths having non-decreasing coordinates. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02–097 Warszawa, Poland. E-mail: [email protected]. 1 2 KRZYSZTOF ZIEMIAŃSKI A semi-cubical set K is a sequence of disjoint sets (K[n])n≥0 , equipped with face maps dεi : K[n] → K[n − 1], where n ≥ 0, i ∈ {1, . . . , n} and ε ∈ {0, 1}, that satisfy pre-cubical relations, i.e., dεi dηj = dηj−1 dεi 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 Cell(K) or by K if it does not lead to confusion. It is partially ordered by inclusion, i.e. c ⊆ c′ if c is the image of c′ under some composition of face maps. Every cube c ∈ K[n] has the initial vertex d0 (c) = d01 . . . d01 (c) and the final vertex d1 (c) = d11 . . . d11 (c), where n face maps appear in both compositions. The geometric realization of a semi-cubical set K is a d-space a K[n] × I~n /(dεi (c), x) ∼ (c, δiε (x)), (2.1) |K| = n≥0 δiε (s1 , . . . , sn−1 ) where = (s1 , . . . , si−1 , ε, si , . . . , sn−1 ). A path α ∈ P (|K|) is directed if there exist numbers 0 = t0 < t1 < · · · < tl = 1, cubes ci ∈ K[ni ] and directed paths βi in I~ni such that α(t) = (ci , βi (t)) for t ∈ [ti−1 , ti ]. For a semi-cubical set K and a pair of its vertices v, w ∈ K[0], a cube chain in K from v to w in K is a sequence of cubes c = (c1 , . . . , cl ), ci ∈ K[ni ], ni > 0, that satisfies the following conditions: • d0 (c1 ) = v, • d1 (cl ) = w, • d1 (ci ) = d0 (ci+1 ) for i ∈ {1, . . . , l − 1}. The set of all cube chains in K from v to w is denoted by Ch(K)w v . There is a natural partial order on Ch(K)w v 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 6= c′ are cubes of K, then {d0 (c), d1 (c)} = 6 0 ′ {d (c ), d1 (c′ )}. Under this assumption, the following holds: Theorem 2.1 ([13, Theorems 1.2 and 1.3]). Let v, w ∈ K[0] be vertices of K. (a) There is a homotopy equivalence w P~ (|K|)w v ≃ | Ch(K)v |, w where | Ch(K)w v | denotes the geometric realization of the nerve of Ch(K)v . w (b) | Ch(K)v | carries a natural structure of a regular CW-complex with closed cells having the form w | Ch≤c (K)| for c ∈ Ch(K)w v , where Ch≤c (K) ⊆ Ch(K)v is a subposet of cube chains that are finer than c.  In this paper we restrict to the case when K is a semi-cubical subset of a standard cube. The standard n-cube n is a semi-cubical set whose k–cubes n [k] are sequences (e1 , . . . , en ), ei ∈ {0, 1, ∗} having exactly k entries equal to ∗. A face map dεi converts the i–th occurrence of ∗ into ε. It is easy to see that the geometric realization of n is d-homeomorphic to the directed cube I~n . Furthermore, every semi-cubical subset of 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 n . Thus, for convenience, the coordinates will be indexed by an arbitrary finite ordered set A rather than by {1, . . . , n}. In the case when A is non-empty, m ∈ A denotes its maximal element and A′ = A \ {m}. Let #X denote the cardinality of a finite set X. Definition 2.2. The standard A–cube A is a semi-cubical set such that • A [k] is the set of all functions c : A → {0, 1, ∗} such that #(c−1 (∗)) = k. • For c ∈ A [k], if c−1 (∗) = {b1 < b2 < · · · < bk }, then ( ε if a = bi ε di (c)(a) = c(a) otherwise. An A–cubical complex is a semi-cubical subset of A . DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 3 We identify |A | with the directed A–cube I~A ; thus, the geometric realization of an A–cubical complex is a subspace of I~A . Let us introduce a notation for cubes of A . For subsets B1 , B∗ , B0 ⊆ A such that A = B1 ∪˙ B∗ ∪˙ B0 , let c(B1 , B∗ , B0 ) be a cube of A such that   0 for a ∈ B0 (2.2) c(B1 , B∗ , B0 )(a) = ∗ for a ∈ B∗   1 for a ∈ B1 . The dimension of c(B1 , B∗ , B0 ) equals #B∗ , and c(B0 , B∗ , B1 ) ⊆ c(B0′ , B∗′ , B1′ ) if and only if B0′ ⊆ B0 , B1′ ⊆ B1 and B∗ ⊆ B∗′ . For a subset B ⊆ A and ε ∈ {0, 1}, let K|εB ⊆ B be the set of functions c : B → {0, 1, ∗} such that the function ( c(a) for a ∈ B (2.3) A ∋ a 7→ ε for a 6∈ B belongs to K. Clearly, K|εB is a B–cubical complex. After passing to geometric realizations, the restriction of K corresponds to the intersection with the suitable face of the directed A–cube, i.e., there is a homeomorphism ∼ |K| ∩ {(ta )a∈A : ∀a∈A\B ta = ε}. (2.4) |K|ε | = B 3. Ordered partitions and cube chains in A–complexes Definition 3.1. An ordered partition of a set A is a sequence λ = B1 |B2 | . . . |Bl(λ) of non-empty S disjoint subsets of A such that A = li=1 Bi . We say that an ordered partition µ = C1 | . . . |Cl(µ) is finer than λ if there exists a sequence of integers 0 = r(0) < r(1) < r(2) < · · · < r(l(λ)) = l(µ) such that Bi = Cr(i−1)+1 ∪ Cr(i−1)+2 ∪ · · · ∪ Cr(i) for all i ∈ {1, . . . , l(λ)}. Let PA be the poset of all ordered partitions of A, with a partial order such that µ ≤ λ if and only if µ is finer than λ. We will define an isomorphism between PA and the poset Ch(A )10 , where 0, 1 ∈ A [0] stand for the constant functions having values 0 and 1 respectively. Pick a cube chain c = (c1 , . . . , cl ) ∈ Ch(A )10 , A and let Bic = c−1 i (∗). For every cube c ∈  and a ∈ A we have ( c(a) for c(a) 6= ∗ ε d (c)(a) = ε for c(a) = ∗. Thus, from the condition d1 (ci ) = d0 (ci+1 ) follows that • ci (a) = 0 implies ci−1 (a) = 0, • ci (a) = 1 implies ci+1 (a) = 1, • ci (a) = ∗ implies ci−1 (a) = 0 and ci+1 (a) = 1. Moreover, • c1 (a) 6= 1 for all a ∈ A, since d0 (c1 ) = 0, • cl (a) 6= 0 for all a ∈ A, since d1 (cl ) = 1. l Thus, for every a ∈ A, the value ∗ appears exactly once in the sequence (ci (a))i=1 ; all preceding elements are 0 and all succeeding ones are 1. As a consequence, c determines an ordered partition λc := B1c |B2c | . . . |Blc of A. On the other hand, an ordered partition λ = B1 | . . . |Bl determines a cube chain cλ = (cλ1 , . . . , cλl ), where (3.1) cλi = c(B1 ∪ · · · ∪ Bi−1 , Bi , Bi+1 ∪ · · · ∪ Bl ). 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 4 KRZYSZTOF ZIEMIAŃSKI Proposition 3.2. The map Ch(A )10 ∋ c 7→ B1c | . . . |Blc ∈ PA is an isomorphism of posets. This is a slight reformulation of [13, Proposition 8.1]. Now let K be an A–cubical complex. Define (3.2) PK := {λ ∈ PA : cλ ∈ Ch(K)10 } = {λ ∈ PA : ∀i∈{1,...,l(λ)} cλi ∈ K}. This is the image of Ch(K)10 under the isomorphism in Proposition 3.2. As a consequence, there is a sequence of homotopy equivalences P~ (|K|)10 ≃ | Ch(K)10 | ≃ |PK |. (3.3) The following criterion will be used later; it follows immediately from the definitions. Proposition 3.3. Assume that A = C ∪˙ B1 ∪˙ . . . ∪˙ Bk ∪˙ D is a partition of A and λ ∈ PC , µ ∈ PD . Then the following conditions are equivalent: (a) λ|B1 | . . . |Bk |µ ∈ PK . (b) λ ∈ P(K|0C ), µ ∈ P(K|1D ) and c(C ∪ B1 ∪ · · · ∪ Bi−1 , Bi , Bi+1 ∪ · · · ∪ Bk ∪ D) ∈ K for every i ∈ {1, . . . , k}.  In the remaining part of the paper we will examine the poset PK with 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]. Definition 4.1 ([1]). A poset P is a CW-poset if, for every a ∈ P , |P<a | is homeomorphic to a sphere S d(a)−1 . Elements of a CW-poset will be called cells and the integer d(a) will be called the dimension of a cell a. If a < b ∈ P and d(a) = d(b) − 1, then a will be called a facet of b and we will write a ≺ b. For a CW-poset P , |P | has a natural CW-structure; its closed k–cells have the form |P≤a |, for a ∈ 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. Definition 4.2. Let P be a subposet of a finite CW-poset. • A discrete vector field V on P is a set of pairwise disjoint pairs (called vectors) (a, b), a, b ∈ P such that a is a facet of b. • A flow of V is a sequence of cells of P (4.1) (a1 , b1 , a2 , b2 , . . . , ak , bk , ak+1 ) such that, for all i ∈ {1, . . . , k}, – (ai , bi ) ∈ V, – (ai+1 , bi ) 6∈ V – ai+1 ≺ bi . • A cycle of V is a flow (4.1) such that a1 = ak+1 , k > 0. • A discrete vector field V is a gradient field if it admits no cycle. For a discrete vector field V on a CW-poset P , let [ {a, b} (4.2) Reg(V) = (a,b)∈V DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 5 be the set of regular cells of V, and let Crit(V) = P \ Reg(V) be the set of critical cells of V. If P ⊆ Q and Q is a CW-poset, then V can be regarded as a discrete vector field on Q. In such case, we will write CritP (V) or CritQ (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: Theorem 4.3 ([9, Theorem 11.13]). Assume that P is a CW-poset and V is a gradient field on P . Then there exists a CW-complex W (V) that is homotopy equivalent to |P |, whose d–dimensional cells are in 1-1 correspondence with d–dimensional critical cells of V. For convenience, we will use a notion of discrete Morse function which is slightly different from the original one. Definition 4.4. Let V be a discrete vector field on a finite CW-poset P . A (discrete) Morse function associated to V is a function h : P → H, where H is an ordered set, such that, for every a ≺ b ∈ P , (a) (a, b) ∈ V implies that h(a) > h(b), (b) (a, b) 6∈ V implies that h(a) ≤ h(b). Lemma 4.5. Let V be a discrete vector field on a finite CW-poset P . If there exists a Morse function associated to V, then V is a gradient field. Proof. If (a1 , b1 , . . . , ak , bk+1 ) is a flow in V, then h(a1 ) > h(b1 ) ≥ h(a2 ) > · · · ≥ h(ak ) > h(ak+1 ); thus, a1 6= ak+1 .  We say that a subposet Q of a poset P is closed if, for x ≤ y ∈ P , y ∈ Q implies that x ∈ Q. Lemma 4.6. Let P be finite CW-poset and let Q ⊆ P is a closed subposet. Let VQ , VP \Q be a discrete vector fields on Q and P \ Q respectively. Then every cycle of V = VQ ∪ VP \Q is contained either in Q or in P \ Q. In particular, if both VQ and VP \Q are gradient fields, then V is also a gradient field. Proof. Let (a1 , b1 , . . . , ak , bk , a1 ) be a cycle. By the assumptions, ai ∈ Q implies bi ∈ Q, and bi ∈ Q implies ai+1 ∈ Q, since ai+1 ≺ bi . Thus, either all elements of the cycle are in Q or none is.  We will give now some examples of gradient fields. While the first two examples are not crucial in proving 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 \ {m}. Example 4.7 (Simplices). The A–simplex is a poset ∆A of non-empty subsets of A. The standard vector field on ∆A is ∆ SA = {(B, B ∪ {m}) : ∅ 6= B ⊆ A′ }. This is a gradient field, since ( 0 if m ∈ B A h : ∆ ∋ B 7→ ∈ {0, 1} 1 if m 6∈ B ∆ is {m}. is a Morse function. The only critical cell of SA Example 4.8 (Cubes). Here A = Cell(A ) denotes the poset of cubes of the standard A–cube. Define A,m = {c ∈ A : c(m) = 0} and A,r = A \ A,m . Clearly, A,m is a closed subposet of ′  r m on posets A,m , A,r and and SA , SA A , which is isomorphic to A . We define vector fields SA A  , respectively, inductively as follows. We put S∅m = S∅r = S∅ = ∅, and for A 6= ∅ let 6 KRZYSZTOF ZIEMIAŃSKI  m (a) SA = {(f, g) : (f |A′ , g|A′ ) ∈ SA ′ , f (m) = g(m) = 0}. r (b) SA = {(f, g) : f |A′ = g|A′ , f (m) = 1, g(m) = ∗}  m r (c) SA = SA ∪ SA .  We will prove inductively that SA is a gradient field. This is obvious for A = ∅. By the inductive ′ m  r hypothesis, SA ≃ SA′ is a gradient field on A,m ≃ A , and SA admits a Morse function A,r ∋ c 7→ c(m) ∈ {∗ < 1} A and hence it is also a gradient field. Now Lemma 4.6 implies that S is a gradient field. The only  critical cell of SA is a 0–cell 0 = (0, . . . , 0). Example 4.9 (Product of discrete vector fields). Let P, Q be finite CW-posets, and let V, W be discrete vector fields on P and Q, respectively. Define a discrete vector field V × W on P × Q by (4.3) V × W = {((p, q), (p, q ′ )) : p ∈ P, (q, q ′ ) ∈ W} ∪ {((p, q), (p′ , q)) : (p, p′ ) ∈ V, q ∈ Crit(W)}. We have Crit(V × W) = Crit(V) × Crit(W). Notice that gradient fields V × W and W × V are not, in general, equal. Proposition 4.10. Assume that P, Q are finite CW-posets and V and W are gradient fields in P and Q, respectively. Then V × W is a gradient field on P × Q. Proof. Let ((p1 , q1 ), (p2 , q2 ), . . . , (p2k+1 , q2k+1 ) = (p1 , q1 )) be a cycle of V × W, i.e., ((p2i−1 , q2i−1 ), (p2i , q2i )) ∈ V × W, (p2i+1 , q2i+1 ) ≺ (p2i , q2i ). Assume that pi are not all equal, and let pi(1) , pi(2) , . . . , pi(r) = pi(1) , i(1) < i(2) < · · · < i(r) be all different values of pi . For s ∈ {1, . . . , r − 1}, the dimensions of pi(s) and pi(s+1) ) differ by 1; if dim(pi(s) ) = dim(pi(s+1) ) − 1, then (pi(s) , pi(s+1) ) ∈ V which implies that (pi(s+1) , pi(s+2) ), since no cell may belong to two different vectors, and then pi(s+2) ≺ pi(s+1) . As a consequence, either (pi(1) , pi(2) , . . . , pi(r) = pi(1) ) or (pi(2) , pi(3) , . . . , pi(r) = pi(1) , pi(2) ) is a cycle of V. If all pi ’s are equal, then (q1 , . . . , q2k+1 ) is a cycle of W. In both cases we get a contradiction.   Notice that the standard gradient fields on cubes can be defined alternatively by formulas S{x} =    {(1, ∗)}, SA = S{m} × SA′ . 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 ∈ A is a maximal element and A′ = A \ {m}. We S will write A = B1 ∪˙ . . . ∪˙ Bn when B1 , . . . , Bn are pairwise disjoint family of subsets of A such that Bi = A and the order on every Bi is inherited from A. Recall that PA denotes the poset of ordered partitions of A. PA is a CW-poset whose geometrical realization is a permutahedron on letters A [12, p. 18]. For a cell λ = B1 |B2 | . . . |Bl ∈ PA , we have dim(λ) = (#B1 − 1) + (#B2 − 1) + · · · + (#Bl − 1). Denote (5.1) m PA := PA′ |m = {λ|m : λ ∈ PA′ }, r m PA = P A \ PA . m is a closed subposet of PA , which is isomorphic to PA′ . Clearly, PA m r m m r r on PA inductively in the ∪ VA and VA = VA on PA , VA on PA Define discrete vector fields VA r m following way. We put V∅ = V∅ = V∅ = ∅ and, for a A 6= ∅, (5.2) m VA = VA′ |m = {(π|m, ̺|m) : (π, ̺) ∈ VA′ } (5.3) r VA = {(π|m|B|̺, π|m ∪ B|̺) : π ∈ PC , ̺ ∈ PD , B 6= ∅, A′ = C ∪˙ B ∪˙ D}. DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 7 Proposition 5.1. The only critical cell of VA is uA = a1 |a2 | . . . |an , {a1 < a2 < · · · < an } = A having dimension 0. r r Proof. Immediately from the definition follows that Reg(VA ) = PA and m CritPAm (VA ) = CritPA′ (VA′ )|m = {uA′ |m} = {uA }. m r ) ∪ CritPAr (VA ) = uA . Thus, CritPA (VA ) = CritPAm (VA  Proposition 5.2. VA is a gradient field. m m ∼ Proof. This is obvious for A = ∅, so assume otherwise. There is an isomorphism (PA , VA ) = m m (PA′ , VA′ ); hence, by the inductive hypothesis, VA is a gradient field on PA . By Lemma 4.6, it r r remains to prove that VA is a gradient field on PA . ˙ on the set H = Z × Z+ in the following way: (s, t) ≤ ˙ (s′ , t′ ) if and only if one of Define an order ≤ the following conditions is satisfied: • s > s′ , • s = s′ and 1 6= t ≤ t′ , • s = s′ and t′ = 1. This is the lexicographic order on the product, with the inverse order on Z and the order 2 < 3 < 4 < ··· < 1 r on Z+ . Every element λ ∈ PA can be written uniquely as (5.4) λ = π|B|̺, where m ∈ B, π ∈ P(C), ̺ ∈ P(D), A = C ∪˙ B ∪˙ D. Let (5.5) hA (λ) = (#C, #B). r PA We will prove that hA : → H is a Morse function. Assume that µ ≺ λ and λ has a presentation r (5.4). If µ = π ′ |B|̺ for π ′ ≺ π ∈ PC , or µ = π|B|̺′ for ̺′ ≺ ̺ ∈ PD , then (µ, λ) 6∈ VA and hA (λ) = hA (µ). Assume otherwise, i.e., that µ = π|B1 |B2 |̺ for B1 ∪˙ B2 = B, B1 , B2 6= ∅. Consider the following cases: r • B1 = {m}. Then (µ, λ) ∈ VA , and ˙ (#C, #B) = hA (λ). hA (µ) = (#C, 1) > r • {m} ( B1 . Then (µ, λ) 6∈ VA and ˙ (#C, #B) = hA (λ), hA (µ) = (#C, #B1 ) < since #B1 > 1. r • m ∈ B2 . Then (µ, λ) 6∈ VA and ˙ (#C, #B) = hA (λ). hA (µ) = (#C + #B1 , #B2 ) < This proves that r VA is a gradient field.  The following picture illustrates the gradient field VA for A = {1, 2, 3}. 1|3|2 3|1|2 1|2|3 (5.6) 3|2|1 2|1|3 2|3|1 8 KRZYSZTOF ZIEMIAŃSKI 6. A gradient field on PK In this Section, we construct a gradient field on PK , for any finite ordered set A and an A–cubical complex K ⊆ A . The starting point is the restriction of VK to PA , i.e., VK = VA |PK = {(λ, µ) ∈ VA : λ, µ ∈ PK }. In general, the discrete vector field VK has critical cells that are not critical cells of VA : if (λ, µ) ∈ VA , λ ∈ PK and µ 6∈ PK , then λ ∈ Crit(VK ). We will add some vectors to VK to reduce the number of critical cells. r r m m m Denote PK = PA ∩ PK , PK = PA ∩ PK . Note that PK is a closed subposet of PK , which is empty ′ if c(A , ∅, m) 6∈ K and otherwise there is an isomorphism of CW-posets PK|0 ′ ∋ λ 7→ λ|m ∈ PK . (6.1) A Definition 6.1. A cube c(C, B ∪ {m}, D) ∈ A is a branching cube of K if • c(C, m ∪ B, D) 6∈ K, • c(C, m, B ∪ D), c(C ∪ m, B, D) ∈ K. These conditions imply that B 6= ∅. Sequences (C, B, D) such that c(C, B ∪ {m}, D) is a branching cube of K will be called branching sequences of K. Let Br(K) be the set of all branching sequences of K. For (C, B, D) ∈ Br(K) let r R(C,B,D) = {π|m|B|̺ : π ∈ PK|0C , ̺ ∈ PK|1D } ⊆ PK (6.2) and let (6.3) RK = [ r R(C,B,D) ⊆ PK . (C,B,D)∈Br(K) Clearly, the posets R(C,B,D) are pairwise disjoint, and there is an isomorphism of posets (6.4) R(C,B,D) ∼ = PK|0C × PK|1D , which shifts the dimensions of elements by #B − 1. r r (V Proposition 6.2. For an A–cubical complex K, we have CritPK K ) = RK . r Proof. If λ = π|m|B|̺ ∈ R(C,B,D) , then (π|m|B|̺, π|m ∪ B|̺) ∈ VA and c(C, m ∪ B, D) 6∈ K; as a r r r r (V r (V consequence, π|m ∪ B|̺ 6∈ PK and then λ ∈ CritPK ). This proves that RK ⊆ CritPK K K ). r r r r r (V r Assume that λ ∈ CritPK K ). Since CritPA (VA ) = ∅, there exists µ ∈ PA such that (λ, µ) ∈ VA r r and µ 6∈ PK . The definition of VA implies that λ = π|m|B|̺, µ = π|m ∪ B|̺ for A = C ∪˙ m ∪˙ B ∪˙ D, π ∈ PC , ̺ ∈ PD . By 3.3, λ ∈ PK implies that π ∈ PK|0C , ̺ ∈ PK|1D and c(C, m, B ∪ D), c(C ∪ m, B, D) ∈ K. Thus, since µ 6∈ PK , c(C, m ∪ B, D) cannot belong to K. As a consequence, (C, B, D) ∈ Br(K) and then λ ∈ R(C,B,D) ⊆ RK .  As a consequence, there is a decomposition (6.5) m ˙ r PK = PK ∪ Reg(VK ) ∪˙ RK . We will define inductively discrete vector fields on the components of this decomposition; they are empty if A = ∅, and otherwise they are inductively defined by the following formulas: m : • On PK m WK = WK|0 ′ |m = {(λ|m, µ|m) : (λ, µ) ∈ WK|0 ′ } A A m m and µ ∈ PK|0 ′ , then also = ∅. Notice that if λ|m ∈ PK if c(∅, A′ , m) ∈ K; otherwise, WK A m µ|m ∈ PK , which guarantees that this definition is valid. r r . ) we take VK • On Reg(VK DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 9 • For (C, B, D) ∈ Br(K), let Y(C,B,D) be a discrete vector field on R(C,B,D) Y(C,B,D) = WK|0C |m|B|WK|1D = {(π|m|B|̺, π|m|B|̺′ ) : π ∈ PK|0C , (̺, ̺′ ) ∈ WK|1D } ∪ {(π|m|B|̺, π ′ |m|B|̺) : (π, π ′ ) ∈ WK|0C , ̺ ∈ Crit(WK|1D )}. This is isomorphic to the product discrete vector field WK|0C × WK|1D on PK|0C × PK|1D via the isomorphism PK|0C × PK|1D ∋ (λ, µ) 7→ λ|m|B|µ ∈ R(C,B,D) of the underlying posets. • On RK : YK = Finally, we define r WK = r VK ∪ YK and [ Y(C,B,D) . m r m r = WK ∪ VK ∪ YK . WK = WK ∪ WK (6.6) An easy inductive argument shows that VK ⊆ WK . r r Recall (5.5) that hA : PA → H is a weak Morse function associated to VA . r r Proposition 6.3. If (λ, µ) ∈ YK , then hA (λ) = hA (µ). As a consequence, (λ, µ) ∈ WK implies that ˙ hA (µ). hA (λ) ≥ Proof. We have hA (λ) = hA (µ) = (#C, 1) for (λ, µ) ∈ Y(C,B,D) .  Proposition 6.4. WK is a gradient field. m Proof. Proof by induction with respect to the cardinality of A. By the inductive hypothesis, WK ≃ m m WK|0 ′ is a gradient field on PK ≃ PK|0 ′ . Since PK is closed in PK , by 4.6 it remains to prove that A A r r WK is a gradient field on PK . Assume that (λ1 , µ1 , λ2 , µ2 , . . . , µk , λ1 ) r WK . is a cycle of By 6.3, all values hA (λi ), hA (µi ) are equal; thus, (λi , µi ) ∈ YK for all i, since hA r is a weak Morse function of VK . For every i, λi and µi must lie in the same component R(Ci ,Bi ,Di ) . Moreover, Ci+1 = Ci and Bi+1 ⊆ Bi ; this implies that the triples (Ci , Bi , Di ) are equal for all i. Thus, this cycle is a cycle of Y(C,B,D) for some (C, B, D) ∈ Br(K); this leads to a contradiction since YC,B,D is isomorphic to WK|0C × WK|1D , which is a gradient field by the inductive hypothesis.  As a consequence of [13, Theorem 1.2] and Theorem 4.3 we obtain Corollary 6.5. For an A–cubical complex K ⊆ A , the space P~ (K)10 is homotopy equivalent to a CW-complex whose k–cells correspond to k–dimensional critical cells of WK . The following inductive formula for critical cells of WK is an immediate consequence of the definition of WK : Proposition 6.6. Let K be an A–cubical complex. If A = ∅, then CritPK (WK ) = PK ; if A 6= ∅ ( {λ|m ∈ PK : λ ∈ Crit(WK|0 ′ )} if c(∅, A′ , m) ∈ K m A m (W ) = CritPK K ∅ otherwise [ r r (W {π|m|B|̺ : π ∈ Crit(WK|0C ), ̺ ∈ Crit(WK|1D )}, CritPK K) = (C,B,D)∈Br(K) m r m (W r (W CritPK (WK ) = CritPK K ) ∪ CritPK K) In the next Section we will obtain an explicit formula for the critical cells of WK . 10 KRZYSZTOF ZIEMIAŃSKI 7. Explicit formula for the critical cells For any finite ordered set B = {b1 < b2 < · · · < bl } define τB , κB ∈ PB by (7.1) τB = b1 |b2 | . . . |bl , κB = bl |{b1 , b2 , . . . , bl−1 }; for κB we require that B has at least two elements. Definition 7.1. A critical sequence in an A–cubical complex K is a pair of sequences of subsets ((Ej )qj=1 , (Fj )qj=0 ) of A such that (a) A = E1 ∪˙ . . . ∪˙ Eq ∪˙ F0 ∪˙ F1 ∪˙ . . . ∪˙ Fq . (b) The critical cell σ((Ej ), (Fj )) := τF0 |κE1 |τF1 |κE2 | . . . |τFq−1 |κEq |τFq ∈ PA associated to ((Ej ), (Fj )) belongs to PK . (c) For every j ∈ {1, . . . , q}, either Fj−1 = ∅ or max(Fj−1 ) < max(Ej ), (d) For every j ∈ {1, . . . , q}, c(Cj , Ej , Dj ) 6∈ K, where (7.2) Cj = F0 ∪ E1 ∪ F1 ∪ E2 ∪ · · · ∪ Ej−1 ∪ Fj−1 Dj = Fj ∪ Ej+1 ∪ Fj+1 ∪ Ej+2 ∪ · · · ∪ Eq ∪ Fq If (b) is satisfied, this is equivalent to the condition τF0 |κE1 | . . . |τFj−1 |Ej |τFj | . . . |κEq |τFq 6∈ PK . Let CrSeq(K) be the set of all critical sequences in K. We do not require that the sets Fj are non-empty but the conditions (b) and (d) imply that every Ej has at least two elements. The dimension of a critical sequence ((Ej )qj=1 , (Fj )qj=0 ) is (7.3) dim((Ej )qj=1 , (Fj )qj=0 ) = q X (#Ej − 2), j=1 which is equal to the dimension of the associated critical cell. Let CrSeqd (K) denote the set d– dimensional critical sequences. Let CrCell(K) ⊆ PK be the set of critical cells in K, i.e., the cells that are associated to a critical sequence, and let CrCelld (K) ⊆ CrCell(K) be the subset of d–dimensional cells. Notice that there are bijections CrSeq(K) ∼ = CrCell(K) and CrSeqd (K) ∼ = CrCelld (K) since every critical cell λ determines a unique critical sequence ((Ej ), (Fj )) ∈ CrSeq(K) such that λ = σ((Ej ), (Fj )), and the dimensions of critical sequences and of the associated critical cells coincide. Proposition 7.2. For every A–cubical complex K and d ≥ 0, we have Critd (WK ) = CrCelld (K) ∼ = CrSeqd (K). Proof. This is obvious if A = ∅, so we assume otherwise and proceed inductively. Assume that ((Ej )qj=1 , (Fj )qj=0 ) ∈ CrSeq(K); we will show that λ := σ((Ej ), (Fj )) ∈ Crit(WK ). There are two cases to consider: • There exists r such that m ∈ Er . Then r−1 0 ((Ej )r−1 j=1 , (Fj )j=0 ) ∈ CrSeq(K|Cj ) ((Ej )qj=r+1 , (Fj )qj=r ) ∈ CrSeq(K|1Dj ) where Cj , Dj are defined as in (7.2). From the inductive hypothesis, r−1 0 π := σ((Ej )r−1 j=1 , (Fj )j=0 ) ∈ Crit(K|Cj ) and ̺ := σ((Ej )qj=r+1 , (Fj )qj=r ) ∈ Crit(K|1Dj ). The conditions (b) and (d) imply that (Cj , Ej \ {m}, Dj ) ∈ Br(K). Therefore, from 6.6 follows that λ = σ((Ej )qj=1 , (Fj )qj=0 ) = π|κEj |̺ = π|m|Ej \ {m}|̺ ∈ Crit(WK ). DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 11 • m 6∈ Br for every r. Then the condition (c) guarantees that m ∈ Fq . Let λ′ = σ((Ej )qj=1 , (F0 , F1 , . . . , Fq−1 , Fq \ {m})); it is easy to check that λ′ ∈ CrCell(K|0A′ ). As above, from the inductive hypothesis and 6.6 follows that λ = λ′ |m ∈ Crit(WK ). Now assume that λ ∈ Crit(WK ). Again, by 6.6 there are two cases: • λ = λ′ |m for λ′ ∈ Crit(WK|0 ′ ). By the inductive hypothesis, λ′ ∈ CrCell(K|0A′ ) and then A λ′ = σ((Ej )qj=1 , (Fj )qj=0 ) for ((Ej ), (Fj )) ∈ CrSeq(K). Clearly, ((Ej )qj=1 , (F0 , . . . , Fq−1 , Fq ∪ {m})) ∈ CrSeq(K). Thus, λ ∈ CrCell(K), since it is the critical cell of the sequence above. • λ = π|m|B|̺ for (C, B, D) ∈ Br(K), π ∈ Crit(WK|0C ), ̺ ∈ Crit(WK|1D ). By the inductive hypothesis, π = σ((Ejπ )qj=1 , (Fjπ )qj=0 ), ̺ = σ((Ej̺ )rj=1 , (Fj̺ )rj=0 ), for ((Ejπ ), (Fjπ )) ∈ CrSeq(K|0C ), ((Ej̺ ), (Fj̺ )) ∈ CrSeq(K|1D ). Now λ is associated to a sequence ((E1π , . . . , Eqπ , B ∪ {m}, E1̺ , . . . , Er̺ ), (F0π , . . . , Fqπ , F0̺ , . . . , Fr̺ )), which is critical in K; the only non-trivial fact to check is that either Eqπ = ∅ or max(Eqπ ) < max(B ∪ {m}), which is guaranteed since m is a maximal element of A. We have shown that Crit(Wk ) ∼  = CrSeq(K) and it is clear that dimension is preserved. Theorem 7.3. Let K be an A–cubical complex. Then |PK | ≃ P~ (|K|)10 is homotopy equivalent to a CW-complex XK that has exactly # CrSeqd (K) cells of dimension d. Proof. By 6.4, WK is a gradient field and by 7.2 the number of critical cells of dimension d equals to # CrSeqd (K). The conclusion follows from 4.3.  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, 14] . We recall the definition here and show that Euclidean cubical complexes can be regarded as A–cubical complexes. ~ n is a subset having the form Definition 8.1. An elementary cube in R ~ n : a ≤ x ≤ b} = {(x1 , . . . , xn ) ∈ R ~ n : ∀ni=1 ai ≤ xi ≤ bi }, [a, b] = {x ∈ R where a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ Zn and bi − ai ∈ {0, 1} for all i. The integer |b − a| = P n i=1 (bi − ai ) is the dimension of the cube [b, l]. A set dir([a, b]) = {i ∈ {1, . . . , n} : bi − ai = 1} is the set of directions of the cube [a, b]. ~ n is a family of elementary cubes in R ~ n that is closed with A Euclidean cubical complex K in R respect to taking subsets. K can be regarded as a semi-cubical set: K[d] is the set of all d–dimensional elementary cubes of K, and the face maps are defined as follows. If [a, b] ∈ K[d] and dir([a, b]) = {r(1) < · · · < r(d)}, then dεi ([a, b]) = [a + εer(i) , b − (1 − ε)er(i) ], where ei = (0, . . . , 0, 1, 0, . . . , 0) and 1 stands at the i–th place. This corresponds to taking the lower or the upper face in the i–th direction of the cube. Remark. The geometric realization of a Euclidean complex K regarded as a semi-cubical set is homeomorphic, in a canonical way, with a sum of this cubes regarded as subsets of Rn . 12 KRZYSZTOF ZIEMIAŃSKI Let K be a finite Euclidean cubical complex. Without loss of generality, we can assume that K is contained in a hyperrectangle [0, k], k = (k1 , . . . , kn ) ∈ Zn , since K can be shifted if necessary. Define a poset (8.1) Ak = {(1, 1) < (1, 2) < · · · < (1, k1 ) < (2, 1) < · · · < (2, k2 ) < · · · < (n, 1) < · · · < (n, kn )} For an elementary cube [a, b] ⊆ [0, k] having dimension d, define ik ([a, b]) ∈ Ak [d] by   0 for bi < j, (8.2) ik ([a, b])(i, j) = 1 for j ≤ ai ,   ∗ for ai < j = bi . This definition is valid since bi − ai ∈ {0, 1} for all i, and defines an injective semi-cubical map [0, k] → Ak ; it is elementary to check that this commutes with the face maps. Denote ⊞k := ik ([0, k]) ⊆ Ak . (8.3) For an Euclidean cubical subcomplex K ⊆ [0, k] obviously ik (K) ⊆ ⊞k ⊆ Ak . Thus, K can be regarded as an Ak –cubical complex. 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 Wik (K) , with possibly another set of critical cells. The following observation is elementary but will be frequently used Proposition 8.2. Let λ = B1 | . . . |Bl ∈ PAk . The following conditions are equivalent: (a) λ ∈ P⊞k . (b) For every i ∈ {1, . . . , n} and j < j ′ ∈ {1, . . . , ki }, if (i, j) ∈ Br and (i, j ′ ) ∈ Br′ , then r < r′ .  For a subset B ⊆ Ak , let B̄ be a multiset that contains only elements from {1, . . . , n}, and every i ∈ {1, . . . , n} is contained in B̄ with multiplicity #{j ∈ {1, . . . , ki } : (i, j) ∈ E}. B̄ is the image of B under the projection Ak ∋ (i, j) 7→ i ∈ {1, . . . , n}, with multiplicities preserved. In terms of characteristic functions, we have (8.4) χB̄ (i) = ki X χB (i, j). j=1 Proposition 8.3. If λ = B1 | . . . |Bl ∈ P⊞k , then, for every r ∈ {1, . . . , l}, B̄r is a set. Proof. This follows from Proposition 8.2.  Let [k] denote the multiset having characteristic function k, i.e., such that contains i ∈ {1, . . . , n} exactly ki times. An ordered partition of [k] is a sequence µ = C1 |C2 | . . . |Cl , where Ci are multisets P with all elements in {1, . . . , n}, such that li=1 χCi = k. An ordered partition µ is proper if all multisets Ci are sets, i.e., χCi ≤ 1. Let Rk be the poset of ordered partitions of k, ordered by refinement, and let Rpr k ⊆ Rk be the subposet of proper partitions. Proposition 8.4. If λ = B1 | . . . |Bl ∈ P⊞k , then λ̄ = B̄1 | . . . |B̄l ∈ Rpr k . Proof. For i ∈ {1, . . . , n} we have l X r=1 χB̄r (i) = ki l X X r=1 j=1 χBr (i, j) = ki X χS l (i, j) r=1 Br j=1 Thus, λ̄ is an ordered partition of [k], and by 8.3 this is proper. = ki X χAk (i, j) = ki . j=1  Proposition 8.5. For every proper ordered partition E1 |E2 | . . . |El of [k] there exists a unique ordered partition λ = B1 | . . . |Bl ∈ P⊞k such that Er = B̄r for all r ∈ {1, . . . , l}. DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 13 Proof. Define (8.5) Br = {(i, j) : i ∈ Er and j = 1 + #{s ∈ {1, . . . , r − 1} : i ∈ Es }}. Clearly, B̄r = Er and λ = B1 | . . . |Bl is a partition of Ak , which satisfies the condition 8.2.(b). Thus, λ ∈ P⊞k . On the other hand, if i ∈ Er , then Br must contain a pair (i, j) and 8.2.(b) enforces that j = 1 + #{s ∈ {1, . . . , r − 1}}.  As a consequence, the formula (8.6) Uk : P⊞k ∋ λ = B1 |B2 | . . . |Bl 7→ λ̄ = B̄1 |B̄2 | . . . |B̄l ∈ Rk . defines an isomorphism of posets. For a Euclidean cubical complex K ⊆ [0, k] let RK = Uk (Pik (K) ). (8.7) Proposition 8.6. For a Euclidean cubical complex K ⊆ [0, k], we have RK := {E1 | . . . |El ∈ Rpr k : ∀r∈{1,...,l} [ r−1 X χE s , s=1 r X χEs ] ∈ K}. s=1 Proof. The definition of ik implies that (8.8) c(B1 ∪ · · · ∪ Br−1 , Br , Br+1 ∪ · · · ∪ Bl ) = ik ([ r−1 X s=1 χB̄s , r X χB̄s ]). s=1 The conclusion follows.  Proposition 8.7. Assume that ((Ej )qj=1 , (Fj )qj=0 ) ∈ CrSeq(ik (K)). Then, for all j, Ēj is a set. Proof. The associated critical cell σ((Ej ), (Fj )) belongs to Pik (K) and has the form λ|(r, s)|Ej′ |µ, where (r, s) = max(Ej ), Ej′ = Ej \ (r, s). Then, by 8.3, Ēj′ is a set. If max(Ēj′ ) = r, then (r, s′ ) ∈ Ej′ for some s′ . But 8.2 implies that s < s′ , which contradicts the assumption that (r, s) = max(Ej ). Thus,  r > max(Ēj′ ) and then Ēj = {r} ∪ Ēj′ is a set. For a ≤ b ∈ Zn , the minimal line from a to b is a Euclidean cubical complex La,b such that (8.9) |La,b | = n [ {a1 } × · · · × {ai−1 } × [ai , bi ] × {bi+1 } × · · · × {bn } ⊆ Rn . i=1 j q Definition 8.8. A route to k, where 0 ≤ k ∈ Zn , is a pair of sequences ((aj )q+1 j=1 , (b )j=0 ) of points of n Z such that (a) (b) (c) (d) b0 = 0 aq+1 = k, bj ≤ aj+1 for 0 ≤ j ≤ q, 0 < bj − aj ≤ 1 for 0 < j ≤ q. Let K ⊆ [0, k] be a Euclidean complex. A critical route in K is a route to k such that (e) Lbj ,aj+1 ⊆ K for 0 ≤ j ≤ q, (f) [aj , bj ] 6∈ K for 0 < j ≤ q, (g) [aj , aj + emj ], [aj + emj , bj ] ∈ K for 0 < j ≤ q, where mj = max(dir([aj , bj ])). 14 KRZYSZTOF ZIEMIAŃSKI The following picture illustrates all critical routes in an exemplary Euclidean cubical set. b1 a2 a3 a2 a1 b2 b 1 = a2 (8.10) b0 a1 b0 b1 a1 b0 We will prove that there is 1–1 correspondence between critical sequences in ik (K) and critical routes in K. For ((Ej )qj=1 , (Fj )qj=0 ) ∈ CrSeq(ik (K)) define (8.11) aj = χF̄0 + χĒ1 + χF̄1 + · · · + χĒj−1 + χF̄j−1 bj = χF̄0 + χĒ1 + χF̄1 + · · · + χĒj−1 + χF̄j−1 + χĒj . j q We will check that ((aj )q+1 j=1 , (b )j=0 ) satisfies the conditions (a)-(g) of Definition 8.8. Points (a) and (c) and obvious, and (b) follows from 7.1.(a), since χĀk = k. Points (e) and (g) follow from 7.1.(b) and the definitions of τFj and κEj (7.1), respectively. Finally, point (d) follows from 8.7 and (f) follows from 7.1.(d). Thus, ((aj ), (bj )) is a critical route in K. j q On the other hand, if ((aj )q+1 j=1 , (b )j=0 ) is a critical route in K, then we define sequences of subsets of Ak : (8.12) Ej = {(i, bji ) : i ∈ dir([aj , bj ])} = {(i, r) : i ∈ {1, . . . , n}, aji < r ≤ bji } (8.13) Fj = {(i, r) : i ∈ {1, . . . , n}, bij−1 < r ≤ aji }. The argument similar as above shows that ((Ej )qj=1 , (Fj )qj=0 ) is a critical sequence in an Ak –cubical complex ik (K). Both constructions preserve the dimension. Let Rtd (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. Let K ⊆ [0, k] be a Euclidean cubical complex. For every d ≥ 0 there are bijections CritdPi k (K) (Wik (K) ) ≃ CrSeqd (ik (K)) ≃ Rtd (K).  Here follows the main theorem of this Section. Theorem 8.10. Let k ∈ Zn≥0 and let K ⊆ [0, k] be a Euclidean cubical complex. Then P~ (|K|)k0 is homotopy equivalent to a CW-complex XK which has exactly # Rtd (K) cells of dimension d. Proof. This follows from Theorem 7.3 and Proposition 8.9.  9. Applications At the first glance, it is not clear how efficient is the description of the space of directed paths on an A–cubical complex provided in Theorems 7.3 and 8.10. In this Section we describe three cases in which this description is optimal, i.e., the cells of the CW-complex XK correspond to the generators of the homology of P~ (|K|)10 . ”Not (s + 1)–equal configuration spaces”. For integers 0 < s < n, the ”not (s + 1)-equal” configuration space is (9.1) Conf n,s (R) = {(ti )ni=1 ∈ Rn : ∀t∈R #{i : ti = t} ≤ s}. Homology groups of these spaces were calculated by Björner and Welker [2]. We will reprove their result here. DIRECTED PATH SPACES VIA DISCRETE VECTOR FIELDS 15 The q-skeleton of a semi-cubical complex K is a cubical complex K(q) such that ( K[d] for d ≤ q (9.2) K(q) [d] = ∅ for d > q. and the face maps of K(q) are inherited from K. Fix n > 0 and let A = {1 < 2 < · · · < n}; we will write n instead of {1<2<···<n} . Proposition 9.1. For 0 < s ≤ n, the following spaces are homotopy equivalent: (a) P~ (|n |)10 , (s) (b) |Pn(s) |, (c) Conf n,s (R). Proof. This is a consequence of Section 2 and [10, Section 2.2].  The space P~ (|n(s) |)10 plays an important role in concurrency, since it is the execution space of a PV-program consisting of n processes and each of then uses once a resource having capacity s. Proposition 9.2. Fix s > 1. Then CrSeqd (n(s) ) = ∅ if (s − 1) does not divide d; otherwise, CrSeqq(s−1) (n(s) ) = {((Ej )qj=1 , (Fj )qj=0 ) : ∀j #Ej = s + 1 and max(Fj−1 ) < max(Ej )}, where E1 ∪˙ . . . ∪˙ Eq ∪˙ F0 ∪˙ . . . ∪˙ Fq = {1, . . . , n}. Proof. The condition (b) in Definition 7.1 implies that #Ej ≤ s + 1, and the condition (d) implies that #Ej > s for all j.  As a consequence, we obtain Proposition 9.3. For 0 < s ≤ n, P~ (n(s) )10 is homotopy equivalent to a CW-complex that has exactly # CrSeqd (n(s) ) cells of dimension d. If s > 2, then ( Zb(n,s,q) n 1 ~ Hd (P ((s) )0 ) = 0 for d = q(s − 1), otherwise, where b(n, s, q) := # Critq(s−1) (n(s) ). Proof. The first statement follows from 7.3 and 9.2. If s > 1, no cells having consecutive dimensions appear, which implies the second statement.  For s = 2, a calculation of the homology groups requires checking that the incidence numbers of cells having consecutive dimensions always vanish. This can be done using methods from, for example, [9, Chapter 11]. We omit these technical calculations here. Generalized not (s + 1)–equal configuration spaces. For k = (k1 , . . . , kn ) ∈ Zn+ define a generalized ”not (s + 1)–equal” configuration space: j ki 1 i Conf k,s (R) = {(tji )j=1,...,k i=1,...,n : ∀i ti < · · · < ti and ∀t∈R #{(i, j) : ti = t} ≤ s}. (9.3) This is a generalization of ”not (s+1)–equal” configuration spaces, since Conf n,s (R) = Conf (1,...,1),s (R). The following proposition is an analogue of 9.1, and its proof is similar. Proposition 9.4. For k ∈ Zn+ , the following spaces are homotopy equivalent: (a) P~ ([0, k](s) )k0 , (b) P~ (|⊞k |)1 , (s) 0 (c) |P⊞k |, (s) (d) Conf k,s (R).  16 KRZYSZTOF ZIEMIAŃSKI In terms of PV-programs, the space P~ ([0, k](s) )k0 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 ki times. j q Proposition 9.5. Fix s > 0. A route ((aj )q+1 j=1 , (b )j=0 ) to k is a critical route in [0, k](s) if and only j j if dim([a , b ]) = s + 1 for j ∈ {1, . . . , q}. In particular, the dimension of the critical route equals q(s − 1). Proof. The condition 8.8.(e) is satisfied since s > 0, and the condition dim([aj , bj ]) = s+1 is equivalent to the conditions 8.8.(f)-(g).  Immediately from Theorem 8.10 follows the analogue of Proposition 9.3. Proposition 9.6. Let bk (n, s, q) be the number of q(s − 1)–dimensional routes in [0, k](s) . Then P~ ([0, k](s) )k0 is homotopy equivalent to a CW-complex that has exactly bk (n, s, q) cells of dimension q(s − 1) and no cells having dimension non-divisible by (s − 1). As a consequence, ( Zbk (n,s,q) for d = q(s − 1), n 1 Hd (P~ ((s) )0 ) = 0 otherwise, for s > 2. This recovers results obtained by Meshulam and Raussen in [10, Section 5.3]. Directed path spaces on K for [0, k](n−1) ⊆ K ⊆ [0, k]. This case was considered in [11]. Proposition 9.7. Assume that n ≥ 2, 0 ≤ k ∈ Zn and that K is a Euclidean cubical complex such j q that [0, k](n−1) ⊆ K ⊆ [0, k]. Then a route to k ((aj )q+1 j=1 , (b )j=0 ) is a critical route in K if and only if [aj , bj ] 6∈ K for j ∈ {1, . . . , n}. In particular, this implies that aj = bj − 1 for j ∈ {1, . . . , n} and that the dimension of the critical route equals q(n − 2). Proof. The conditions (e) and (g) in Definition 8.8 are trivially satisfied so only the condition (f) remains.  Note that there is 1-1 correspondence between critical routes in K and cube sequences in K defined j q 1 q in [11, Section 1.4]: if ((aj )q+1 j=1 , (b )j=0 ) is a critical route in K, then [b , . . . , b ] is a cube sequence j q and, inversely, a cube sequence [b1 , . . . , bq ] determines a critical route ((bj − 1)q+1 j=1 , (b )j=0 ) (where 0 q+1 b = 0, b = k + 1). Thus, the main theorem of [11] (Theorem 1.1) follows immediately from Theorem 8.10 if n 6= 3 since there are no critical routes having consecutive dimensions. 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