aa r X i v : . [ m a t h . K T ] O c t Fat realization and Segal’s classifying space
Yi-Sheng WangOctober 12, 2017
Abstract
In this paper, we give a new proof of a well-known theorem due to tomDieck that the fat realization and Segal’s classifying space of an internalcategory in the category of topological spaces are homotopy equivalent.
Contents G -cocycles. . . . . . . . . . 33 A classification theorem . . . . . . . . . . . . . . . 74 Fat realization and Segal’s classifying space . . . . . . . 10A candidate for a homotopy inverse to π . . . . . . . . 14References . . . . . . . . . . . . . . . . . . . . . 17 Motivated by bundle theory, foliation theory, and delooping theory, classifyingspaces of topological groups and groupoids were intensively studied during the60s, 70s and 80s. Since then, many different constructions of classifying spacesof topological groups and groupoids have been introduced, for example, theMilnor construction, the Segal construction, fat realization, geometric realiza-tion and so on [Mil56a], [Mil56b], [DL59], [Sta63a], [Sta63b], [Sta63c], [Seg68],[Hae71], [Bot72], [Seg74], [tD74]. Some of them have even been generalized toany internal categories in
Top , the category of topological spaces. For topolog-ical groups, most of the constructions give rise to homotopy equivalent spaces.However, for general internal categories in
Top , the relation between them is notalways clear. In this paper, we shall focus on the comparison between the Segalconstruction and fat realization of internal categories in
Top . Main results tom Dieck’s theorem [tD74, Proposition 2] asserts that, given any simplicialspace X · , the projection π : || X × S · || → || X · || (1)is a homotopy equivalence, where S · is the semi-simplicial set given by strictlyincreasing sequences of natural numbers. The idea of his proof is to constructa map || X · || → || X × S · || and show it is a homotopy inverse to the projection1 . However, to find a well-defined map from || X · || to || X · × S · || and show itis a homotopy inverse to π turn out to be quite difficult and complicated . Toget around this inconvenience, we employ Quillen’s theorem A . Our approachis more conceptual, but it works only when X k has the homotopy type of aCW-complex and X · = Ner · C , where Ner · C is the nerve of an internal categoryin Top . Theorem 1.1.
Given C an internal category in Top such that its nerve
Ner · C has the homotopy type of a CW -complex at each degree, then the canonicalprojection π : || Ner · C × S · || → || Ner · C|| is a homotopy equivalence.
Combing Theorem 1.1 with the fact that Segal’s classifying space | Ner · C N | is homeomorphic to the space || Ner · C × S · || , where C N is Segal’s unraveledcategory of C over the natural numbers N , one can easily deduce the followinguseful corollary. Corollary 1.2.
If, in addition to the assumptions in Theorem 1.1, the simplicialspace
Ner · C is proper, then the forgetful functor C N → C induces a homotopy equivalence | Ner · C N | → | Ner · C| . The space | Ner · C N | is more natural from the point of view of bundle theory,whereas, category-theoretically, the geometric realization | Ner · C| is easier tohandle.In the third section, following Stasheff’s approach [Bot72, Appendices B and C ], we work out a detailed proof of a classification theorem for bundles withstructures in a topological groupoid. Theorem 1.3.
Given a topological groupoid G and a topological space X , thereexists a - correspondence between the set of homotopy classes of maps from X to | Ner · G N | and the set of homotopy classes of numerical G -structures on X . It is not a new theorem, and we also claim no originality for the approachpresented here as it is essentially the proof of Theorem D , a special case of Theo-rem 1.3, in [Bot72, Appendix C ]. In fact, Stasheff has indicated that his methodcan be applied to more general cases (see [Mos76, p.126] and [Bot72, Theorem E in Appendix C ]). A similar classification theorem in terms of Milnor’s construc-tion can be found in [Hae71]. It is because we need a classification theorem interms of Segal’s classifying space in [Wan17], Theorem 1.3 is discussed in detailshere. The homotopy inverse ρ given in [tD74] appears not to be well-defined. See the appendixfor more details. utline of the paper In the second section, we review some background notions on topological groupoids,denoted by G , and G -structures on topological spaces. The third section dis-cusses how Stasheff’s approach [Bot72, Appendix B and C ] can be generalized toarbitrary topological groupoids. The forth section, where the novelty of the pa-per is, is independent of the previous two sections, and we shall apply Quillen’stheorem A to prove Theorem 1.1 there.As the homotopy inverse ρ : || X · || → || X · × S · || in [tD74] appears not to bewell-defined, in the appendix, we use a different construction, which is due toS. Goette, to find a well-defined map τ : || X · || → || X · × S · || and explain why itis a promising candidate for a homotopy inverse to π . Notation and convention
Throughout the paper, we shall use the Quillen equivalences between the modelcategory of simplicial sets sSets and the Quillen model category of topologicalspaces
Top given by the singular functor and geometric realization functor: | − | : sSets ⇆ Top : Sing · . A simplicial space is proper if and only of it is a cofibrant object in the Reedymodel category associated to the Strøn model structure on
Top .Given
X, Y ∈ Top , [
X, Y ] denotes the set of homotopy classes of maps from X to Y .Recall that the natural numbers N is an ordered set and hence can be viewedas a category.We have chosen to work with the category of topological spaces, but the re-sults in this paper hold for other convenient categories of topological spaces suchas the category of k -spaces (Kelly spaces) or the category of weakly Hausdorff k -spaces. Acknowledgment
I would like to thank S. Goette for suggesting the approach using Quillen’stheorem A . The construction of the map τ in Appendix is also due to him. G -cocycles Definition 2.1.
A topological groupoid G is an internal category in Top equippedwith the inverse map i : M ( G ) → M ( G ) and the identity-assigning map e : O ( G ) → M ( G ) such that s ◦ i = t : M ( G ) → O ( G ) and t ◦ i = s : M ( G ) → O ( G ) ;and the diagrams below commute M ( G ) O ( G ) M ( G ) × M ( G ) M ( G ) × O ( G ) M ( G ) M ( G ) e D (id , i ) s ◦ ( G ) O ( G ) M ( G ) × M ( G ) M ( G ) × O ( G ) M ( G ) M ( G ) e D ( i, id) t ◦ where s and t are the source and target maps, respectively, D is the diagonalmap D( x ) = ( x, x ) and ◦ : M ( G ) × O ( G ) M ( G ) → M ( G ) is the composition map. Given a topological groupoid G and a topological space X , we can define a G -structure on X . Definition 2.2.
1. A G -cocycle on X is a collection { U α ; f βα } α,β ∈ I , where { U α } α ∈ I is an open cover of X and f βα is a map f βα : U α ∩ U β → M ( G ) that satisfies f γβ ◦ f βα = f γα . In particular, f αα factors through O ( G ) , meaning f αα : U α → O ( G ) e −→ M ( G ) . Hence we often omit the repetition of α and just write f α , thinking of itas a map from U α to O ( G ) .2. Two G -cocycles { U α ; f βα } α,β ∈ I and { V γ ; g δγ } γ,δ ∈ J are isomorphic if andonly if there exists a map φ γα : U α ∩ V γ → M ( G ) , for each α ∈ I and γ ∈ J , such that g δγ ◦ φ γα = φ δβ ◦ f βα , or equivalently, the union { U α , V γ ; f βα , g δγ , φ γα } α,β ∈ I ; γ,δ ∈ J constitutes a G -cocycle on X . Note that the index sets I and J are oftenomitted when there is no risk of confusion.An isomorphism class of G -cocycles on X is called a G -structure, and theset of G -structures on X is denoted by H ( X, G ) .3. Two G -structures u, v ∈ H ( X, G ) are said to be homotopic if and only ifthere exists a G -structure w ∈ H ( X × I, G ) such that i ∗ w = u and i ∗ w = v ,where i : X = X × { } ֒ → X × I and i : X = X × { } ֒ → X × I . G ( X ) denotes the set of homotopy classes of G -structures, and it is acontravariant functor from Top to Sets , the category of sets.
Remark 2.3.
In this remark, we shall expand on the definition above. . Isomorphisms of G -cocycles constitute an equivalence relation on the setof G -cocycles. Suppose the G -cocycles { U α ; f βα } and { V γ ; g δγ } are iso-morphic through φ γα and the G -cocycles { V γ ; g δγ } and { W ǫ ; h ηǫ } are iso-morphic through ψ γǫ —namely, { U α ; f βα } φ γα ≃ { V γ ; g δγ } ψ ǫγ ≃ { W ǫ ; h ηǫ } . Then we can define ρ ǫα,γ := ψ ǫγ ◦ φ γα : U α ∩ W ǫ ∩ V γ → M ( G ) , for each α, γ and ǫ . Since they are compatible when γ varies, there is awell-defined map ρ ǫα : U α ∩ W ǫ → M ( G ) . On the other hand, from the definition of ρ ǫα,γ , we have the identity h ηǫ ◦ ρ ǫα,δ = ρ ηβ,γ ◦ f βα on W η ∩ U α ∩ W ǫ ∩ U β ∩ V γ ∩ V δ , for all γ and δ , and hence { ρ ǫα } constitute an isomorphism between the G -cocycles { U α ; f βα } and { W ǫ ; h ηǫ } .2. The notion of homotopy between G -structures on X gives an equivalencerelation on H ( X, G ). To see this, it suffices to observe that, for any twoisomorphic G -cocycles { U α ; f βα } α,β ∈ I φ γα ≃ { U ′ γ ; f ′ δγ } γ,δ ∈ J , there is a G -cocycle { ˆ U µ , ˆ f νµ } ν,µ ∈ I ∪ J on X × I which is given by ˆ U α := U α × (1 / , U γ := U ′ γ × [0 , / . and ˆ f βα ( x, t ) := f βα ( x ) t > / on ˆ U α ∩ ˆ U β ˆ f γδ ( x, t ) := f γδ ( x ) t < / on ˆ U δ ∩ ˆ U γ ˆ φ γα ( x, t ) := φ γα ( x ) 1 / < t < / on ˆ U α ∩ ˆ U ′ γ . To define numerable G -structures on a topological space. We first recall thedefinition of a partition of unity. Definition 2.4.
Given a topological space X and an open cover { U α } α ∈ I , then { λ α } α ∈ I is a partition of unity subordinate to the open cover { U α } α ∈ I if andonly if1. λ α is a map λ α : X → [0 , with supp( λ α ) ⊂ U α , for each α ∈ I .2. For every point x ∈ X , there exists a neighborhood V x of x such that,when restricted to this neighborhood, λ α = 0 for all but finite α ∈ I . . For every x ∈ X , X α ∈ I λ α ( x ) = 1 . A numerable open cover is an open cover that admits a partition of unitysubordinate to it.
Not every open cover admits a partition of unity, for example, the line withtwo origins.
Definition 2.5.
1. A numerable G -cocycle on X is a G -cocycle { U α ; f βα } with { U α } a numerable open cover.2. Two numerable G -cocycles are isomorphic if and only if they are isomor-phic as G -cocycles. An isomorphism class of numerable G -cocycles is calleda numerable G -structure on X . The set of G -structures on X is denotedby H nu ( X, G ) .3. Two numerable G -structures are homotopic if and only if they are homo-topic as G -structures via a numerable G -structure on X × I . The set ofhomotopy classes of numerable G -structures is denoted by G nu ( X ) , and itis a contravariant functor from Top to Sets . The following lemmas imply that, when G is a topological group, G nu ( X ) = H nu ( X, G ). Lemma 2.6.
Let G be a topological group and assume w is a numerable principle G -bundle on X × I . Then there is a bundle morphism w → π ∗ w , where π is thecomposition X × I ( x,t ) ( x, −−−−−−−→ X × { } ֒ → X × I Proof.
See [Hus94, Theorem 9 . Lemma 2.7.
Let G be a topological group. Then there is a - correspondencebetween H ( X, G ) and { principle G -bundles } / iso..Proof. See [Swi02, Theorem 11 . Lemma 2.8.
Given a numerable open cover { V j } of a topological space X ,there exists a countable numerable open cover { W n } n ∈ N such that, for each n , W n is an union of some open sets, each of which is contained in some membersof the original open cover { V j } .Proof. This lemma is due to Milnor (see [Jam84, Thm.7.27] or [Hus94, Propo-sition 12 .
1] for a detailed proof).A G -cocycle { U α , f βα } on X is countable and numerable if and only if { U α } has countably many members and is numerable. Two countable numerable G -cocycles are isomorphic if and only if they are isomorphic as G -cocycles. Twocountable numerable G -structures—isomorphism classes of countable numerable G -cocycles—are homotopic if and only if they are homotopic through a countablenumerable G -structure on X × I . 6 orollary 2.9.
1. Given a numerable G -structure on X , there exists a countablenumerable G -cocycle representing this G -structure.2. Given two homotopic numerable G -structures, there exists a countablenumerable G -cocycle on X × I such that its restrictions to X × { } and X × { } represent the two given G -structures.In particular, the set of homotopy classes of numerable G -structures is thesame as the set of homotopy classes of countable numerable G -structures. On the other hand, in most cases, there is no loss of generality by assum-ing the open cover in a G -cocycle is numerable. The ensuing corollary resultsfrom the fact that every open cover of a paracompact Hausdorff space admits asubordinate partition of unity. Corollary 2.10.
Suppose X is paracompact Hausdorff, then every G -cocycleon X is a numerable G -cocycle, and any two numerable G -structures on X arehomotopic if and only if they are homotopic through a numerable G -structureon X × I . In other words, classifying G -cocycles on a paracompact Hausdorffspace X is equivalent to classifying numerable G -cocycles on X .Proof. It is because the product of a paracompact space and a compact spaceis paracompact.
This section discusses a classification theorem for numerable G -structures, andthe method presented here is taken from [Bot72, Theorem D in Appendix C ],where Stasheff classifies Haefliger’s structures and indicates that his approachcan be applied to any topological groupoids.Recall first the construction of classifying spaces in [Bot72, Appendix B ]. Definition 3.1.
Given C an internal category in Top , the associated classifyingspace is defined to be the quotient space B C := a α :[ k ] ֒ → N Ner k C × △ kα / ∼ , where α can be viewed a strictly increasing sequence of natural numbers { i
Given a topological groupoid G , there is a - correspondence: [ X, B G ] ↔ G nu ( X ) , for every X ∈ Top . roof. Firstly, we note there is a canonical open cover of B G given by thepreimage U j := t − j ((0 , t j is induced by the projection a i <...
1] defined by w i ( x, s ) := max { , t i ( x ) − s X j i l On the other hand, given a countable numerable open cover { U α } of a topo-logical space X and a subordinate partition of unity { λ α } , one can define thespace X U := a α :[ k ] → N U α × △ kα / ∼ , where α = { i < ... < i k } ⊂ N and the equivalence relation is( x ; i , ..., j , ..., i k ) ∼ ( x ; i , ..., i j − , i j +1 , ..., i k ) , for any x ∈ U i ...i j ...i k . There is a homotopy equivalence λ from X to X U givenby λ : X → X U x ∈ U i ...i k ( x ; λ i ( x ) , ...., λ i k ( x )) , whose homotopy inverse is the canonical projection p : X U → X. It is clear that p ◦ λ = id X and there is an obvious linear homotopy connectingid X U and λ ◦ p . In this way, we see that the homotopy type of X U is independentof partition of unities.Now, in view of Corollary 2.9, we may assume all numerable G -cocycleson X are countable, and they are homotopic if they are homotopic through a8ountable numerable G -cocycles on X × I . Given a countable numerable G -cocycle u = { U α , g βα , λ α } , we have the composition X λ −→ X U Bu −−→ B G , (3)where the map Bu is induced from the assignment( x ; t i , ..., t i k ) ( g i i ( x ) , g i i ( x ) , ..., g i k i k − ( x ); t i , t i , ..., t i k ) . Suppose two G -cocycles u = { U α , g βα , λ α } and v = { V γ , f δγ , µ γ } on X arehomotopic, then their induced maps Bu ◦ λ and Bv ◦ µ are also homotopic. Thiscan be seen from the diagram below: XX × IX X U X W X V B G λι ι νµ BuBwBv where w = { W ǫ , h ηǫ , ν ǫ } is a countably numerable G -cocycle on X × I connecting u and v , meaning ι ∗ w = u and ι ∗ w = v (see Remark 2.3). We may also assumethat { ι ∗ ν ǫ } = { λ α } and { ι ∗ ν ǫ } = { µ α } .Thus, there is a well-defined map of setsΨ : G nu ( X ) → [ X, B G ] . Since G nu ( X ) is a contravariant functor, given any map X → B G , by pullingback the universal G -cocycle on B G , one obtains a G -cocycle on X . It is alsoclear that pullback G -cocycles along homotopic maps are homotopic, and hencethere is a well-defined map of setsΦ : [ X, B G ] → G nu ( X ) . To see Φ is the inverse of Ψ, we first note that Ψ ◦ Φ = id is obvious as thecollection { v i ( x, s ) } defined in equation (2) connects { t i } and { v i (1 , − ) } andhence gives the homotopy betweenΨ ◦ Φ( f )( x ) = ( g i i ( x ) , ..., g i k i k − ( x ); v i (1 , x ) , ..., v i k (1 , x ))and f ( x ) = ( g i i ( x ) , ..., g i k i k − ( x ); t i ( x ) , ..., t i k ( x )) , for any map f : X → B G . Secondly, we observe that, given u = { U α , g βα , λ α } a G -cocycle on X , the pullback G -cocycle along Bu ◦ λ is simply a restriction of u , namely that the pullback open cover is a subcover of { U α } , and thus u and( Bu ◦ λ ) ∗ u are isomorphic. 9 Fat realization and Segal’s classifying space
In this section, we shall employ a variant of Quillen’s theorem A [Wal83, Sec-tion 1 .
4] to show Segal’s classifying space and the fat realization of an internalcategory in
Top are homotopy equivalent under some mild conditions.We first recall the construction of Segal’s classifying space [Seg68, Section3].
Definition 4.1.
Let C be an internal category in Top . Then the associatedunraveled category C N is defined as the subcategory of C × N obtained by deletingthose morphisms ( f, i ≤ i ) when f = id ; and Segal’s classifying space of C is thegeometric realization of the associated unraveled category C N , namely | Ner · C N | . Lemma 4.2.
Let S · denote the semi-simplicial space given by S k := { strictly increasing maps from [ k ] to N } . Then there are homeomorphisms || Ner · C × S · || ≃ B C ≃ |
Ner · C N | . Proof.
The first homeomorphism (left) is clear as S · is just another way ofinterpreting triangles {△ kα } , where α is a strictly increasing sequence of naturalnumbers of length k + 1.For the second homeomorphism, we observe that the inclusion a α :[ k ] ֒ → N Ner k C × △ kα ֒ → a k Ner k C N × △ k ( c → ... → c k , t, i < ... < i k ) (cid:0) ( c , i ) → ... → ( c k , i k ) , t (cid:1) descends to a homeomorphism B C ≈ −→ | Ner · C N | . Now we can state our main theorem (compare with [tD74, Proposition 2]).
Theorem 4.3.
The canonical projection B C ≃ |
Ner · C N | ≃ || Ner · C × S · || π −→ || Ner · C|| is a homotopy equivalence, provided
Ner k C has the homotopy type of a CW -complex, for every k .Proof. Firstly, we observe that there is a commutative diagram for any simplicialspace X · with X k having the homotopy type of a CW-complex, for each k :10 X N · || X N ,p · | || X · × S · |||| ( X · × S · ) p || || X · |||| X p · ||| X p · | ∼ ∼ β ∼≀ πγ (4)where arrows with the symbol ∼ stand for homotopy equivalences and the (semi-) simplicial space Y p · is the properization of a (semi-) simplicial space Y · , namelylevel-wisely applying the singular functor and geometric realization to Y · : Y pk := | Sing · Y k | . The simplicial space Y N · in the diagram above is given by Y N n := a k ≤ ... ≤ k n Y l , where l is the number of the distinct members in { k , ..., k n } . The degeneratemap s N i : Y N n − → Y N n is given by identities, sending the copy of Y l indexed by k ≤ ... ≤ k i ≤ ... ≤ k n to another copy indexed by k ≤ ... ≤ k i = k i ≤ ... ≤ k n .To define its face maps, we first group together the members in { k , ..., k n } thatare the same. Meaning, given a sequence of increasing sequence k ≤ ... ≤ k n that contains l distinct numbers, we partition it into l groups: z}|{ ... < z}|{ ... < ... < l z}|{ ... . (5)Then we define d N i : Y N n → Y N n − to be d N i | Y l := idwhen k i belongs to the group of more than one member, or otherwise d N i | Y l := d j : Y l → Y l − , where Y l is indexed by the given sequence and k i belongs to the j -the group infigure (5). It is not difficult to see from the construction that Y · × S · can beobtained by throwing away the degenerate part of Y N · —namely, those compo-nents indexed by non-strictly increasing numbers. Furthermore, if Y · = Ner · C ,we have Y N · = Ner · C N .Now, we should expand on the homotopy equivalences in diagram (4). Firstly,since X N · is proper (with respect to the Strøm model structure) and both X N k X N ,pk have the homotopy type of CW-complexes, the upper right arrow isa homotopy equivalence [GJ99, VII, Proposition 3 . | Y N | ≃ || Y · × S · || for any simplicial space Y · , we im-mediately get the upper left arrow is also a homotopy equivalence. Secondly,following from the fact that X p · → X · is a level-wise homotopy equivalence andProposition A. || X p · || → || X · || is also a homotopyequivalence. Hence, in view of diagram (4), it suffices to show the map γ : | X N ,p · | → | X p · | (6)is a homotopy equivalence. Because both simplicial spaces involved in the map(6) are proper, it is enough to prove the mapSing k X N · → Sing k X · induces a homotopy equivalence, for every k . Now, in the case where X · =Ner · C , we have X N · is the nerve of C N and the map (6) is given by the naturalforgetful functor C N →C ( c, k ) c ( c → d, k ≤ l ) c → d Because the nerve (Ner · ) and unraveling ( C 7→ C N ) constructions commute withthe singular functor, the mapSing k Ner · C N → Sing k Ner · C is identical to Ner · Sing k C N → Ner · Sing k C . Therefore, if one can show the functor C N → C induces a homotopy equivalence, for any discrete category C , then we are done.Let’s pause for a moment and recall the variant of Quillen’s theorem A in[Wal83, Sec.1.4]: Given a map of simplicial space f : X · → Y · , if, for any y : △ n · → Y · , the space | f / ( △ n · , y ) · | is contractible, then f induces a homotopyequivalence, where f / ( △ n · , y ) · is the pullback of △ n · → Y · ← X · . Using this version of Quillen’s theorem A , we know if one can prove thespace | γ/ ( △ n · , y ) · | is contractible, for every y : △ n · → Ner · C , then the theorem follows.To show this, we note first that every simplex y : △ n · → Ner · C factorsthrough a non-degenerate one as illustrated below:12 / ( △ n · , y = x ◦ p ) · △ n · γ/ ( △ m · , x ) · △ m · Ner · C N Ner · C p x ,non-deg. γy (7)In view of commutative diagram (7) and the fact that p is a trivial fibration andthe category of simplicial sets sSets is a proper model category, , we may assume y is non-degenerate. In this case, y : △ n · → Ner · C is induced from a functor[ n ] → C , and hence, the pullback simplicial set γ/ ( △ n · , y ) · can be identifiedwith Ner · ([ n ] N ) and the map γ/ ( △ n · , y ) · → △ n · can be realized by the forgetfulfunctor [ n ] N π −→ [ n ].Now, we claim the forgetful functor [ n ] N π −→ [ n ] induces a homotopy equiva-lence. Consider the full subcategory [ n ] N , ′ of [ n ] N which consists of objects ( k, l )with k ≤ l . There is a natural projection π : [ n ] N → [ n ] N , ′ , ( k, l ) ( k, k ) k ≥ l, ( k, l ) ( k, l ) k ≤ l. Suppose ι : [ n ] N , ′ → [ n ] N is the canonical inclusion, then π ◦ ι = id is obvious,and on the other hand, there is a natural transformation φ : id ι ◦ π givenby φ ( k, l ) : ( k, l ) → ( k, k ) k ≥ l ( k, l ) id −→ ( k, l ) k ≤ l. Thus, the functors ι and π are inverse equivalences of categories.Similarly, there is a natural projection π : [ n ] N , ′ → [ n ]( k, l ) k, which, along with the canonical inclusion ι : [ n ] → [ n ] N , ′ k ( k, k ) , gives an equivalence of categories. More precisely, we have π ◦ ι = id and thenatural transformation φ : ι ◦ π id given by φ ( k, l ) : ( k, k ) → ( k, l ) , for every ( k, l ) ∈ [ n ] N , ′ .Thus, we have shown the commutative diagram of equivalences of categories:13 n ] N , ′ [ n ] N [ n ] π ιπ π ι and in particular, the space | Ner · [ n ] N | is contractible. AppendixA candidate for a homotopy inverse to π In [tD74, p.7], in order to prove the projection π : || X · × S · || → || X · || is a homotopy equivalence, a map ρ is constructed, and it is given by the as-signment( y ; t , ..., t n ) ∈ X n × △ n ( y, < ... < n ; s ,n ( t , ..., t n ) , ..., s n,n ( t , ..., t n )) ∈ X n × S n × △ n , (8)where s j,n ( t , ..., t n ) := ( j + 1) X E max(0 , min j ∈ E t j − max j E t j )and E runs through all subsets of [ n ] with j + 1 elements. However, this assign-ment does not respect face maps. In fact, the first and second components inthe assignment (8) should also depend on t , ..., t n .Here we present a construction of a well-defined map || X · || → || X · × S · || andconjecture that it is a homotopy inverse to the map π . Theorem.
There exists a map τ : || X · || → || X · × S · || such that π ◦ τ is homotopyequivalent to id .Proof. To define the map τ , we first recall that, given a simplex △ n := | Ner · [ n ] | ,any sequences of subsets of [ n ] A ⊂ ... ⊂ A k represents a k -simplex in the barycentric subdivision of △ n , denoted by Sd △ n .Then, we consider the following assignment τ n,k : X n × Sd k △ n → X k × S k × △ k ( x, A ⊂ ... ⊂ A k , t ) ( u ∗ x, | A | < | A | < ... < | A k | , t ) , k △ n stands for the set of k -simplices in Sd △ n , t is a point in a k -simplex, and | A i | is the size of A i . The map u ∗ is given by the assignment[ k ] → [ n ](1 , , ..., k ) (max( A ) , max( A ) , ..., max( A k )) ⊂ [ n ] , and max( A i ) stands for the maximal element in the set A i . It is clear that { τ n,k } k ∈ [ n ] induces a map τ n : X n × Sd △ n → || X · × S · || . To see it respects the face maps in X · , we assume y = d ∗ i x ∈ X n − andexpress the image of the simplex( x, A ⊂ A ⊂ ... ⊂ A k )with A k ⊂ [ n ] \ { i } under τ n in || X · × S · || by( u ∗ x, | A | < ... < | A k | ) (9)and the image of the simplex ( y, B , ..., B k )with d i | B j : B j ≃ −→ A j for each j under τ n − in || X · × S · || by( v ∗ y, | B | < ... < | B k | ) . (10)Now, the second components in simplices (9) and (10) are clearly the same,in view of the assumption d i | B j : B j ≃ −→ A j , and the same assumption also implies the compositions(1 , , ..., k ) (max( B ) , ..., max( B k )) ⊂ [ n − d i −→ [ n ];(1 , , ...., k ) (max( A ) , ..., max( A k )) ⊂ [ n ]are identical. Hence, we have u ∗ x = v ∗ d ∗ i x = v ∗ y , meaning that the firstcomponents in simplices (9) and (10) are also identical.The homotopy between π ◦ τ ≃ id is very easy to describe. It is given by thelinear homotopy from the identity to the last vertex map. Pictorially, it lookslike the following: (11)15 onjecture. Is τ a homotopy inverse to the map π ? Remark.
Though the idea is not so complicated, we find it very hard to writedown the homotopy between τ ◦ π and id in details. Observe first that, given anysimplicial space Y · , there is a natural filtration ∅ ⊂ || Y · || (0) ⊂ ... ⊂ || Y · || ( k ) ⊂ ... ⊂ || Y · || given by truncating the simplicial space Y · . Our strategy is to define a homotopy h n : || X · × S · || ( n ) × I → || X · × S · || ( n +1) between τ ◦ π | || X · × S · || ( n ) and id , for each n such that the diagram below commutes: || X · × S · || ( n ) × I || X · × S · || ( n +1) × I || X · × S · || ( n +1) || X · × S · || ( n +2) (12) Then, passing to the colimit, we get the required homotopy.The homotopy h is simply the homotopy given by the -simplex ( x, < k ) ,where x ∈ X . For general n , we decompose the homotopy h n into two parts.Given a non-degenerate x ∈ X n , there is a canonical embedding △ n × [ n, + ∞ ) ֒ → || X · × S · || , and the first part of h n is the linear homotopy to the projection △ n × [ n + ǫ, + ∞ ) → △ n × { n + ǫ } with any thing below { n + ǫ } intact, where ǫ > . The following illustrates thecase n = 1 . Homotopy I ⇓ x The second part of h n is nothing but a thickened version of the homotopy illus-trated in figure (11) except that instead of moving the vertices simultaneously,we start with the vertex of largest depth and down to the one of the smallestdepth. The figure below illustrates the case n = 1 . ⇓⇓ One still needs to take care of degenerate simplices in X · , and it seems to bea cumbersome task to write down the homotopy of those degenerate simplices,even though it is possible to describe it in low-dimension. We are hoping for abetter way to approach this problem. References [Bot72] Raoul Bott. Lectures on characteristic classes and foliations.
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