Topology of the immediate snapshot complexes
aa r X i v : . [ c s . D C ] A p r TOPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES
DMITRY N. KOZLOVA bstract . The immediate snapshot complexes were introduced as combinatorial modelsfor the protocol complexes in the context of theoretical distributed computing. In the pre-vious work we have developed a formal language of witness structures in order to defineand to analyze these complexes.In this paper, we study topology of immediate snapshot complexes. It is known thatthese complexes are always pure and that they are pseudomanifolds. Here we prove twofurther independent topological properties. First, we show that immediate snapshot com-plexes are collapsible. Second, we show that these complexes are homeomorphic to closedballs. Specifically, given any immediate snapshot complex P (¯ r ), we show that there existsa homeomorphism ϕ : ∆ | supp ¯ r |− → P (¯ r ), such that ϕ ( σ ) is a subcomplex of P (¯ r ), whenever σ is a simplex in the simplicial complex ∆ | supp ¯ r |− .
1. W itness structures and immediate snapshot protocol complexes
Modeling protocol complexes for the immediate snapshot read / write distributedprotocols. A crucial ingredient in the topological approach to theoretical distributed computing,see Herlihy et al, [HKR], is associating a simplicial complex, called the protocol com-plex , to every distributed protocol, once the computational model is fixed. In this paper,we study topology of standard full-information protocol complexes in one of the centralmodels of computation.Let us fix the computational model to be the immediate snapshot read / write model,which was originally introduced by Borowsky and Gafni in [BG]. Roughly, this meansthat the processes can write their values to the assigned memory registers, and they canread the entire memory in one atomic step (snapshot read). The execution of the protocolmust have a layer structure, where in each layer a group of processes becomes active,the processes in this group atomically write their values to the memory, after this theyatomically read the entire memory. Importantly, there are no further restrictions on howthese layers get activated during the protocol execution.In our previous work, [Ko14b], we introduced combinatorial models for the protocolcomplexes for the standard protocols in that chosen computational model, called immediatesnapshot complexes . For this, we needed to define new combinatorial structures, called witness structures , and study their structure theory, including various operations, such as ghosting . We have proved that the immediate snapshot complexes provide the correctmodel for these protocol complexes, and started to study their topology.The standard protocols are naturally enumerated by finite sequences of nonnegativeintegers, which we called round counters , denoted ¯ r . Accordingly, the immediate snapshotcomplexes themselves were denoted P (¯ r ). In [Ko14b] it was proved that the complexes Key words and phrases. collapses, distributed computing, combinatorial algebraic topology, immediate snap-shot, protocol complexes. P (¯ r ) are always pseudomanifolds with boundary, and the combinatorics of the boundarysubcomplex was described.In this paper, we improve our understanding of topology of P (¯ r ) significantly. We re-fine the notion of canonical subcomplex decomposition of P (¯ r ) from [Ko14b], and givea complete combinatorial description of the incidence relations in this stratification. Thisgives us a good approach to understanding the inner structure of P (¯ r ). In particular, it isstraightforward to prove the contractibility of P (¯ r ) by pairing the combinatorial descrip-tion of this incidence structure with the standard result in combinatorial topology, calledthe Nerve Lemma , see [Ko07]. As a first topological property we show a stronger result:namely, that the complexes P (¯ r ) are always collapsible. The collapsing sequence is alsoexplicitly described.It takes much more e ff ort to derive the second topological property of P (¯ r ), namely thefact that each such complex is homeomorphic to a closed ball of dimension | supp ¯ r | − P (¯ r ), there exists a homeomorphism ϕ : ∆ | supp ¯ r |− → P (¯ r ), such that ϕ ( σ ) is a subcomplex of P (¯ r ), whenever σ is a simplex inthe simplicial complex ∆ | supp ¯ r |− .The work presented here is the rigorous workout of the second part of the preprint[Ko14a]. The detailed expansion of the first part of [Ko14a] has already appeared in[Ko14b], where we laid the combinatorial groundwork for the topological results of thispaper. We spend the rest of this section reminding the notations of [Ko14b] and resultsproved there. Our presentation here is quite condensed and the reader is referred to [Ko14b]for further details. We remark that topology of protocol complexes for related computa-tional models has been studied by many authors, see e.g., [Ha04, HKR, HS, Ko12, Ko13].Furthermore, we recommend Attiya and Welch, [AW], for an in-depth background on the-oretical distributed computing.Fundamentally, this paper can be viewed a stand-alone article, written in a rigorousmathematical fashion, making it possible, in principle, to be read independently. However,we strongly recommend that the reader consults [Ko14b], before starting reading this one.Furthermore, in order both to facilitate researchers who are mainly interested in distributedcomputing, as well as topologists interested in more distributed computing background, weshall comment throughout the text, explaining the distributed computing intuition behindthe mathematical concepts.1.2. Round counters.
To start with, we review some of the standard terminology which we will use. We let Z + denote the set of nonnegative integers { , , , . . . } . For a natural number n we shalluse [ n ] to denote the set { , . . . , n } , with a convention that [ − = ∅ . For a finite subset S ⊂ Z + , such that | S | ≥
2, we let smax S denote the second largest element, i.e., smax S : = max( S \ { max S } ). Finally, for a set S and an element a , we set χ ( a , S ) : = , if a ∈ S ;0 , otherwise.Furthermore, whenever ( X i ) ti = is a family of topological spaces, we set X I : = T i ∈ I X i .Also, when no confusion arises, we identify one-element sets with that element, and write,e.g., p instead of { p } .Next, we proceed to the combinatorial enumeration of all standard protocols, togetherwith relation terminology. This is accomplished by the introduction of the so-called roundcounters . OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 3
Definition 1.1.
Given a function ¯ r : Z + → Z + ∪ {⊥} , we consider the set supp ¯ r : = { i ∈ Z + | ¯ r ( i ) , ⊥} . This set is called the support set of ¯ r.A round counter is a function ¯ r : Z + → Z + ∪ {⊥} with a finite support set. Obviously, a round counter can be thought of as an infinite sequence ¯ r = (¯ r (0) , ¯ r (1) , . . . ),where, for all i ∈ Z + , either ¯ r ( i ) is a nonnegative integer, or ¯ r ( i ) = ⊥ , such that only finitelymany entries of ¯ r are nonnegative integers. We shall frequently use a short-hand notation¯ r = ( r , . . . , r n ) to denote the round counter given by¯ r ( i ) = r i , for 0 ≤ i ≤ n ; ⊥ , for i > n . Definition 1.2.
Given a round counter ¯ r, the number P i ∈ supp ¯ r ¯ r ( i ) is called the cardinality of ¯ r, and is denoted | ¯ r | . The sets act ¯ r : = { i ∈ supp ¯ r | ¯ r ( i ) ≥ } and pass ¯ r : = { i ∈ supp ¯ r | ¯ r ( i ) = } are called the active and the passive sets of ¯ r. Distributed Computing Context 1.3.
Since we consider full-information protocols only,they can be described by specifying the number of rounds each process executes the write-read sequence. Mathematically, these protocols are indexed by round counters. Givena round counter ¯ r, the set supp ¯ r indexes the participating processes, and is required tobe finite. The symbol ⊥ means that the process does not participate. Accordingly, the set pass ¯ r indexes the passive processes, i.e., those, which formally take part in the execution,but which do not actually perform any active steps, while the set act ¯ r indexes the processeswhich execute at least one step. The following special class of round counters is important for our study.
Definition 1.4.
For an arbitrary pair of disjoint finite sets A , B ⊆ Z + we define a roundcounter χ A , B given by χ A , B ( i ) : = , if i ∈ A ;0 , if i ∈ B . Furthermore, for an arbitrary round counter ¯ r, we set χ (¯ r ) : = χ act ¯ r , pass ¯ r . We note that supp ¯ r = supp ( χ (¯ r )). In the paper we shall also use the short-hand notation χ A : = χ A , ∅ .We define two operations on the round counters. To start with, assume ¯ r is a roundcounter and we have a subset A ⊆ Z + . We let ¯ r \ A denote the round counter defined by(¯ r \ A )( i ) = ¯ r ( i ) , if i < A ; ⊥ , if i ∈ A . We say that the round counter ¯ r \ A is obtained from ¯ r by the deletion of A . Note thatsupp (¯ r \ A ) = supp (¯ r ) \ S , act (¯ r \ A ) = act (¯ r ) \ A , and pass (¯ r \ A ) = pass (¯ r ) \ A .Furthermore, we have χ (¯ r \ A ) = χ (¯ r ) \ A . Finally, we note for future reference that for A ⊆ C ∪ D we have(1.1) χ C , D \ A = χ C \ A , D \ A . DMITRY N. KOZLOV
For the second operation, assume ¯ r is a round counter and we have a subset S ⊆ act ¯ r .We let ¯ r ↓ S denote the round counter defined by(¯ r ↓ S )( i ) = ¯ r ( i ) , if i < S ;¯ r ( i ) − , if i ∈ S . We say that the round counter ¯ r ↓ S is obtained from ¯ r by the execution of S . Note thatsupp (¯ r ↓ S ) = supp ¯ r , act (¯ r ↓ S ) = { i ∈ act ¯ r | i < S , or ¯ r ( i ) ≥ } , and pass (¯ r ↓ S ) = pass (¯ r ) ∪ { i ∈ S | ¯ r ( i ) = } . However, in general we have χ (¯ r ) ↓ S , χ (¯ r ↓ S ). Distributed Computing Context 1.5.
The replacement of ¯ r with ¯ r \ A yields a new proto-col, where all processes from A have been banned from participation. The replacement of ¯ r with ¯ r ↓ S corresponds to letting processes from S execute one round, and then runningthe remaining protocol with new inputs.
For an arbitrary round pointer ¯ r and sets S ⊆ act ¯ r , A ⊆ supp ¯ r we set(1.2) ¯ r S , A : = (¯ r ↓ S ) \ A = (¯ r \ A ) ↓ ( S \ A ) . In the special case, when A ∩ S = ∅ , the identity (1.2) specializes to(1.3) ¯ r S , A : = (¯ r ↓ S ) \ A = (¯ r \ A ) ↓ S . When A = ∅ , we shall frequently use the short-hand notation ¯ r S instead of ¯ r S , A , in otherwords, ¯ r S = ¯ r ↓ S . Again, for future reference, we note that for S ⊆ C , we have(1.4) χ C , D ↓ S = χ C \ S , D ∪ S . Witness structures and the ghosting operation.
Next, we describe the basic terminology which we will need to define the immediate snap-shot complexes.
Definition 1.6. A witness prestructure is a finite sequence of pairs of finite subsets of Z + ,denoted σ = (( W , G ) , . . . , ( W t , G t )) , with t ≥ , satisfying the following conditions: (P1) W i , G i ⊆ W , for all i = , . . . , t; (P2) G i ∩ G j = ∅ , for all i , j ∈ [ t ] , i < j; (P3) G i ∩ W j = ∅ , for all i , j ∈ [ t ] , i ≤ j.A witness prestructure is called stable if in addition the following condition is satisfied: (S) if t ≥ , then W t , ∅ .A witness structure is a witness prestructure satisfying the following strengthening ofcondition (S): (W) the subsets W , . . . , W t are all nonempty. Definition 1.7.
We define the following data associated to an arbitrary witness prestruc-ture σ = (( W , G ) , . . . , ( W t , G t )) : • the set W ∪ G is called the support of σ and is denoted by supp σ ; • the ghost set of σ is the set G ( σ ) : = G ∪ · · · ∪ G t ; • the active set of σ is the complement of the ghost setA ( σ ) : = supp ( σ ) \ G ( σ ) = W \ ( G ∪ · · · ∪ G t ); • the dimension of σ is dim σ : = | A ( σ ) | − = | W | − | G | − · · · − | G t | − . For brevity of some formulas, we set W − : = W ∪ G = supp σ . OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 5
Definition 1.8.
For a prestructure σ and an arbitrary p ∈ supp σ , we set Tr( p , σ ) : = { ≤ i ≤ t | p ∈ W i ∪ G i } , and call it the trace of p. Furthermore, for all p ∈ supp σ , we set last ( p , σ ) : = max {− ≤ i ≤ t | p ∈ W i } . When the choice of σ is unambiguous, we shall simply write Tr( p ) and last ( p ). Thefollowing definition provides an alternative approach to witness structures using traces. Definition 1.9. A witness prestructure is a pair of finite subsets A , G ⊆ Z + together witha family { Tr( p ) } p ∈ A ∪ G of finite subsets of Z + , satisfying the following condition: (T) 0 ∈ Tr( p ) , for all p ∈ A ∪ G.A witness prestructure is called stable if it satisfies an additional condition: (TS) if A = ∅ , then Tr( p ) = { } , for all p ∈ G, else max p ∈ A last ( p ) ≥ max [ p ∈ G Tr( p ) . Set t : = max p ∈ A last ( p ) . A stable witness prestructure is called witness structure if thefollowing strengthening of Condition (TS) is satisfied: (TW) for all ≤ k ≤ t either there exists p ∈ A such that k ∈ Tr( p ) , or there exists p ∈ Gsuch that k ∈ Tr( p ) \ max Tr( p ) . We shall call the form of the presentation of the witness prestructure as a triple( A , G , { Tr( p ) } p ∈ A ∪ G ) its trace form . Distributed Computing Context 1.10.
The witness structure is a mathematical objectmodelling the information which the processes have during the execution of the full-information protocol. Let us explain the distributed computing intuition behind this no-tation.The set supp σ indexes all processes which are participating in the protocol. The pro-cesses indexed by the set W are of two di ff erent types. Those, whose view is included in σ , and those, who have only been passively witnessed by others. The processes of the firsttype are indexed by the set A ( σ ) , the other ones are indexed by the union G ∪ · · · ∪ G t .The set G indexes those processes from supp σ which have not be witnessed by anybodyin this particular execution.The fact, that p ∈ W k is to be interpreted as “the active participation of process p inround k has been witnessed”. This can happen in two ways, either p itself is active in thisexecution, or p is being passively witnessed and this is not the last occurence of p. Thefact that p ∈ G k means that process p has been passively witnessed and this is the lastoccurence of p.We refer the reader to [Ko14b, Section 6] , where connection between witness structuresand witness posets is explained. Next, we proceed to describe various operations in witness structures and prestructures.To start with, any stable witness prestructure can be turned into a witness structure, whichis called its canonical form . Definition 1.11.
Assume σ = (( W , G ) , . . . , ( W t , G t )) is an arbitrary stable witness pre-structure. Set q : = |{ ≤ i ≤ t | W i , ∅}| . Pick = i < i < · · · < i q = t, such that DMITRY N. KOZLOV { i , . . . , i q } = { ≤ i ≤ t | W i , ∅} . We define the witness structure C ( σ ) = (( W , G ) , ( e W , e G ) , . . . , ( e W q , e G q )) , which is called the canonical form of σ , by setting (1.5) e W k : = W i k , e G k : = G i k − + ∪ · · · ∪ G i k , for all k = , . . . , q , Furthermore, any witness prestructure can be made stable using the following operation.
Definition 1.12.
Let σ = (( W , G ) , . . . , ( W t , G t )) be a witness prestructure, setq : = max { ≤ i ≤ t | W i * G ( σ ) } . The stabilization of σ is the witness prestructure st( σ ) whose trace form is ( A ( σ ) , G ( σ ) , { Tr( p ) | [ q ] } p ∈ supp σ ) .More generally, assume S ⊆ A ( σ ) , and setq : = max { ≤ i ≤ t | W i * S ∪ G ( σ ) } . The stabilization of σ modulo S is the witness prestructure st S ( σ ) whose trace form is ( A ( σ ) \ S , G ( σ ) ∪ S , { Tr( p ) | [ q ] } p ∈ supp σ ) . Combining stabilization modulo a set with taking the canonical form yields a new oper-ation, called ghosting , which will be of utter importance for the combinatorial descriptionof the incidence structure in the immediate snapshot complexes.
Definition 1.13.
For an arbitrary witness structure σ , and an arbitrary S ⊆ A ( σ ) , wedefine Γ S ( σ ) : = C (st S ( σ )) . We say that Γ S ( σ ) is obtained from σ by ghosting S .
Distributed Computing Context 1.14.
The operation of ghosting the set of processes Scorresponds to excluding their views from the knowledge that the witness structure en-codes. Clearly, the occurences of processes from S will not vanish from the witness struc-ture Γ S ( σ ) altogether, but these processes will cease being active, and whatever we willsee of them will just be the residual information passively witnessed by other processes. The main property of ghosting which one needs for proving the well-definedness of theimmediate snapshot complexes is that it behaves well with respect to iterations.
Proposition 1.15.
Assume σ is a witness structure, and S , T ⊆ A ( σ ) , such that S ∩ T = ∅ .Then we have Γ T ( Γ S ( σ )) = Γ S ∪ T ( σ ) , expressed functorially we have Γ T ◦ Γ S = Γ S ∪ T . Immediate snapshot complexes.
We have now introduced su ffi cient terminology in order to describe our main objects ofstudy. Definition 1.16.
Assume ¯ r is a round counter. We define an abstract simplicial complexP (¯ r ) , called the immediate snapshot complex associated to the round counter ¯ r, as fol-lows. The vertices of P (¯ r ) are indexed by all witness structures σ = ( { p } , G , { Tr( q ) } q ∈{ p }∪ G ) ,satisfying these three conditions: (1) { p } ∪ G = supp ¯ r; (2) | Tr( p ) | = r ( p ) + ; (3) | Tr( q ) | ≤ r ( q ) + , for all q ∈ G.We say that such a vertex has color p. In general, the simplices of P (¯ r ) are indexed by allwitness structures σ = ( A , G , { Tr( q ) } q ∈ A ∪ G ) , satisfying: (1) A ∪ G = supp ¯ r; (2) | Tr( q ) | = r ( q ) + , for all q ∈ A; (3) | Tr( q ) | ≤ r ( q ) + , for all q ∈ G. OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 7
The empty witness structure (( ∅ , supp ¯ r )) indexes the empty simplex of P (¯ r ) . When conve-nient, we identify the simplices of P (¯ r ) with the witness structures which index them.Let σ be a non-empty witness structure satisfying the conditions above. The set ofvertices V ( σ ) of the simplex σ is given by { Γ A ( σ ) \{ p } ( σ ) | p ∈ A } . Taking boundaries of simplices in P (¯ r ) corresponds to ghosting of the witness struc-tures. This is only natural taking into account the intuition from the distributed computingcontext 1.14. Proposition 1.17.
Assume ¯ r is the round counter, and assume σ and τ are simplices ofP (¯ r ) . Then τ ⊆ σ if and only if there exists S ⊆ A ( σ ) , such that τ = Γ S ( σ ) . The first property of the simplicial complexes P (¯ r ), which is quite easy to see, is thatthese complexes are pure of dimension | supp ¯ r | −
1. Furthermore, zero values in the roundcounter have a simple topological interpretation.
Proposition 1.18. ([Ko14b, Proposition 4.4]).
Assume ¯ r = ( r (0) , . . . , r ( n )) and ¯ r ( n ) = .Let ¯ q denote the truncated round counter ( r (0) , . . . , r ( n − . Consider a cone over P ( ¯ q ) ,which we denote P ( ¯ q ) ∗ { a } , where a is the apex of the cone. Then we have (1.6) P (¯ r ) ≃ P ( ¯ q ) ∗ { a } , where ≃ denotes the simplicial isomorphism. For brevity, we set P n : = P (1 , . . . , | {z } n + ). It turns out that the standard chromatic subdivisionof an n -simplex, see [Ko12], is a special case of the immediate snapshot complex. Proposition 1.19. ([Ko14b, Proposition 4.10]).
The immediate snapshot complex P n andthe standard chromatic subdivision of an n-simplex χ ( ∆ n ) are isomorphic as simplicialcomplexes. Explicitly, the isomorphism can be given by (1.7) Φ : (( B , . . . , B t )( C , . . . , C t )) W C C . . . C t [ n ] \ W B \ C B \ C . . . B t \ C t , where W = B ∪ · · · ∪ B t . Recall the following property of pure simplicial complexes, strengthening the usualconnectivity.
Definition 1.20.
Let K be a pure simplicial complex of dimension n. Two n-simplices of Kare said to be strongly connected if there is a sequence of n-simplices so that each pair ofconsecutive simplices has a common ( n − -dimensional face. The complex K is said tobe strongly connected if any two n-simplices of K are strongly connected. Clearly, being strongly connected is an equivalence relation on the set of all n -simplices. Proposition 1.21. ([Ko14b, Proposition 5.6]).
For an arbitrary round counter ¯ r, the sim-plicial complex P (¯ r ) is strongly connected. The next definition describes a weak simplicial analog of being a manifold.
Definition 1.22.
We say that a strongly connected pure simplicial complex K is a pseudo-manifold if each ( n − -simplex of K is a face of precisely one or two n-simplices of K.The ( n − -simplices of K which are faces of precisely one n-simplex of K form a simplicialsubcomplex of K, called the boundary of K, and denoted ∂ K. It was shown in [Ko14b], that immediate snapshot complexes are always pseudomani-folds with boundary.
DMITRY N. KOZLOV
Theorem 1.23. ([Ko14b, Proposition 5.9]).
For an arbitrary round counter ¯ r, the sim-plicial complex P (¯ r ) is a pseudomanifold, where ∂ P (¯ r ) is the subcomplex consisting of allsimplices σ = (( W , G ) , . . . , ( W t , G t )) , satisfying G , ∅ .
2. A canonical decomposition of the immediate snapshot complexes
Definition and examples.
The canonical decomposition of the immediate snapshot complexes has been introducedin [Ko14b]. In order to better understand the topology of these complexes, we need togeneralize that definition and look at finer strata.
Definition 2.1.
Assume ¯ r is a round counter. • For every subset S ⊆ act ¯ r, let Z S denote the set of all simplices σ = (( W , G ) ,. . . , ( W t , G t )) , such that S ⊆ G . • For every pair of subsets A ⊆ S ⊆ act ¯ r, let Y S , A denote the set of all simplices σ = (( W , G ) , . . . , ( W t , G t )) , such that R = S and A ⊆ G . Furthermore, setX S , A : = Y S , A ∪ Z S We shall also use the following short-hand notation: X S : = X S , ∅ . This case has beenconsidered in [Ko14b], where it was shown that X S is a simplicial subcomplex of P (¯ r ) foran arbitrary S . Distributed Computing Context 2.2.
The subcomplexes X S correspond to the subset ofexecutions which start with the processes indexed by the set S executing simultaneously.This explains, from the point of view of distributed computing, why the protocol complexdecomposes into these strata. On the other extreme, clearly Z S = X S , S for all S . When A * S , we shall use theconvention Y S , A = ∅ . Note, that in general the sets Y S , A need not be closed under takingboundary. Proposition 2.3.
The sets X S , A are closed under taking boundary, hence form simplicialsubcomplexes of P (¯ r ) . Proof.
Let σ = (( W , G ) , . . . , ( W t , G t )) be a simplex in X S , A , and assume τ ⊂ σ . By Propo-sition 1.17 there exists T ⊆ A ( σ ), such that τ = Γ T ( σ ). By Proposition 1.15 it is enough toconsider the case | T | =
1, so assume T = { p } , and let τ = (( e W , e G ) , . . . , ( e W ˜ t , e G ˜ t )).By definition of X S , A we have either σ ∈ Z S or σ ∈ Y S , A . Consider first the case σ ∈ Z S ,so S ⊆ G . Since e G ⊇ G , we have τ ∈ Z S .Now, assume σ ∈ Y S , A . This means W ∪ G = S and A ⊆ G . Again e G ⊇ G implies A ⊆ e G . (cid:3) In particular, X S and Z S are simplicial subcomplexes of P (¯ r ), for all S . When we aredealing with several round counters, in order to avoid confusion, we shall add ¯ r to thenotations, and write X S , A (¯ r ), X S (¯ r ), Y S , A (¯ r ), Z S (¯ r ). We shall also let α S , A (¯ r ) denote theinclusion map α S , A (¯ r ) : X S , A (¯ r ) ֒ → P (¯ r ) . The strata of the canonical decomposition as immediate snapshot complexes.
The first important property of the simplicial complexes X S , A is that they themselves canbe interpreted as immediate snapshot complexes. Here, and in the rest of the paper, weshall use to denote simplicial isomorphisms. OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 9
Proposition 2.4.
Assume A ⊆ S ⊆ act ¯ r, then there exists a simplicial isomorphism γ S , A (¯ r ) : X S , A (¯ r ) P (¯ r S , A ) . Proof.
We start by considering the case A = ∅ . Pick an arbitrary simplex σ = (( W , G ) ,. . . , ( W t , G t )) belonging to X S . By the construction of X S , we either have W ∪ G = S , or S ⊆ G . If W ∪ G = S , then set γ S ( σ ) : = W \ G W . . . W t G ∪ G G . . . G t , else S ⊆ G , in which case we set γ S ( σ ) : = W \ S W W . . . W t G ∪ S G \ S G . . . G t . Reversely, assume τ = (( V , H ) , . . . , ( V t , H t )) is a simplex of P (¯ r S ). Note, that in anycase, we have S ⊆ V ∪ H . If V ∩ S , ∅ , we set ρ S ( τ ) : = V ∪ ( H ∩ S ) V ∩ S V . . . V t H \ ( H ∩ S ) H ∩ S H . . . H t , else S ⊆ H , and we set ρ S ( τ ) : = V ∪ S V V . . . V t H \ S H ∪ S H . . . H t . It is immediate that γ S and ρ S preserve supports, A ( − ), G ( − ), and hence also the di-mension. Furthermore, we can see what happens with the cardinalities of the traces. Forall elements p which do not belong to S , the cardinalities of their traces are preserved.For all elements in S , the map γ S decreases the cardinality of the trace, whereas, the map ρ S increases it. It follows that γ S and ρ S are well-defined as dimension-preserving mapsbetween sets of simplices.To see that γ S preserves boundaries, pick a top-dimensional simplex σ = ( W , S , W , . . . , W t ) in X S and ghost the set T . Assume first S * T . In this case not all ele-ments in S are ghosted. Assume now that S ⊆ T . This implies that γ S is well-defined asa simplicial map. Finally, a direct verification shows that the maps γ S and ρ S are inversesof each other, hence they are simplicial isomorphisms.Let us now consider the case when A is arbitrary. The simplicial complex X S , A is a sub-complex of X S consisting of all simplices σ satisfying the additional condition A ⊆ G .The image γ S ( X S , A ) consists of all τ = (( V , H ) , . . . , ( V t , H t )) in P (¯ r S , A ) satisfying A ⊆ H .The map Ξ : γ S ( X S , A ) → P (¯ r S , A ), taking τ to (( V , H \ A ) , ( V , H ) , . . . , ( V t , H t )), is obvi-ously a simplicial isomorphism, hence the composition γ S , A = Ξ ◦ γ S : X S , A → P (¯ r S , A ) isa simplicial isomorphism as well. (cid:3) Note that, in particular, γ A , A ( σ ) = W \ A W W . . . W t G G \ A G . . . G t . The statement of Proposition 2.4 for the example ¯ r = (2 , , X S ∪ A , A (¯ r ) to X A , A (¯ r ) is the same as the relation of the stratum X S (¯ r \ A ) to P (¯ r \ A ). Proposition 2.5.
Assume ¯ r is an arbitrary round counter, and S , A ⊂ act ¯ r, such thatS ∩ A = ∅ , then the following diagram commutes (2.1) X A , A (¯ r ) X S ∪ A , A (¯ r ) P (¯ r \ A ) X S (¯ r \ A ) P (¯ r S , A ) , i γ A , A (¯ r ) γ S ∪ A , A (¯ r ) α S (¯ r \ A ) γ S (¯ r \ A ) where i denotes the strata inclusion map. Proof.
To start with, note that ¯ r S , A = (¯ r ↓ S ) \ A = (¯ r ↓ ( S ∪ A )) \ A , so the diagram (2.1) iswell-defined. To see that it is commutative, pick an arbitrary σ = (( W , G ) , . . . , ( W t , G t )).We know that either A ⊆ G and W ∪ G = S ∪ A , or A ∪ S ⊆ G . On one hand, we have( γ A , A (¯ r ) ◦ i )( σ ) = W \ A W W . . . W t G G \ A G . . . G t . On the other hand, we have γ S ∪ A , A (¯ r )( σ ) = W \ G W . . . W t G ∪ G \ A G . . . G t , if A ⊆ G , W ∪ G = S ∪ A ; W \ ( S ∪ A ) W W . . . W t G ∪ S G \ ( S ∪ A ) G . . . G t , if A ∪ S ⊆ G . Applying γ S (¯ r \ A ) − we can verify that γ A , A (¯ r ) ◦ i = α S (¯ r \ A ) ◦ γ S (¯ r \ A ) − ◦ γ S ∪ A , A (¯ r ). (cid:3) Corollary 2.6.
For any A ⊆ act ¯ r, we have (2.2) X A , A (¯ r ) = [ ∅ , S ⊆ act ¯ r \ A X S ∪ A , A (¯ r ) = [ A ⊂ T ⊆ act ¯ r X T , A (¯ r ) . Proof.
Since P (¯ r \ A ) = S ∅ , S ⊆ act ¯ r \ A X S (¯ r \ A ), the equation (2.2) is an immediate conse-quence of the commutativity of the diagram (2.1). (cid:3) The incidence structure of the canonical decomposition.
Clearly, P (¯ r ) = ∪ S X S (¯ r ). We describe here the complete combinatorics of intersectingthese strata. Proposition 2.7.
For all pairs of subsets A ⊆ S ⊆ supp ¯ r and B ⊆ T ⊆ supp ¯ r we have:X S , A ⊆ X T , B if and only if at least one of the following two conditions is satisfied: • S = T and B ⊆ A, • T ⊆ A. We remark that it can actually happen that both conditions in Proposition 2.7 are satisfied.This happens exactly when S = T = A . Proof of Proposition 2.7.
First we show that T ⊆ A implies X S , A ⊆ X T , B . Take σ ∈ X S , A .If σ ∈ Z S , then we have the following chain of implications: S ⊆ G ⇒ A ⊆ G ⇒ T ⊆ G ⇒ σ ∈ Z T . If, on the other hand, σ ∈ Y S , A , we also have A ⊆ G , implying T ⊆ G ,hence σ ∈ Z T .Next we show that if S = T and B ⊆ A , then X S , A ⊆ X S , B . Clearly, we just need to showthat Y S , A ⊆ X S , B . Take σ ∈ Y S , A , then we have the following chain of implications: R = SA ⊆ G ⇒ R = TB ⊆ G ⇒ σ ∈ Y T , B . OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 11
This proves the if part of the proposition.To prove the only if part, assume X S , A ⊆ X T , B . If S , A , set τ : = supp ¯ r S \ A p . . . p t ∅ A ∅ . . . ∅ , else S = T , and we set τ : = supp ¯ r p p . . . p t ∅ S ∅ . . . ∅ , where in both cases p , . . . , p t is a sequence of elements from supp ¯ r \ A , with each element p occurring ¯ r ( p ) times. Clearly, in the first case, τ ∈ Y S , A , and in the second case τ ∈ Z S ,hence τ ∈ X T , B = Z T ∪ Y T , B . This means that either T ⊆ A , or S = T and B ⊆ A . (cid:3) Lemma 2.8.
Assume A ⊆ S ⊆ supp ¯ r and B ⊆ T ⊆ supp ¯ r. We have (1) Z S ∩ Z T = Z S ∪ T , (2) Y S , A ∩ Z T = Y S , A ∪ T , (3) Y S , A ∩ Y T , B = Y S , A ∪ B , if S = T , ∅ , otherwise . Proof.
To show (1), pick σ ∈ Z S ∩ Z T . We have S ⊆ G and T ⊆ G , hence S ∪ T ⊆ G ,and so σ ∈ Z S ∪ T .To show (2), pick σ ∈ Y S , A ∩ Z T . We have R = S , A ⊆ G , and T ⊆ G . It follows that R = S and A ∪ T ⊆ G , so σ ∈ Y S , A ∪ T .Finally, to show (3), pick σ ∈ Y S , A ∩ Y T , B . On one hand, σ ∈ Y S , A means R = S and A ⊆ G , on the other hand, σ ∈ Y T , B means R = T and B ⊆ G . We concludethat Y S , A ∩ Y T , B = ∅ if S , T . Otherwise, we have R = S = T and A ∪ B ⊆ G , so σ ∈ Y S , A ∪ B . (cid:3) Proposition 2.9.
For all pairs of subsets A ⊆ S ⊆ supp ¯ r and B ⊆ T ⊆ supp ¯ r we have thefollowing formulae for the intersection:X S , A ∩ X T , B = X S , A ∪ B , if S = T ; (2.3) X T , S ∪ B , if S ⊂ T ; (2.4) Z S ∪ T = X S ∪ T , S ∪ T , if S * T and T * S . (2.5)
Proof.
In general, we have(2.6) X S , A ∩ X T , B = ( Z S ∩ Z T ) ∪ ( Z S ∩ Y T , B ) ∪ ( Y S , A ∩ Z T ) ∪ ( Y S , A ∩ Y T , B ) = Z S ∪ T ∪ Y T , S ∪ B ∪ Y S , T ∪ A ∪ Y S , A ∪ B , if S = T ; Z S ∪ T ∪ Y T , S ∪ B ∪ Y S , T ∪ A , otherwise . Assume first that S = T . In this case Y T , S ∪ B = Y S , T ∪ A = Z S , hence the equation (2.6)translates to X S , A ∩ X T , B = Z S ∪ Y S , A ∪ B = X S , A ∪ B .Let us now consider the case S ⊂ T . We have Y S , T ∪ A = ∅ , hence (2.6) translates to X S , A ∩ X T , B = Z T ∪ Y T , S ∪ B = X T , S ∪ B .Finally, assume S * T and T * S . Then Y T , S ∪ B = Y S , T ∪ A = ∅ , hence (2.6) says X S , A ∩ X T , B = Z S ∪ T . (cid:3) For convenience we record the following special cases of Proposition 2.9.
Corollary 2.10.
For S , T we haveX S ∩ X T = X T , S , if S ⊂ T , Z S ∪ T , otherwise, (2.7) X S ∩ Z T = Z S ∪ T . Proof.
The first formula is a simple substitution of A = B = ∅ in (2.4) and (2.5). To see(2.7), substitute A = ∅ , B = T in (2.4) to obtain X S , ∅ ∩ X T , T = X T , S ∪ T , if S ⊂ TZ S ∪ T , otherwise = Z T , if S ⊂ TZ S ∪ T , otherwise = Z S ∪ T . (cid:3) We invite the reader to trace the intersections formulae from Corollary 2.10 for the example¯ r = (2 , , X ≃ P (1 , , X ≃ P (2 , , X ≃ P (1 , , X ∩ X = X , ≃ P (1 , X ≃ P (2 , , X ≃ P (2 , , X ≃ P (1 , , X ≃ P (1 , , igure P (2 , , Remark 2.11.
Corollary 2.10 implies that every stratum X S , A can be represented as anintersection of two strata of the type X S , with only exception provided by the strata X S , S ,when | S | = . Corollary 2.12.
Assume S , . . . , S t ⊆ [ n ] , such that S S i , for all i = , . . . , t. Thefollowing two cases describe the intersection X S ∩ · · · ∩ X S t : (1) if S ⊃ S i , for all i = , . . . , t, then X S ∩ · · · ∩ X S t = X S , S ∪···∪ S t ; (2) if there exists ≤ i ≤ t, such that S S i , then X S ∩ · · · ∩ X S t = Z S ∪ S ∪···∪ S t = X S ∪ S ∪···∪ S t , S ∪ S ∪···∪ S t . Proof.
Assume first that S ⊃ S i , for all i = , . . . , t . By iterating (2.4) we get X S ∩ · · · ∩ X S t = X S , ∅ ∩ X S , ∅ ∩ · · · ∩ X S t , ∅ = X S , S ∩ X S , ∅ ∩ · · · ∩ X S t , ∅ = X S , S ∪ S ∩ X S , ∅ ∩ · · · ∩ X S t , ∅ = · · · = X S , S ∪···∪ S t . This proves (1).To show (2), we can assume without loss of generality, that S S . By (2.5) we have X S ∩ X S = Z S ∪ S . By iterating (2.7) we get Z S ∪ S ∩ X S ∩ · · · ∩ X S t = Z S ∪ S ∪ S ∩ X S ∩ · · · ∩ X S t = X S ∪ S ∪···∪ S t , which finishes the proof. (cid:3) OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 13
The boundary of the immediate snapshot complexes and its canonical decompo-sition.Definition 2.13.
Let ¯ r be an arbitrary round counter, and assume V ⊂ supp ¯ r. We defineB V (¯ r ) to be the simplicial subcomplex of P (¯ r ) consisting of all simplices σ = (( W , G ) , . . . , ( W t , G t )) , satisfying V ⊆ G . The fact that B V (¯ r ) is a well-defined subcomplex of P (¯ r ) is immediate from the defini-tion of the ghosting operation. We shall let β V (¯ r ) denote the inclusion map β V (¯ r ) : B V (¯ r ) ֒ → P (¯ r ) . Proposition 2.14.
For an arbitrary round counter ¯ r, and any V ⊂ supp ¯ r, the map δ V (¯ r ) given by δ V (¯ r ) : (( W , G ) , . . . , ( W t , G t )) (( W , G \ V ) , . . . , ( W t , G t )) is a simplicial isomorphism between simplicial complexes B V (¯ r ) and P (¯ r \ V ) . Proof.
The map δ v (¯ r ) is simplicial, and it has a simplicial inverse which adds V to G . (cid:3) Given an arbitrary round counter ¯ r , A ⊆ S ⊆ act ¯ r , and V ⊂ supp ¯ r , such that S ∩ V = ∅ ,we set X S , A , V (¯ r ) : = X S , A (¯ r ) ∩ B V (¯ r ) . We can use the notational convention B ∅ (¯ r ) = P (¯ r ), which is consistent with Defini-tion 2.13. In this case we get X S , A , ∅ (¯ r ) = X S , A (¯ r ), fitting well with the previous notations.The diagram (2.8) in the next proposition means that we can naturally think about X S , A , V (¯ r ) both as X S , A (¯ r \ V ) as well as B V (¯ r S , A ), or abusing notations we write B V ∩ X S , A = X S , A ( B V ) = B V ( X S , A ). Proposition 2.15.
Assume ¯ r is an arbitrary round counter, V ⊂ supp ¯ r, A ⊆ S ⊆ act ¯ r,and V ∩ S = ∅ . Then there exist simplicial isomorphisms ϕ and ψ making the followingdiagram commute: (2.8) P (¯ r ) X S , A (¯ r ) P (¯ r S , A ) B V (¯ r ) X S , A , V (¯ r ) B V (¯ r S , A ) P (¯ r \ V ) X S , A (¯ r \ V ) P (¯ r S ∪ V , A ∪ V ) , αβ γ j β i δ ϕψ δα γ where i and j denote inclusion maps. Proof.
Note that X S , A , V (¯ r ) consists of all simplices σ = (( W , G ) , . . . , ( W t , G t )), such that V ⊆ G , A ⊆ G , and either W ∪ G = S , or S ⊆ G . The fact that V and S are disjointensures that these conditions do not contradict each other. We let ϕ be the restrictionof γ S , A (¯ r ) : X S , A (¯ r ) → P (¯ r S , A ) to X S , A , V (¯ r ). Furthermore, we let ψ be the restriction of δ V (¯ r ) : B V (¯ r ) → P (¯ r \ V ) to X S , A , V (¯ r ). (cid:3) The commuting diagram (2.9) in the next proposition shows how the stratum X S , A (¯ r )can be naturally interpreted as a part of the boundary of the stratum X S , B (¯ r ), whenever B ⊆ A ⊆ S ⊆ act ¯ r . Proposition 2.16.
Assume B ⊆ A ⊆ S ⊆ act ¯ r, then the following diagram commutes (2.9) X S , B (¯ r ) X S , A (¯ r ) P (¯ r S , B ) B A \ B (¯ r S , B ) P (¯ r S , A ) i γ S , B (¯ r ) γ S , A (¯ r ) β A \ B (¯ r S , B ) δ A \ B (¯ r S , B ) where i denotes the inclusion map. Proof.
Take σ = (( W , G ) , . . . , ( W t , G t )) ∈ X S , A (¯ r ). On one hand we have( γ S , B (¯ r ) ◦ i )( σ ) = W \ G W . . . W t G ∪ G \ B G . . . G t , if W ∪ G = S , A ⊆ G ; W \ S W W . . . W t G ∪ S \ B G \ S G . . . G t , if S ⊆ G . On the other hand, we have( γ S , A (¯ r ))( σ ) = W \ G W . . . W t G ∪ G \ A G . . . G t , if W ∪ G = S , A ⊆ G ; W \ S W W . . . W t G ∪ S \ A G \ S G . . . G t , if S ⊆ G . Since applying δ A \ B (¯ r S , B ) − will add A \ B to G ∪ G \ A , resp. G ∪ S \ A , above and A ⊆ S , A ⊆ G , we conclude that( γ S , B (¯ r ) ◦ i )( σ ) = ( β A \ B (¯ r S , B ) ◦ δ A \ B (¯ r S , B ) − ◦ γ S , A (¯ r ))( σ ) . Which is the same as to say that the diagram (2.9) commutes. (cid:3)
The combinatorial structure of the complexes P ( χ A , B ) . Let us analyze the simplicial structure of P ( χ A , B ). Set k : = | A | − m : = | B | . By (1.6)the simplicial complex P ( χ A , B ) is isomorphic to the m -fold suspension of P ( χ A ). On theother hand, we saw before that P ( χ A ) is isomorphic to the standard chromatic subdivisionof ∆ k . The simplices of the m -fold suspension of χ ( ∆ k ) (which is of course homeomorphicto ∆ m + k ) are indexed by tuples ( S , ( B , . . . , B t )( C , . . . , C t )), where S is any subset of B ,and the sets B , . . . , B t , C , . . . , C t satisfy the same conditions as in the combinatorial de-scription of the simplicial structure of χ ( ∆ k ). In line with (1.7), the simplicial isomorphismbetween P ( χ A , B ) and the m -fold suspension of χ ( ∆ k ) can be explicitly given by( S , ( B , . . . , B t )( C , . . . , C t )) W C . . . C t ( A ∪ B ) \ W B \ C . . . B t \ C t , where W = S ∪ B ∪ · · · ∪ B t . In particular, up to the simplicial isomorphism, the complex P ( χ A , B ) depends only on m and k .The simplices of P ( χ A , B ) are indexed by all witness structures σ = (( W , G ) , . . . , ( W t , G t )) satisfying the following conditions:(1) W ∪ G = A ∪ B ;(2) W ∩ A = W ∪ · · · ∪ W t ∪ G ∪ · · · ∪ G t ;(3) the sets W , . . . , W t , G , . . . , G t are disjoint. OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 15
It was shown in [Ko12] that there is a homeomorphism τ A : P ( χ A ) −→ (cid:27) ∆ A , such that for any C ⊆ A the following diagram commutes(2.10) P ( χ A ) B A \ C ( χ A ) P ( χ C ) ∆ A ∆ C β A \ C ( χ A ) (cid:27) τ A δ A \ C ( χ A ) (cid:27) τ C i where i : ∆ C ֒ → ∆ A is the standard inclusion map. In general, given a pair if sets ( A , B ),we take the | B | -fold suspension of the map τ A to produce a homeomorphism τ A , B : P ( χ A , B ) −→ (cid:27) ∆ A ∪ B . Definition 2.17.
When A ∪ B = C ∪ D, we set τ ( χ A , B , χ C , D ) : = τ − C , D ◦ τ A , B , clearly, we get a homeomorphism τ ( χ A , B , χ C , D ) : P ( χ A , B ) −→ (cid:27) P ( χ C , D ) . We know that this map is a simplicial isomorphism when restricted to B S ( χ A , B ), for all S ⊆ ( A ∩ C ) ∪ ( B ∩ D ), i.e., we have the following commutative diagram(2.11) B S ( χ A , B ) B S ( χ C , D ) P ( χ A , B ) P ( χ C , D ) τ ( χ A , B , χ C , D ) β S ( χ A , B ) β S ( χ C , D ) τ ( χ A , B , χ C , D ) (cid:27) When C ⊆ A , we have B ⊆ D , so the condition for S becomes S ⊆ B ∪ C . Furthermore, ifin addition T = E ∪ F , we have τ ( χ A , B , χ A , B ) ◦ τ ( χ A , B , χ A , B ) = τ ( χ A , B , χ A , B ) . When A ⊆ C ∪ D , he identity (1.1) implies that we have a simplicial isomorphism β V ( χ C . D ) : B V ( χ C , D ) −→ (cid:27) P ( χ C \ A , D \ A ) . Furthermore, when S ⊆ C , the identity (1.4) implies that we have a simplicial isomorphism X S ( χ C . D ) : γ S ( χ C , D ) −→ (cid:27) P ( χ C \ S , D ∪ S ) . Proposition 2.18.
Assume A ∪ B = C ∪ D and V ⊆ A ∪ B, then the following diagramcommutes (2.12) P ( χ A , B ) B V ( χ A , B ) P ( χ A , B \ V ) P ( χ C , D ) B V ( χ C , D ) P ( χ C , D \ V ) β V ( χ A , B ) (cid:27) τ ( χ A , B , χ C , D ) δ V ( χ A , B ) τ ( χ A , B \ V , χ C , D \ V ) (cid:27) β V ( χ C , D ) δ V ( χ C , D ) Proof.
Consider the diagram on Figure 2.2. Both the upper and the lower part ofthis diagram are versions of (2.10), hence, they commute. Together, they form the dia-gram (2.12). (cid:3) P ( χ A , B ) B V ( χ A , B ) P ( χ A , B \ V ) ∆ A ∪ B ∆ A ∪ B \ V P ( χ C , D ) B V ( χ C , D ) P ( χ C , D \ V ) β V ( χ A , B ) (cid:27) τ A , B δ V ( χ A , B ) τ A \ V , B \ V (cid:27)(cid:27) τ C , D τ C \ V , D \ V (cid:27) β V ( χ C , D ) δ V ( χ C , D ) F igure opology of the immediate snapshot complexes Immediate snapshot complexes are collapsible pseudomanifolds.
Consider a quite general situation, where X is an arbitrary topological space, and { X i } i ∈ I is a finite family of subspace of X covering X , that is I is finite and X = ∪ i ∈ I X i . Definition 3.1. ([Ko07, Definition 15.14]).
The nerve complex N of a covering { X i } i ∈ I isa simplicial complex whose vertices are indexed by I, and a subset of vertices J ⊆ spansa simplex if and only if the intersection ∩ i ∈ J X i is not empty. The nerve complex can be useful because of the following fact.
Lemma 3.2. (Nerve Lemma, [Ko07, Theorem 15.21, Remark 15.22]).
Assume K is asimplicial complex, covered by a family of subcomplexes K = { K i } i ∈ I , such that ∩ i ∈ J K i isempty or contractible for all J ⊆ I, then K is homotopy equivalent to the nerve complex N ( K ) . Corollary 3.3.
For an arbitrary round counter ¯ r, the simplicial complex P (¯ r ) is con-tractible. Proof.
We use induction on | ¯ r | . If | ¯ r | =
0, then P (¯ r ) is just a simplex, hence contractible. Weassume that | ¯ r | ≥
1, and view the canonical decomposition P (¯ r ) = ∪ S ⊆ act ¯ r X S (¯ r ) as a cov-ering of P (¯ r ). By Proposition 2.4, Corollary 2.12, and the induction assumption, all theintersections of the subcomplexes X S (¯ r ) with each other are either empty or contractible.This means, that we can apply the Nerve Lemma 3.2, with K = P (¯ r ), I = act ¯ r \ {∅} , and K i ’s are X S (¯ r )’s.Now, by Corollary 2.12 we see that X act ¯ r ∩ X S = X act ¯ r , S , ∅ for all S ⊂ act ¯ r . Itfollows that the nerve complex of this decomposition as a cone with apex at act ¯ r ∈ I .Since the nerve complex is contractible, it follows from the Nerve Lemma 3.2 that P (¯ r ) iscontractible as well. (cid:3) While contractibility is a property of topological spaces, there is a stronger combina-torial property called collapsibility , see [Co73], which some simplicial complexes mayhave.
Definition 3.4.
Let K be a simplicial complex. A pair of simplices ( σ, τ ) of K is calledan elementary collapse if the following conditions are satisfied: • τ is a maximal simplex, • τ is the only simplex which properly contains σ .A finite simplicial complex K is called collapsible , if there exists a sequence ( σ , τ ) , . . . , ( σ t , τ t ) of pairs of simplices of K, such that • this sequence yields a perfect matching on the set of all simplices of K, OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 17 • for every ≤ k ≤ t, the pair ( σ k , τ k ) is an elementary collapse in K \ { σ , . . . , σ k − ,τ , . . . , τ k − } . When ( σ, τ ) is an elementary collapse, we also say that σ is a free simplex.We have shown in Proposition 1.23 that for any round counter ¯ r the simplicial complex P (¯ r ) is a pseudomanifold with boundary ∂ P (¯ r ). Setint P (¯ r ) : = [ σ ∈ P (¯ r ) , σ < ∂ P (¯ r ) int σ, and, for all A ⊆ S ⊆ act ¯ r , set ∂ X S , A (¯ r ) : = γ S , A (¯ r ) − ( ∂ P (¯ r S , A )) , int X S , A (¯ r ) : = γ S , A (¯ r ) − (int P (¯ r S , A )) . Proposition 3.5.
Assume ¯ r is an arbitrary round counter, A ⊂ S ⊆ act ¯ r, and V ⊆ supp ¯ r \ S .The simplicial complex ∂ X S , A , V (¯ r ) is the subcomplex of X S , A , V (¯ r ) consisting of all simplices σ = (( W , V ) , ( S \ A , A ) , . . . , ( W t , G t )) . Proof.
Pick σ ∈ X S , A , V , and set ρ to be the composition of the simplicial isomorphisms X S , A , V (¯ r ) → B V (¯ r S , A ) → P (¯ r S ∪ V , A ∪ V ) from the commutative diagram (2.8).Assume first that W ∪ G = S , then ρ ( σ ) = (( W \ G , ( G ∪ G ) \ ( A ∪ V )) , ( W , G ) , . . . , ( W t , G t )) . Clearly ρ ( σ ) < ∂ P (¯ r S ∪ V , A ∪ V ) if and only if ( G ∪ G ) \ ( A ∪ V ) = ∅ , i.e., G ∪ G ⊆ A ∪ V .Since we know that A ⊆ G , V ⊆ G , this means that G = V and G = A , which implies W = S \ A .Assume now that S ⊆ G , then we have ρ ( σ ) = (( W \ S , ( G ∪ S ) \ ( A ∪ V )) , ( W , G \ S ) , ( W , G ) , . . . , ( W t , G t )) . Here we have ρ ( σ ) < ∂ P (¯ r S ∪ V , A ∪ V ) if and only if ( G ∪ S ) \ ( A ∪ V ) = ∅ , which is impossible,since V ∩ S = ∅ , and A ⊂ V . (cid:3) Corollary 3.6.
The simplicial complex P (¯ r ) can be decomposed as a disjoint union ofthe simplex ∆ pass ¯ r = ((pass ¯ r , act ¯ r )) , and the sets int X S , A , V , where ( S , A , V ) range over alltriples satisfying A ⊂ S ⊆ act ¯ r and V ⊆ supp ¯ r \ S .Specifically, for a simplex σ ∈ P (¯ r ) , σ = (( W , G ) , . . . , ( W t , G t )) , we have: if t = ,then σ ⊆ ∆ pass ¯ r , else int σ ⊆ int X W \ G , G , G . Proof.
Immediate from Proposition 3.5. (cid:3)
Lemma 3.7.
Assume ¯ r is a round counter, and p ∈ supp ¯ r, then there exists a sequenceof elementary collapses reducing the simplicial complex P (¯ r ) to the subcomplex ( ∂ P (¯ r )) \ int B p (¯ r ) . Proof.
The proof is again by induction on | r | . The case | r | = P (¯ r ) ∪ int B p (¯ r ). Let Σ denote the set of all strata X S , A , where A ⊂ S ⊆ act ¯ r , together with all strata X S , A , p , where A ⊂ S ⊆ act ¯ r , p < S . By Corollary 3.6, the union of the interiors of the strata in Σ isprecisely int P (¯ r ) ∪ int B p (¯ r ).We describe our collapsing as a sequence of steps. At each step we pick a certain pairof strata ( Y , X ), where Y ⊂ X , which we must “collapse”. Then, we use one of the previousresults to show that as a simplicial pair ( Y , X ) is isomorphic to ( B t (¯ r ′ ) , P (¯ r ′ )), for someround counter ¯ r ′ , such that | ¯ r ′ | < | ¯ r | . By induction assumption this means that there isa sequence of simplicial collapses which removes int X ∪ int Y . Finally, we order these pairs of strata with disjoint interiors ( Y , X ) , . . . , ( Y d , X d ) such that for every 1 ≤ i ≤ d ,every simplex σ ∈ P (¯ r ), such that int σ ⊆ int X i ∪ int Y i , and every τ ⊃ σ , such thatdim τ = dim σ +
1, we have(3.1) int τ ⊆ int X ∪ · · · ∪ int X i ∪ int Y ∪ · · · ∪ int Y i . This means, that at step i we can collapse away the pair of strata ( Y i , X i ) (i.e., collapseaway those simplices whose interior is contained in int X i ∪ int Y i ) using the proceduregiven by the induction assumption, and that these elementary collapses will be legal in P (¯ r ) \ ( X ∪ · · · ∪ int X i − ∪ int Y ∪ · · · ∪ int Y i − ) as well.Our procedure is now divided into 3 stages. At stage 1, we match the strata X S , A , p with X S , A , for all A ⊂ S ⊆ act ¯ r , such that p < S . It follows from the commutativity of thediagram (2.8) that each pair of simplicial subcomplexes ( X S , A , p , X S , A ) is isomorphic to thepair ( B p (¯ r S , A ) , P (¯ r S , A )). We have | ¯ r S , A | ≤ | ¯ r | − | S | < | ¯ r | , hence by induction assumption, thispair can be collapsed. As a collapsing order we choose any order which does not decreasethe cardinality of the set A . Take σ such that int σ ⊆ int X S , A , p ∪ int X S , A . By Proposition 3.5this means that σ = (( W , T ) , ( S \ A , A ) , . . . ), where either T = ∅ , or T = { p } . Take τ ⊃ σ ,such that dim τ = dim σ +
1. Then by Proposition 1.17(b) there exists q ∈ A ( τ ), such that σ = Γ q ( τ ). A case-by-case analysis of the ghosting construction shows that int τ ⊆ int X ,where X is one of the following strata: X S , A , X S , A , p , X q , X q , ∅ , p , X S , A \{ q } , X S , A \{ q } , p . Since theorder in which we do collapses does not decrease the cardinality of A , the interiors of thelast 4 of these strata have already been removed, hence the condition (3.1) is satisfied.At stage 2, we match X S with X S , S \{ p } , for all S ⊆ act ¯ r , such that p ∈ S , | S | ≥
2. By commutativity of the diagram (2.9), the pair ( X S , S \{ p } , X S ) is isomorphic to( B S \{ p } (¯ r S ) , P (¯ r S )). This big collapse can easily be expressed as a sequence of elementarycollapses, though in a non-canonical way. For this, we pick any q ∈ S \ { p } . It exists, sincewe assumed that | S | ≥
2. Then we match pairs ( X S , A ∪{ q } , X S , A ), for all A ⊆ S \ { q } . Again,by commutativity of the diagram (2.9), this pair is isomorphic to ( B q (¯ r S , A ) , P (¯ r S , A )). Theorder in which we arrange S does not matter for the collapsing order. Once S is fixed,the collapsing order inside does not decrease the cardinality of A . As above, take σ suchthat int σ ⊆ int X S , A ∪{ q } ∪ int X S , A , take τ ⊃ σ , such that dim τ = dim σ +
1, and take r ∈ A ( τ ), such that σ = Γ q ( τ ). By Proposition 3.5 we have σ = (( W , ∅ ) , ( S \ A , A ) , . . . ), or σ = (( W , ∅ ) , ( S \ ( A ∪ { q } ) , A ∪ { q } ) , . . . ). Note, that both q and r are di ff erent from p , butwe may have q = r . Again, a case-by-case analysis of the ghosting construction shows thatint τ ⊆ int X , where X is one of the following strata: X S , A , X S , A ∪{ q } , X S , A \{ r } , X S , A ∪{ q }\{ r } , X q , X r . Again, since collapsing order does not decrease the cardinality of A , the condition (3.1)is satisfied.At stage 3, we collapse the pair ( X p , p , X p ). Let us be specific. First, by Corol-lary 2.6 we know that X p , p = S { p }⊂ S ⊆ act ¯ r X S , p , and it follows from Proposition 3.5 thatint X p , p = S { p }⊂ S ⊆ act ¯ r int X S , p . By commutativity of the diagram (2.9), the pair ( X p , p , X p )is isomorphic to ( B p (¯ r p ) , P (¯ r p )), hence it can be collapsed using the induction assumption.Clearly, the entire procedure exhausts the set Σ , and we arrive at the simplicial complex( ∂ P (¯ r )) \ int B p (¯ r ). (cid:3) Corollary 3.8.
For an arbitrary round counter ¯ r, the simplicial complex P (¯ r ) is collapsible. Proof.
Iterative use of Lemma 3.7. (cid:3)
Homeomorphic gluing.
OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 19
Definition 3.9.
We say that a simplicial complex K is simplicially homeomorphic toa simplex ∆ A , where A is some finite set, if there exists a homeomorphism ϕ : ∆ A → K,such that for every simplex σ ∈ ∆ A , the image ϕ ( σ ) is a subcomplex of K. When we say that a CW complex is finite we shall mean that it has finitely many cells.
Definition 3.10.
Let X and Y be finite CW complexes. A homeomorphic gluing data between X and Y consists of the following: • a family ( A i ) ti = of CW subcomplexes of X, such that X = ∪ ti = A i , • a family ( B i ) ti = of CW subcomplexes of Y, such that Y = ∪ ti = B i , • a family of homeomorphisms ( ϕ i ) ti = , ϕ i : A i → B i ,satisfying the compatibility condition: if x ∈ A i ∩ A j , then ϕ i ( x ) = ϕ j ( x ) . Given finite CW complexes X and Y , together with homeomorphic gluing data( A i , B i , ϕ i ) ti = from X to Y , we define ϕ : X → Y , by setting ϕ ( x ) : = ϕ i ( x ), whenever x ∈ A i . The compatibility condition from Definition 3.10 implies that ϕ ( x ) is independentof the choice of i , hence the map ϕ : X → Y is well-defined. Lemma 3.11. (Homeomorphism Gluing Lemma).
Assume we are given finite CW complexes X and Y, and homeomorphic gluing data ( A i , B i , ϕ i ) ti = , satisfying an additional condition: (3.2) if ϕ ( x ) ∈ B i , then x ∈ A i , then the map ϕ : X → Y is a homeomorphism.
Proof.
First it is easy to see that ϕ is surjective. Take an arbitrary y ∈ Y , then there exists i such that y ∈ B i . Take x = ϕ − i ( y ), clearly ϕ ( x ) = y .Let us now check the injectivity of ϕ . Take x , x ∈ X such that ϕ ( x ) = ϕ ( x ). Thereexists i such that x ∈ A i . Then ϕ ( x ) = ϕ i ( x ) ∈ B i , hence ϕ ( x ) ∈ B i . Condition (3.2)implies that x ∈ A i . The fact that x = x now follows from the injectivity of ϕ i .We have verified that ϕ is bijective, so ϕ − : Y → X is a well-defined map. We shallnow prove that ϕ − is continuous by showing that ϕ takes closed sets to closed sets. Tostart with, let us recall the following basic property of the topology of CW complexes: asubset A of a CW complex X is closed if and only if its intersection with the closure ofeach cell in X is closed. Sometimes, one uses the terminology weak topology of the CWcomplex. This property was an integral part of the original J.H.C. Whitehead definition ofCW complexes, see, e.g., [Hat02, Proposition A.2.] for further details.Let us return to our situation. We claim that A ⊆ X is closed, if and only if A ∩ A i isclosed in A i , for each i = , . . . , t . Note first that since A i is itself closed, a subset S ⊆ A i isclosed in X if and only if it is closed in A i , so we will skip mentioning where the sets areclosed. Clearly, if A is closed, then A ∩ A i is closed for all i = , . . . , t . On the other hand,assume A ∩ A i is closed for all i . Let σ be a closed cell of X , we need to show that A ∩ σ is closed. Since X = ∪ ti = A i , and A i ’s are CW subcomplexes of X , there exists i , such that σ ⊆ A i . Then A ∩ σ = A ∩ ( A i ∩ σ ) = ( A ∩ A i ) ∩ σ , but ( A ∩ A i ) ∩ σ is closed since A ∩ A i is closed. Hence A ∩ σ is closed and our argument is finished. Similarly, we can show that B ⊆ X is closed, if and only if B ∩ B i is closed, for each i = , . . . , t .Pick now a closed set A ⊆ X , we want to show that ϕ ( A ) is closed. To start with, for all i the set A ∩ A i is closed, hence ϕ i ( A ∩ A i ) ⊆ B i is also closed, since ϕ i is a homeomorphism.Let us verify that for all i we have(3.3) ϕ i ( A ∩ A i ) = ϕ ( A ) ∩ B i . Assume y ∈ ϕ i ( A ∩ A i ). On one hand y ∈ ϕ i ( A i ), so y ∈ B i , on the other hand, y = ϕ i ( x ),for x ∈ A , so y ∈ ϕ ( A ). Reversely, assume y ∈ ϕ ( A ) and y ∈ B i . Then y = ϕ ( x ) ∈ B i , socondition (3.2) implies that x ∈ A i , hence y ∈ ϕ ( A ∩ A i ), which proves (3.3). It follows that ϕ ( A ) ∩ B i is closed for all i , hence ϕ ( A ) itself is closed. This proves that ϕ − is continuous.We have now shown that ϕ − : Y → X is a continuous bijection. Since X and Y are bothfinite CW complexes, they are compact Hausdor ff when viewed as topological spaces.It is a basic fact of set-theoretic topology that a continuous bijection between compactHausdor ff topological spaces is automatically a homeomorphism, see e.g., [Mun, Theorem26.6]. (cid:3) The following variations of the Homeomorphism Gluing Lemma 3.11 will be useful forus.
Corollary 3.12.
Assume we are given finite CW complexes X and Y, and homeomorphicgluing data ( A i , B i , ϕ i ) ti = , satisfying an additional condition: (3.4) for all I ⊆ [ t ] : ϕ : A I → B I is a bijection.Then the map ϕ : X → Y is a homeomorphism.
Proof.
Clearly, we just need to show that the condition (3.4) implies the condition (3.2).Assume y = ϕ ( x ), y ∈ B i , and x < A i . Let I be the maximal set such that y ∈ B I . Thecondition (3.4) implies that there exists a unique element ˜ x ∈ A I , such that ϕ ( ˜ x ) = y . Inparticular, ˜ x ∈ A i , hence x , ˜ x . Even stronger, if x ∈ A i , for some i ∈ I , then x , ˜ x ∈ A i ,hence x = ˜ x , since ϕ i is injective. So x i < A i , for all i ∈ I . Hence, there exists j < I , suchthat x ∈ A j , which implies ϕ ( x ) ∈ B j , yielding a contradiction to the maximality of theset I . (cid:3) Corollary 3.13.
Assume we are given CW complexes X and Y, a collection ( A i ) ti = of CWsubcomplexes of X, a collection ( B i ) ti = of CW subcomplexes of Y, and a collection ( ϕ I ) I ⊆ [ t ] of maps such that • X = ∪ ti = A i , Y = ∪ ti = B i ; • for every I ⊆ [ t ] , the map ϕ I : A I → B I is a homeomorphism; • for every J ⊇ I the following diagram commutes (3.5) A J B J A I B I ϕ J (cid:27) ϕ I (cid:27) Then ( A i , B i , ϕ i ) ti = is a homeomorphic gluing data, and the map ϕ : X → Y defined by thisdata is a homeomorphism.
Proof.
For arbitrary 1 ≤ i , j ≤ t , commutativity of (3.5) implies that also the followingdiagram is commutative A i A { i , j } A j B i B { i , j } B j ϕ { i } (cid:27) ϕ { i , j } (cid:27) ϕ { j } (cid:27) In other words, for any x ∈ A i ∩ A j , we have ϕ { i } ( x ) = ϕ { i , j } ( x ) = ϕ { j } ( x ). It followsthat ( A i , B i , ϕ { i } ) ti = is a homeomorphic gluing data. Since for all I ⊆ [ t ], the map ϕ I is OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 21 a homeomorphism, it is in particular bijective, so conditions of Corollary 3.12 are satisfied,and the defined map ϕ is a homeomorphism. (cid:3) Main Theorem.
The fact that the protocol complexes in the immediate snapshotread / write shared memory model are homeomorphic to simplices has been folklore knowl-edge in the theoretical distributed computing community, [Her]. The next theorem providesa rigorous mathematical proof of this fact. Theorem 3.14.
For every round counter ¯ r there exists a homeomorphism Φ (¯ r ) : P (¯ r ) (cid:27) −→ P ( χ (¯ r )) , such that (1) for all V ⊂ supp ¯ r the following diagram commutes: (3.6) P (¯ r \ V ) B V (¯ r ) P (¯ r ) P ( χ (¯ r \ V )) B V ( χ (¯ r )) P ( χ (¯ r )) δ V (¯ r ) (cid:27) Φ (¯ r \ V ) β V (¯ r ) Φ (¯ r ) (cid:27) δ V ( χ (¯ r )) β V ( χ (¯ r )) (2) for all S ⊆ act ¯ r the following diagram commutes: (3.7) X S (¯ r ) P (¯ r S ) P ( χ (¯ r S )) P ( χ (¯ r ) S ) X S ( χ (¯ r )) P (¯ r ) P ( χ (¯ r )) γ S (¯ r ) α S (¯ r ) Φ (¯ r S ) (cid:27) τ (cid:27) γ S ( χ (¯ r )) Φ (¯ r ) (cid:27) α S ( χ (¯ r )) where τ = τ ( χ (¯ r S ) , χ (¯ r ) S ) .In particular, the complex P (¯ r ) is simplicially homeomorphic to ∆ supp ¯ r . Proof.
Our proof is a double induction, first on | supp ¯ r | , then, once | supp ¯ r | is fixed, onthe cardinality of the round counter ¯ r . As a base of the induction, we note that the case | supp ¯ r | = | supp ¯ r | is fixed,and | ¯ r | =
0, we take Φ (¯ r ) to be the identity map. In this case the simplicial complexes P (¯ r )and P ( χ (¯ r )) are simplices. The diagram (3.6) commutes, since also Φ (¯ r \ V ) is the identitymap. The condition (2) of the theorem is void, since act ¯ r = ∅ . As a matter of fact, moregenerally, Φ (¯ r ) can be taken to be the identity map whenever ¯ r = χ (¯ r ), that is whenever¯ r ( i ) ∈ { , } , for all i ∈ supp ¯ r .We now proceed to prove the induction step, assuming that | ¯ r | ≥
1. For every pair ofsets A ⊆ S ⊆ act ¯ r , such that S , ∅ , we define a map ϕ S , A (¯ r ) : X S , A (¯ r ) −→ X S , A ( χ (¯ r )) , as follows(3.8) ϕ S , A (¯ r ) : X S , A (¯ r ) P (¯ r S , A ) P ( χ (¯ r S , A )) P ( χ (¯ r ) S , A ) X S , A ( χ (¯ r )) , γ S , A (¯ r ) Φ (¯ r S , A ) (cid:27) τ (cid:27) γ S , A ( χ (¯ r )) where τ = τ ( χ (¯ r S , A ) , χ (¯ r ) S , A ). Since | ¯ r S , A | ≤ | ¯ r S | = | ¯ r | − | S | < | ¯ r | , the map Φ (¯ r S , A ) is alreadydefined by induction, so ϕ S , A (¯ r ) is well-defined by the sequence (3.8). Obviously, the map ϕ S , A is a homeomorphism for all pairs S , A .We want to use Corollary 3.13 to construct the global homeomorphism Φ (¯ r ) by gluingthe local ones ϕ S , A (¯ r ). In our setting here, the notations of Corollary 3.13 translate to X = P (¯ r ), Y = P ( χ (¯ r )), A I ’s are X S , A (¯ r )’s, B I ’s are X S , A ( χ (¯ r ))’s, and ϕ I ’s are ϕ S , A (¯ r )’s. To satisfy the conditions of Corollary 3.13, we need to check that the following diagramcommutes whenever X S , A ⊆ X T , B (3.9) X S , A (¯ r ) X S , A ( χ (¯ r )) X T , B (¯ r ) X T , B ( χ (¯ r )) , ϕ S , A (¯ r ) (cid:27) i j ϕ T , B (¯ r ) (cid:27) where i and j denote the inclusion maps.Note, that by Proposition 2.7, we have X S , A ⊆ X T , B if and only if either S = T and B ⊆ A ,or T ⊆ A . Consider first the case S = T , B = ∅ . Consider the diagram on Figure 3.1.The leftmost pentagon is the diagram (2.9), which commutes by Proposition 2.16. The P (¯ r S , A ) P ( χ (¯ r S , A )) P ( χ (¯ r ) S , A ) X S , A (¯ r ) B A (¯ r S ) B A ( χ (¯ r S )) B A ( χ (¯ r ) S ) X S , A ( χ (¯ r )) X S (¯ r ) P (¯ r S ) P ( χ (¯ r S )) P ( χ (¯ r ) S ) X S ( χ (¯ r )) Φ (cid:27) τ (cid:27) γ δβ δβ βδ γγ Φ (cid:27) τ (cid:27) γ F igure r is replaced with ¯ r S . Since | ¯ r S | = | ¯ r | − | S | < | ¯ r | , this diagram commutes by induction. The next hexagon is the diagram (2.12), for χ C , D = χ (¯ r S ), χ C , D = χ (¯ r ) S , and we use the fact that χ (¯ r S ) \ A = χ (¯ r S , A ). Finally, therightmost pentagon is also the commuting diagram (2.9), where ¯ r is replaced with χ (¯ r ).Since removing the 3 inner terms of the diagram on Figure 3.1 yields the diagram (3.9)with S = T , B = ∅ , we conclude that (3.9) commutes in this special case.Consider now the case S = T , B ⊆ A . We have inclusions X S , A ֒ → X S , B ֒ → X S , andit is easy to see that the commutativity of the diagram (3.9) for the inclusion X S , A ֒ → X S , B follows from the commutativity of the diagrams (3.9) for the inclusions X S , A ֒ → X S and X S , B ֒ → X S . Hence we are done with the proof of this case.Let us now prove the commutativity of the diagram (3.9) for the inclusion X S , A ֒ → X T , B ,when T ⊆ A . Assume first that A = T = B , ∅ , and consider the diagram on Figure 3.2,where e S = S \ A . P (¯ r S , A ) P ( χ (¯ r S , A )) P ( χ (¯ r ) S , A ) X S , A (¯ r ) X e S (¯ r \ A ) X e S ( χ (¯ r \ A )) X S , A ( χ (¯ r )) X A , A (¯ r ) P (¯ r \ A ) P ( χ (¯ r \ A )) X A , A ( χ (¯ r )) , Φ (cid:27) τ (cid:27) γ αγ αγ γγ Φ (cid:27) γ F igure OPOLOGY OF THE IMMEDIATE SNAPSHOT COMPLEXES 23
A few of the maps in the diagram on Figure 3.2 need to be articulated. To start with, wehave the identity ¯ r A , A = ¯ r \ A , explaining the simplicial isomorphism γ A , A (¯ r ) : X A , A (¯ r ) → P (¯ r \ A ). Similarly, χ (¯ r ) A , A = χ (¯ r \ A ) explains γ A , A ( χ (¯ r )) : X A , A ( χ (¯ r )) → P ( χ (¯ r \ A )).Furthermore, by (1.2) we have ¯ r \ A ↓ e S = ¯ r ↓ S \ A = ¯ r S , A and χ (¯ r \ A ) ↓ e S = χ (¯ r ) \ A ↓ e S = χ (¯ r ) S , A . These identities explain the presence of the maps γ e S (¯ r \ A ) : X e S (¯ r \ A ) → P (¯ r S , A ),and γ e S ( χ (¯ r \ A )) : X e S ( χ (¯ r \ A )) → P ( χ (¯ r ) S , A ).Let us look at the commutativity of the diagram on Figure 3.2. The middle heptagonis the diagram (3.7) with ¯ r \ A instead of ¯ r and e S instead of S ; where we again use theidentity ¯ r \ A ↓ e S = ¯ r ↓ S \ A . Since | supp (¯ r \ A ) | = | supp ¯ r | − | A | < | supp ¯ r | , the inductionhypothesis implies that this heptagon commutes. The leftmost pentagon is (2.1), with e S instead of S , whereas the rightmost pentagon is (2.1) as well, this time with e S instead of S , and χ (¯ r ) instead of ¯ r . They both commute by Proposition 2.5. Again, removing the 2inner terms from the diagram on Figure 3.2 will yield the diagram (3.9) with A = T = B ,so we conclude that (3.9) commutes in this special case.In general, when T ⊆ A , we have a sequence of inclusions X S , A ֒ → X S , T ֒ → X T , T ֒ → X T , B . Again, it is easy to see that the commutativity of the diagram (3.9) for the inclusion X S , A ֒ → X T , B follows from the commutativity of the diagrams (3.9) for the inclusions X S , A ֒ → X S , T , X S , T ֒ → X T , T , and X T , T ֒ → X T , B . Hence we are done with the proof of thiscase as well.We now know that Φ (¯ r ) is a well-defined homeomorphism between P (¯ r ) and P ( χ (¯ r )).To finish the proof of the main theorem, we need to check the commutativity of the dia-grams (3.6) and (3.7). The commutativity of (3.7) is an immediate consequence of (3.8),and the way Φ (¯ r ) was defined. To show that (3.6) commutes, pick any S ⊆ act ¯ r , whichis disjoint from A , and consider the diagram on Figure 3.3. The maps ϕ and ψ are as in P ( χ (¯ r )) X S ( χ (¯ r )) P ( χ (¯ r S )) P (¯ r ) X S (¯ r ) P (¯ r S ) B V ( χ (¯ r )) B V (¯ r ) X S , ∅ , V (¯ r ) B V (¯ r S ) B V ( χ (¯ r S )) P (¯ r \ V ) X S (¯ r \ V ) P (¯ r S , V ) P ( χ (¯ r \ V )) X S ( χ (¯ r \ V )) P ( χ (¯ r S , V )) αβ ρ (cid:27) ββ (cid:27) Φ α Φ (cid:27) βδ δ ψ ϕ δ δ (cid:27) Φ α Φ (cid:27) α ν (cid:27) F igure ρ and ν are given by ρ : X S ( χ (¯ r )) P ( χ (¯ r ) S ) P ( χ (¯ r S )) γ S ( χ (¯ r )) τ ( χ (¯ r S ) , χ (¯ r ) S ) − and ν : X S ( χ (¯ r \ A )) P ( χ (¯ r ) S , A ) P ( χ (¯ r S , A )) , γ S ( χ (¯ r \ A )) τ ( χ (¯ r S , A ) , χ (¯ r ) S , A ) − P ( χ (¯ r )) X S ( χ (¯ r )) P ( χ (¯ r ) S ) P ( χ (¯ r S )) B V ( χ (¯ r )) B V ( χ (¯ r ) s ) B V ( χ (¯ r S )) P ( χ (¯ r \ V )) X S ( χ (¯ r \ V )) P ( χ (¯ r ) S , V ) P ( χ (¯ r S , V )) αβ γ τ (cid:27) β βδ δ δα γ τ (cid:27) F igure χ (¯ r \ A ) = χ (¯ r ) \ A , χ (¯ r S ) \ A = χ (¯ r S , A ), and χ (¯ r \ A ) S = χ (¯ r ) S , A ,with the latter one relying on the fact that S ∪ A = ∅ .Let us investigate the diagram on Figure 3.3 in some detail. The middle part is preciselythe diagram (2.8), which commutes by Proposition 2.15. We have 4 hexagons surroundthat middle part. The hexagon on the left is the diagram (3.6) itself. The hexagon aboveis precisely the diagram (3.7), so it commutes. The hexagon below is the diagram (3.7)with ¯ r \ A instead of ¯ r , where we use (1.3) again. This diagram commutes by the inductionhypothesis. The hexagon on the right is the diagram (3.6) with ¯ r S instead of ¯ r . Since | ¯ r S | < | ¯ r | , it also commutes by the induction assumption.Let us now show that the diagram obtained from the one on Figure 3.3 by the removalof the 9 inner terms commutes. This diagram can be factorized as shown on Figure 3.4.The left part of the diagram on Figure 3.4 is (2.8) with χ (¯ r ) instead of ¯ r , whereas the rightpart of the diagram on Figure 3.4 is the diagram (2.12) with χ C , D = χ (¯ r S ), χ C , D = χ (¯ r ) S .They both commute, hence so does the whole diagram.Consider now two sequences of maps in the diagram on Figure 3.3:(3.10) B V (¯ r ) ∩ X S (¯ r ) B V (¯ r ) P (¯ r ) P ( χ (¯ r )) β Φ (cid:27) and(3.11) X S , ∅ , V (¯ r ) B V (¯ r ) P (¯ r \ V ) P ( χ (¯ r \ V )) B V ( χ (¯ r )) P ( χ (¯ r )) δ Φ (cid:27) δ β It follows by a simple diagram chase that the commutativities in the diagram on Figure 3.3which we have shown imply the equality of these two maps. This is true for all S , such that S ⊆ act ¯ r and S ∩ V = ∅ . On the other hand, the subcomplexes X S , ∅ , V (¯ r ), where S ⊆ act ¯ r , S ∩ V = ∅ , cover B V (¯ r ). As a matter of fact, the simplicial isomorphisms ψ and δ V (¯ r )show that they induce a stratification which is isomorphic to the stratification of P (¯ r \ V )by X S (¯ r \ V ). The fact that they cover B V (¯ r ) completely implies that the maps (3.10) and(3.11) remain the same after the first term is skipped, which is the same as to say that (3.6)commutes. This concludes the proof. (cid:3) Corollary 3.15.
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