TThe homology of the Higman–Thompson groups
Markus Szymik and Nathalie WahlOctober 2018
We prove that Thompson’s group V is acyclic, answering a 1992 question of Brown in the positive. Moregenerally, we identify the homology of the Higman–Thompson groups V n , r with the homology of the zerothcomponent of the infinite loop space of the mod n − = V , , we can deduce thatthis group is acyclic. Our proof involves establishing homological stability with respect to r , as well asa computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of anytype n .MSC: 19D23, 20J05.Keywords: Higman–Thompson groups, Cantor algebras, homological stability, stable homology. Introduction
About half a century ago, Thompson introduced a group V together with subgroups F (cid:54) T (cid:54) V in order toconstruct examples of finitely presented groups with unsolvable word problem. Thompson’s groups havesince developed a life of their own, relating to many branches of mathematics. The homology of the group Fwas computed by Brown and Geoghegan [BG84]; it is free abelian of rank 2 in all strictly positive degrees.The homology of the group T was computed by Ghys and Sergiescu [GS87]; it is isomorphic to the homologyof the free loop space on the 3-sphere. As for Thompson’s group V itself, Brown [Bro92] proved that it isrationally acyclic and suggested that it might even be integrally so. In the present paper, we prove that V isindeed integrally acyclic.Thompson’s group V fits into the more general family of the Higman–Thompson groups V n , r for n (cid:62) r (cid:62)
1, with V = V , as the first case: A Cantor algebra of type n is a set X equipped with a bijec-tion X n ∼ = X , and the Higman–Thompson group V n , r is the automorphism group of the free Cantor alge-bra C n [ r ] of type n on r generators. The main result of this text is an identification of the homology of all ofthe groups V n , r in terms of a well-known object of algebraic topology: the mod n − M n − . Theorem A.
For any n (cid:62) r (cid:62) n , r → Ω ∞ M n − inducing an isomorphismH ∗ ( V n , r ; M ) ∼ = −→ H ∗ ( Ω ∞ M n − ; M ) . in homology for any coefficient system M on Ω ∞ M n − .Here the space Ω ∞ M n − is the zeroth component of the infinite loop space Ω ∞ M n − that underlies themod n − M n − . Note that the target of the isomorphism does not depend on r .In the case n =
2, the spectrum M n − is contractible and the above result answers Brown’s question:1 a r X i v : . [ m a t h . G R ] O c t orollary B ( Theorem 6.4).
Thompson’s group V = V , is acyclic.In [Bro92], Brown indicates that his argument for the rational acyclicity of V extends to prove rationalacyclicity for all groups V n , r . When n is odd, the group V n , r was known not to be integrally acyclic justfrom the computation of its first homology group, which is Z / n (cid:62) n , r , and at the same timeshowing that integral acyclicity only holds in the special case n = Corollary C ( Theorem 6.5).
For all n (cid:62)
3, the group V n , r is rationally but not integrally acyclic.We give in the last section of the paper some additional explicit consequences of Theorem A. In particu-lar, we confirm and complete the known information about the abelianizations and Schur multipliers of thegroups V n , r (Propositions 6.1 and 6.3), and compute the first non-trivial homology group of V n , r for each n and r (Proposition 6.2). When n is odd, the commutator subgroups V + n , r is an index two subgroup, and ourmethods can also be applied to study this group (Corollary 6.7).The proof of our main theorem rests on two pillars. The first is homological stability: For any fixed n (cid:62) n , r fit into a canonical diagramV n , −→ V n , −→ V n , −→ · · · ( (cid:63) )of groups and (non-surjective) homomorphisms, and we show that the maps V n , r → V n , r + induce isomor-phisms in homology for large r in any fixed homological degree. The definition of Cantor algebras leadsto isomorphisms C n [ r ] ∼ = C n [ r + ( n − )] for all n (cid:62) r (cid:62)
1, giving isomorphisms V n , r ∼ = V n , r +( n − ) forall n (cid:62) r (cid:62)
1. Using these isomorphisms, we obtain that the stabilization maps are actually isomor-phisms in homology in all degrees, see Theorem 3.6.To prove homological stability, we use the framework of [R-WW17]. The main ingredient for stability is theproof of high connectivity of a certain simplicial complex of independent sets in the free Cantor algebra C n [ r ] .It follows from [R-WW17] that homological stability also holds with appropriate abelian and polynomialtwisted coefficients.Our stability theorem can be reformulated as saying that the mapV n , r −→ (cid:91) r (cid:62) V n , r = V n , ∞ is a homology isomorphism, where the union is defined using the maps in the diagram ( (cid:63) ), and our secondpillar is the identification of the homology of V n , ∞ . This is achieved by the identification of the K-theory ofthe groupoid of free Cantor algebras of type n , as we describe now.Let Cantor × n denote the category of free Cantor algebras of type n with morphisms their isomorphisms. Thecategory Cantor × n is symmetric monoidal, and hence has an associated spectrum K ( Cantor × n ) , its algebraicK-theory. We denote by Ω ∞ K ( Cantor × n ) the zeroth component of its associated infinite loop space. Applyingthe group completion theorem, we get a mapBV n , ∞ −→ Ω ∞ K ( Cantor × n ) , defined up to homotopy, that induces an isomorphism in homology with all local coefficient systems on thetarget (see Theorem 5.4). Using a model of Thomason, we identify the classifying space | Cantor × n | with thatof a homotopy colimit Tho n in symmetric monoidal categories build out of the category of finite sets and thefunctor that takes the product with a set of cardinality n : | Cantor × n | (cid:39) | Tho n | , where the equivalence respects the symmetric monoidal structure, see Theorem 4.1. In particular, the twocategories have equivalent algebraic K-theory spectra: K ( Cantor × n ) (cid:39) K ( Tho n ) . The main theorem thenfollows from an identification K ( Tho n ) (cid:39) M n − , Tho n is a homo-topy mapping torus of the functor, defined on the category of finite sets and bijections, that takes the productwith a set of size n . Thinking of the finite sets as the generating sets of free Cantor algebras of a giventype n , this functor implements, for any r , the identification of a Cantor algebra C n [ r ] with the Cantor alge-bra C n [ rn ] = C n [ r + r ( n − )] , which reflects the defining property of Cantor algebras. Now the K-theory ofthe groupoid of finite sets is the sphere spectrum, by the classical Barratt–Priddy–Quillen theorem, and inspectra, this mapping torus equalizes multiplication by n with the identity on the sphere spectrum, whichleads to the Moore spectrum M n − .The paper is organized as follows. In Section 1, we introduce the Cantor algebras and the Higman–Thompsongroups, and we give some of their basic properties that will be needed later in the paper. In Section 2, weshow how the groups V n , r fit into the set-up for homological stability of [R-WW17] and construct the spacesrelevant to the proof of homological stability, which is given in Section 3. The following Section 4 is devotedto the homotopy equivalence | Cantor × n | (cid:39) | Tho n | , which is given as a composition of three homotopy equiv-alences. Section 5 then relates the first of these two spaces to V n , ∞ and the second to the Moore spectrum; thesection ends with the proof of the main theorem. Finally, Section 6 draws the computational consequences ofour main theorem. Throughout the paper, the symbol A will denote a fixed finite set of cardinality n . In this section, we recall some facts we need about Cantor algebras, and their groups of automorphisms, theHigman–Thompson groups. We follow Higman’s own account [Hig74]. See also [Bro87, Sec. 4] for a shortersurvey.Let A be a finite set of cardinality at least 2 and let A ∗ = (cid:71) n (cid:62) A n denote the word monoid on the set A . This is the free monoid generated by A , with unit Ø ∈ A and multipli-cation by juxtaposition. By freeness, a (right) action of the monoid A ∗ on a set S is uniquely determined by amap of sets S × A → S . Such a map has an adjoint S −→ S A = Map ( A , S ) . (1.1) Definition 1.1. A Cantor algebra of type A is an A ∗ –set S such that the adjoint structure map (1.1) is abijection. The morphisms of Cantor algebras of type A are the maps of A ∗ –sets, that is the set maps thatcommute with the action of A .Such objects go also under the name J´onsson–Tarski algebras .For any finite set X , there exists a free Cantor algebra C A ( X ) of type A generated by X . It can be constructedfrom the free A ∗ –set with basis X , namely the setC + A ( X ) : = X × A ∗ = (cid:71) n (cid:62) X × A n by formally adding elements so as to make the adjoint of the map defining the action bijective. (See [Hig74,Sec. 2]; the set C + A ( X ) is denoted X (cid:104) A (cid:105) in [Hig74].) Throughout the paper we will work with free Cantoralgebras, but only the elements of the canonical free A ∗ –set C + A ( X ) ⊂ C A ( X ) will play a direct role.The set A will be fixed throughout the paper to be a set A = { a , . . . , a n } n (cid:62) A comes thus with a canonical isomorphism to the set [ n ] = { , . . . , n } ,but we prefer giving it a different name to emphasize its special role. As for the generating set, we will beparticularly interested in the case X = [ r ] = { , . . . , r } , though we will also make use of the Cantor algebrasand A ∗ –sets C A ( E ) and C + A ( E ) generated by other sets build from [ r ] and A . Note that the elements of C + A [ r ] canonically identify with the vertices of a planar forest consisting of r rooted infinite n –ary trees, as inFigure 1.1. It can be useful to have this picture in mind in what follows, and interpret results using it.Figure 1.1: The set C + A [ ] for | A | = n , r , are the automorphism groups of the freeCantor algebras C A [ r ] : = C A ( X ) for X = [ r ] : Definition 1.2.
Let A = { a , . . . , a n } with n (cid:62) r (cid:62)
1. The
Higman–Thompson group V n , r = Aut ( C A [ r ]) is the automorphism group of the free Cantor algebra of type A on r generators. We need to understand isomorphisms of Cantor algebras. By freeness, a morphism of Cantor algebrasfrom C A ( X ) to a Cantor algebra S is determined by its value on the generating set X . For instance, thecanonical map X × A → C + A ( X ) → C A ( X ) induces a map C A ( X × A ) → C A ( X ) , which one can show is anisomorphism, using the Cantor algebra structure map of C A ( X × A ) . An isomorphism f : C A ( X ) → C A ( Y ) of Cantor algebras can in general take the generating set X to any generating set of Y . In this section, westudy generating sets of Cantor algebras, in particular those called expansions, preparing for Higman’s simpledescription of isomorphisms between free Cantor algebras, which is recalled in the following section. Definition 1.3.
A subset S ⊂ C A ( X ) is called a basis for the free Cantor algebra C A ( X ) if the induced homo-morphism C A ( S ) → C A ( X ) of Cantor algebras is an isomorphism. Definition 1.4.
Given a subset Y of a free Cantor algebra C A ( X ) , an expansion of Y is a subset of C A ( X ) obtained from Y by applying a finite sequence of simple expansions , where a simple expansion replaces oneelement y ∈ Y by the elements { y } × A ⊂ C A ( X ) , its “descendants” in C A ( X ) . For Y ⊂ C A ( X ) , we denoteby E ( Y ) the set of all expansions of Y .If Y was a basis, then so is any of its expansions [Hig74, Lem. 2.3]. In particular, all the expansions ofthe canonical basis X represent bases for C A ( X ) , and these are the bases we will work with. Note that anysuch basis is a finite subset of C + A ( X ) . If we think of C + A ( X ) as the vertices of an infinite | A | –ary forest withroots the elements of X (as in Figure 1.1), an expansion of X is the set of leaves of finite | A | –ary subforest F which “generates” in the sense that the infinite forest is the union of F and the infinite trees attached to theleaves of F . (See Figure 1.2 for an example.) 4igure 1.2: Recall the identification of C A [ ] + for | A | = [ ] , obtained from it by applying a sequence of five simple expansions. This expansion has cardinal-ity 12 = + ( − ) . Lemma 1.5.
The set of finite bases S of C A ( X ) satisfying that S ⊂ C + A ( X ) identifies with the set E ( X ) of allexpansions of X. It is a partially ordered set with the relationY (cid:54) Z ⇐⇒ Z is an expansion of Y . The poset E ( X ) has a least element, namely X.Proof. We first check that the given relation (cid:54) defines a partial ordering on E ( X ) : transitivity follows directlyfrom the definition of expansion and antisymmetry follows using in addition the fact that if Z is an expansionof Y , then | Z | (cid:62) | Y | , with strict inequality if the extension is non-trivial. We have that X ∈ E ( X ) and, bydefinition, X (cid:54) Y for any Y ∈ E ( X ) , so X is a least element.The fact that all expansions of X are bases is given by Lemma 2.3 of [Hig74], and these, by definition, lieinside C + A ( X ) . The fact that any finite basis that is a subset of C + A ( X ) is an expansion of X follows fromLemma 2.4 in [Hig74]: Suppose S is such a basis, and let U = C + A ( S ) ⊂ C + A ( X ) . Then U = C + A ( S ) ∩ C + A ( X ) ,hence satisfies condition (i) in the lemma, which is equivalent to condition (iii) in the lemma, thatis U = C + A ( Z ) for Z some expansion of X . Hence C + A ( Z ) = C + A ( S ) as A ∗ –subsets of C + A ( X ) , which is onlypossible if S = Z because S and Z both generate this free A ∗ –set.In Section 2 and 3, we will work with independent sets, which are subsets of expansions: Definition 1.6.
A finite subset P ⊂ C + A ( X ) is called independent if there exists an expansion E of X suchthat P ⊂ E . We denote by I ( X ) the set of all independent sets of C A ( X ) . If P , Q are independent sets, we willsay that P is independent from Q if they are disjoint and P (cid:116) Q is still independent.We have that E ( X ) ⊂ I ( X ) . We call the elements of I ( X ) : = I ( X ) \ E ( X ) the non-generating independentsets , so that the elements of I ( X ) are precisely the independent sets that are not bases.When X = [ r ] , we write E [ r ] , I [ r ] and I [ r ] for E ( X ) , I ( X ) and I ( X ) . Lemma 1.7.
Let P ∈ I ( X ) be an independent set. The subposet of E ( X ) of expansions of X containing P hasa least element.Proof. Suppose E = Q (cid:116) P is an expansion of X containing P . Consider the subalgebra of C A ( X ) generatedby Q . By [Hig74, Lem. 2.7(ii),(iii)], this subalgebra has a finite generating set G ⊂ C + A ( X ) with the propertythat any element of its intersection with C + A ( X ) is an expansion of an element of G . In particular, Q isnecessarily an expansion of G , and G (cid:116) P is the requested least element.Higman proved the following important fact about bases:5igure 1.3: The black dots define an independent set in C A [ ] + for | A | =
3, identified with the set of verticesof the forest of Figure 1.1, and the white dots complete it to the least expansion of [ ] containing it. Lemma 1.8. [Hig74, Cor. 1]
Any two finite bases of a free Cantor algebra have a common expansion.
In particular, for any X , the poset E ( X ) is directed, that is any pair of elements in E ( X ) have a common upperbound.A consequence of the lemma is that the cardinality of a finite basis for C A ( X ) is congruent to | X | mod-ulo | A | −
1. In fact, for two finite sets X and Y , we have that C A ( X ) ∼ = C A ( Y ) if and only if X = Ø = Y or if X and Y are both non-empty of cardinality congruent modulo | A | −
1; this is the condition that guarantees thatthe two Cantor algebras admit finite bases of the same cardinality.If X = X (cid:116) X is the disjoint union of two finite sets, we have a canonical isomorphismC + A ( X ) ∼ = C + A ( X ) (cid:116) C + A ( X ) and more generally, the set C + A ( X ) splits as a disjoint unionC + A ( X ) = (cid:71) x ∈ X C + A ( { x } ) . In terms of the forests of Figure 1.1, we see that C + A ( X ) is a disjoint union of trees, one for each element of X .Expansions of X are subsets of C + A ( X ) . The following result says that the property of being an expansion canbe checked componentwise. Lemma 1.9.
Let S ⊂ C + A ( X (cid:116) Y ) be a finite subset. Then S ∈ E ( X (cid:116) Y ) if and only if S ∩ C + A ( X ) ∈ E ( X ) and S ∩ C + A ( Y ) ∈ E ( Y ) , where we consider C + A ( X ) and C + A ( Y ) as subsets of C + A ( X (cid:116) Y ) through the identifi-cation C + A ( X (cid:116) Y ) ∼ = C + A ( X ) (cid:116) C + A ( Y ) .Proof. Suppose first that S is an expansion of X (cid:116) Y , so S is obtained from X (cid:116) Y by applying a finite sequenceof simple expansions. Now each simple expansion is either expanding an element of C + A ( X ) or of C + A ( Y ) ,and we see that S = S (cid:116) S , with S the subset obtained by applying to X the expansions of the first type,and S the subset applying to Y the expansions of the second type. As S = S ∩ C + A ( X ) and S = S ∩ C + A ( Y ) ,this gives the first direction. The reverse direction is direct from the definition.In fact, the lemma can be used to show that E ( X ) ∼ = ∏ x ∈ X E ( { x } ) as a poset. 6 , Figure 1.4: Representation of an automorphism of C A [ ] with | A | = ( E , F , λ ) , where E is the set of black dots in the first tree, F the set of black dots in the second tree, and λ is some bijectionbetween these two sets. Given a basis E of C A ( X ) , a basis F of C A ( Y ) and a bijection λ : E → F , there is a unique isomorphism ofCantor algebras f : C A ( X ) → C A ( Y ) that satisfies that f | E = λ . (Figure 1.4 gives an example in terms of for-est.) But such a representing triple ( E , F , λ ) is far from unique. Indeed, any expansion E (cid:48) of E defines a newsuch triple ( E (cid:48) , F (cid:48) , λ (cid:48) ) representing the same isomorphism f simply by taking F (cid:48) = f ( E (cid:48) ) and λ (cid:48) = f | E (cid:48) . Wewill now see that any isomorphism can be represented by a triple ( E , F , λ ) where E ∈ E ( X ) and F ∈ E ( Y ) , andthat there is in fact a canonical representative among such triples. To describe this canonical representative,we use the following partial ordering on the set of all such triples: Definition 1.10.
For f : C A ( X ) → C A ( Y ) an isomorphism of Cantor algebras, defineRep ( f ) = { ( E , F , λ ) | E ∈ E ( X ) , F ∈ E ( Y ) and λ = f | E : E ∼ = −→ F } to be the set of triples representing f with the property that E and F are expansions of X and Y respectively,with poset structure ( E , F , λ ) (cid:54) ( E (cid:48) , F (cid:48) , λ (cid:48) ) ⇐⇒ E (cid:54) E (cid:48) , where the right hand side uses the partial order relation on the set E ( X ) of expansions of X from Lemma 1.5.Note that this defines indeed a partial ordering on Rep ( f ) : the transitivity follows from transitivity in theposet E ( X ) , as does anti-symmetry once one notices that E = E (cid:48) forces F = F (cid:48) and λ = λ (cid:48) given that thetriples represent the same morphism.Note also that having the relation ( E , F , λ ) (cid:54) ( E (cid:48) , F (cid:48) , λ (cid:48) ) in Rep ( f ) forces the relation F (cid:54) F (cid:48) in E ( Y ) ,because F = f ( E ) and F (cid:48) = f ( E (cid:48) ) , and f takes expansions to expansions. Likewise, the map λ (cid:48) under sucha condition is necessarily the map induced by λ on E (cid:48) . Also, as F and λ are determined by E and f , theforgetful map Rep ( f ) → E ( X ) that takes a triple ( E , F , λ ) to E is injective, so Rep ( f ) canonically embeds asa sub-poset of E ( X ) .The main result of the section is the following: Proposition 1.11.
For any isomorphism of Cantor algebras f : C A ( X ) → C A ( Y ) , the poset Rep ( f ) is non-emtpy and has a least element. In other words, any isomorphism f : C A ( X ) → C A ( Y ) can be represented in the above sense by atriple ( E , F , λ ) with E an expansion of X , F an expansion of Y , and with λ : E → F a bijection, and there isa unique minimal such representative, with E minimal with respect to the ordering on E ( X ) .The result is a mild generalization of [Hig74, Lem. 4.1] who gives the case X = Y , and it follows from thesame proof. We give it for completeness as the result is crucial for us. Proof.
Consider the subset U of C + A ( X ) defined by U = C + A ( X ) ∩ f − ( C + A ( Y )) = C + A ( X ) ∩ C + A ( f − ( Y )) .
7y [Hig74, Lem. 2.4], there exists an expansion E of X such that U = C + A ( E ) . As E lies inside U , it satisfiesthat F = f ( E ) is an expansion of Y . Taking ( E , F , λ ) with λ = f | E yields a presentation of f with E and F expansions of X and Y , which shows that Rep ( f ) is non-empty.We also claim that the triple ( E , F , λ ) just constructed is in fact the least element of Rep ( f ) . Indeed, assumethat E (cid:48) is another expansion of X satisfying that f ( E (cid:48) ) is an expansion of Y , and let ( E (cid:48) , f ( E (cid:48) ) , λ (cid:48) ) ∈ Rep ( f ) be the associated triple. Then E (cid:48) must lie in U = C + A ( E ) and hence be an expansion of E by Lemma 1.5 as itis a basis of C A ( X ) and hence also of C A ( E ) . It follows that ( E , F , λ ) (cid:54) ( E (cid:48) , f ( E (cid:48) ) , λ (cid:48) ) in Rep ( f ) . Example 1.12.
Any bijection λ : A ∼ = −→ [ n ] induces an isomorphism C A [ ] ∼ = −→ C A [ n ] which is representedby the triple ( { } × A , [ n ] , λ ) . Its inverse C A [ n ] → C A [ ] is represented by the triple ([ n ] , { } × A , λ − ) .More generally, if f is (minimally) represented by ( E , F , λ ) , then its inverse f − is (minimally) representedby ( F , E , λ − ) . Remark 1.13.
The composition of isomorphisms of Cantor algebras in terms of representatives can be com-puted as follows. If f : C A ( X ) → C A ( Y ) is represented by ( E , F , λ ) and g : C A ( Y ) → C A ( Z ) is representedby ( G , H , µ ) , we have a diagram ˆ E (cid:54) ˆ λ (cid:47) (cid:47) ˆ F = ˆ G (cid:54) (cid:54) ˆ µ (cid:47) (cid:47) ˆ H (cid:54) E (cid:54) λ (cid:47) (cid:47) F G (cid:54) (cid:54) µ (cid:47) (cid:47) H (cid:54) X Y = Y Z for ˆ F = ˆ G a common expansion of F and G (which exists by Lemma 1.8), and ˆ λ and ˆ µ the maps inducedby λ and µ , or equivalently the restrictions of f and g to ˆ E and ˆ G , with ˆ E : = f − ( ˆ F ) and ˆ H : = g ( ˆ G ) . Thenthe triple ( ˆ E , ˆ H , ˆ µ ◦ ˆ λ ) represents the composition g ◦ f . Note that, even if the original representing tripleswere minimal, and ˆ G is chosen minimally, the resulting triple representing the composition will in generalnot be minimal, as can be checked for instance in Example 1.12. We will in this paper work with permutative categories , which are symmetric monoidal categories that arestrictly associative and have a strict unit. By a strict monoidal functor , we will mean a monoidal functor F such that the morphisms F ( x ) ⊕ F ( y ) → F ( x ⊕ y ) are the identity.Let Set denote the category with objects the natural numbers, where we identify the integer r (cid:62) [ r ] = { , . . . , r } , and with morphisms the maps of sets. We denote by Set × its subcategory of iso-morphisms (the bijections). The categories Set and
Set × are both permutative categories with the monoidalstructure ⊕ defined using the sum on objects, and disjoint union on morphisms, using the canonical identifica-tion [ r ] (cid:116) [ s ] ∼ = [ r + s ] . The unit is the empty set [ ] = Ø and the symmetry [ r ] ⊕ [ s ] = [ r + s ] → [ r + s ] = [ s ] ⊕ [ r ] is given by the ( r , s ) block permutation.Let A = { a , . . . , a n } as before. We now define our category Cantor A of finitely generated free Cantor algebrasof type A . To avoid set-theoretical issues, we will only consider the Cantor algebras freely generated by thesets [ r ] for r (cid:62)
0. So the objects of
Cantor A are the natural numbers just like for the category Set , butwith r now identified with the Cantor algebra C A [ r ] . The morphisms in Cantor A are the morphisms of Cantoralgebras as in Definition 1.1. We will denote by Cantor × A the subcategory of isomorphisms in Cantor A .Taking the free Cantor algebra on a given set induces a functorC A : Set −→ Cantor A which is the identity on objects. Indeed, any map of sets [ r ] → [ s ] induces a morphism C A [ r ] → C A [ s ] betweenthe free Cantor algebras, because C A [ r ] is free on [ r ] and [ s ] canonically identifies with a subset of C A [ s ] . Thisassociation is compatible with composition. 8 efinition 1.14. For r < s , we will denote by i L : [ r ] (cid:44) → [ s ] i R : [ r ] (cid:44) → [ s ] the left and right embeddings of [ r ] into [ s ] , i.e. that of [ r ] as the first r (resp. last) r elements of [ s ] . We willlikewise denote by i L , i R : C A [ r ] −→ C A [ s ] the corresponding induced maps C A ( i L ) and C A ( i R ) .We use the functor C A and the permutative structure of Set to define a permutative structure, also denoted ⊕ ,on Cantor A : On objects, we define C A [ r ] ⊕ C A [ s ] : = C A [ r + s ] and for morphisms f : C A [ r ] → C A [ r (cid:48) ] and g : C A [ s ] → C A [ s (cid:48) ] , we define f ⊕ g : C A [ r + s ] → C A [ r (cid:48) + s (cid:48) ] to be the unique morphism defined on the basis [ r + s ] = i L [ r ] (cid:116) i R [ s ] using the map [ r ] f | [ r ] −→ C A [ r (cid:48) ] i L −→ C A [ r (cid:48) + s (cid:48) ] on the first r elements and [ s ] g | [ s ] −→ C A [ s (cid:48) ] i R −→ C A [ r (cid:48) + s (cid:48) ] on the last s elements. Finally, let σ r , s : C A [ r + s ] = C A ([ r ] ⊕ [ s ]) −→ C A ([ s ] ⊕ [ r ]) = C A [ r + s ] be the image under the functor C A of the symmetry [ r ] ⊕ [ s ] → [ s ] ⊕ [ r ] in the category Set . Proposition 1.15.
The sum ⊕ and symmetry σ r , s defined above make Cantor × A into a permutative category,with unit C A [ ] = Ø , with the property that the free Cantor algebra functor C A : Set × → Cantor × A is a strictsymmetric monoidal functor, and that the sum Cantor × A ( C A [ r ] , C A [ r (cid:48) ]) × Cantor × A ( C A [ s ] , C A [ s (cid:48) ]) ⊕ −→ Cantor × A ( C A ([ r + s ] , C A [ r (cid:48) + s (cid:48) ])) is injective. One can likewise show that
Cantor A is a permutative category. We restrict to Cantor × A for simplicity, as thisis the part that is relevant to us.Before proving the result, we interpret the sum ⊕ in terms of representatives.We can reinterpret Lemma 1.9 as saying that the sum of sets [ r ] ⊕ [ s ] = [ r + s ] extends to a sum of expansions: ⊕ : E [ r ] × E [ s ] −→ E [ r + s ] , formally defined by setting E ⊕ F = i L ( E ) (cid:116) i R ( F ) . (By the same lemma, this sum is an isomorphism.) Usingthis sum operation, we have the following: Lemma 1.16.
Let f ∈ Cantor × A ( C A [ r ] , C A [ r (cid:48) ]) and g ∈ Cantor × A ( C A [ s ] , C A [ s (cid:48) ]) be isomorphisms representedby ( E , F , λ ) ∈ Rep ( f ) and ( G , H , µ ) ∈ Rep ( g ) . Then ( E ⊕ G , F ⊕ H , λ ⊕ µ ) represents the sum f ⊕ g. More-over, if ( E , F , λ ) is the least element of Rep ( f ) and ( G , H , µ ) the least element of Rep ( g ) , then the leastelement of Rep ( f ⊕ g ) is ( E ⊕ G , F ⊕ H , λ ⊕ µ ) . roof. The sum f ⊕ g is defined as the unique map taking i L [ r ] ⊂ [ r + s ] to C A [ r (cid:48) + s (cid:48) ] using i L ◦ f and i R [ s ] ⊂ [ r + s ] using i R ◦ g . Now suppose that f is represented by ( E , F , λ ) and g by ( G , H , µ ) . The mapof Cantor algebras h : C A [ r + s ] → C A [ r (cid:48) + s (cid:48) ] represented by ( E ⊕ G , F ⊕ H , λ ⊕ µ ) by definition takes i L ( E ) to i L ( F ) using λ = f | E and i L ( G ) to i L ( H ) using µ = g | G . As i L ( E ) is an expansion of i L [ r ] with i L ( F ) thecorresponding expansion of i L ( f [ r ]) , and the induced map h is a map of Cantor algebras, we necessarily havethat h | i L [ r ] = i L ◦ f , and likewise, h | i R [ s ] = i R ◦ g . Hence h = f ⊕ g .The sum respects the poset structure by Lemma 1.9. Now suppose that ( E , F , λ ) and ( G , H , µ ) are leastelements, and that there exists ( S , T , ν ) < ( E ⊕ G , F ⊕ H , λ ⊕ µ ) strictly smaller in Rep ( f ⊕ g ) . By Lemma 1.9,the subset S : = S ∩ C + A [ r ] is an expansion of [ r ] . As S ⊂ C + A [ r ] ⊂ C + A [ r + s ] ∼ = C + A [ r ] (cid:116) C + A [ s ] is an expansion of i L [ r ] inside C A [ r + s ] , we have ( f ⊕ g )( S ) = i L ◦ f ( S ) inside C A [ r (cid:48) + s (cid:48) ] , as f ⊕ g restrictsto f on this subset. It follows that ( S , f ( S ) , f | S ) is also a representative of f which is smaller or equalto ( E , F , λ ) . Likewise, setting S : = S ∩ C + A [ r ] ⊂ C + A [ r + s ] , we get a representative ( S , g ( S ) , g | S ) of g whichis smaller or equal to ( G , H , µ ) . Given that S = S ⊕ S , if S < E ⊕ G , we must have that either S < E or S < G , contradicting the minimality assumption. Hence the sum ⊕ takes least elements to least elements. Proof of Proposition 1.15.
By the lemma, we can write the sum
Cantor × A ( C A [ r ] , C A [ s ]) × Cantor × A ( C A [ r (cid:48) ] , C A [ s (cid:48) ]) ⊕ −→ Cantor × A ( C A [ r + r (cid:48) ] , C A [ s + s (cid:48) ]) in terms of (minimal) representatives. Functoriality follows then from the fact that C + A [ r + r (cid:48) ] ∼ = C + A [ r ] (cid:116) C + A [ r (cid:48) ] and likewise for s , s (cid:48) , and the fact that composition can be computed componentwise. Associativity of thesum is likewise easily checked in this description of the sum, and C A [ ] is a strict unit. For the symmetry, weneed to check that C A [ r ] ⊕ C A [ s ] f ⊕ g (cid:47) (cid:47) σ r , s (cid:15) (cid:15) C A [ r (cid:48) ] ⊕ C A [ s (cid:48) ] σ r (cid:48) , s (cid:48) (cid:15) (cid:15) C A [ s ] ⊕ C A [ r ] g ⊕ f (cid:47) (cid:47) C A [ s (cid:48) ] ⊕ C A [ r (cid:48) ] commutes for any morphism f , g . In terms of representatives, one checks that if f is represented by ( E , F , λ ) and g by ( G , H , µ ) , then both compositions will be represented by ( E ⊕ G , H ⊕ F , (cid:98) σ r (cid:48) , s (cid:48) ◦ ( λ ⊕ µ )) = ( E ⊕ G , H ⊕ F , ( µ ⊕ λ ) ◦ (cid:98) σ r , s ) for (cid:98) σ r , s the restrictions of the symmetry σ r , s ∈ Aut ( C A [ r + s ]) to E ⊕ G and (cid:98) σ r (cid:48) , s (cid:48) the corresponding restric-tion of σ r (cid:48) , s (cid:48) ∈ Aut ( C A [ r (cid:48) + s (cid:48) ]) to λ ( F ) ⊕ µ ( G ) . So the square commutes and ( Cantor A , ⊕ , σ , C A [ ]) is apermutative category.Finally, injectivity of the sum follows, also using representatives, from the fact that an automorphism f isuniquely represented by its least representative in Rep ( f ) (Proposition 1.11), that sums of minimal represen-tatives are minimal representatives (Lemma 1.16), and that the map is injective on representatives.The following proposition will be useful in Section 2. Proposition 1.17.
Let g : C A [ r + s ] → C A [ r + s ] be an isomorphism and suppose that g restricts to the iden-tity on a given finite basis of C A [ s ] ≡ i R C A [ s ] ⊂ C A [ r + s ] . Then we have g = g ⊕ C A [ s ] for some isomor-phism g : C A [ r ] → C A [ r ] . Here and in the following we often employ the Milnor–Moore notation and denote the identity morphism ofan object by that object. 10 roof.
Let ( F , G , λ ) ∈ Rep ( g ) be a representative of g and let E ⊂ i R C A [ s ] be a finite basis of C A [ s ] fixed by g .As F is an expansion of [ r + s ] , we have that F : = F ∩ i R C + A [ s ] is an expansion of [ s ] (Lemma 1.9). Now F and E have a common expansion ˆ E by Lemma 1.8, which is an expansion of [ s ] as F was an expansionof [ s ] . Let ˆ F = ( F ∩ i L C + A [ r ]) ∪ ˆ E and ˆ G = g ( ˆ F ) and ˆ λ = g | ˆ F . Then ( ˆ F , ˆ G , ˆ λ ) is also a representative of g .Note now that g ( ˆ E ) = ˆ E and ˆ λ | ˆ E = id as g respects expansions and restricted to the identity on E . It followsthat ˆ G = ( ˆ G ∩ i L C + A [ r ]) ∪ ˆ E for ( ˆ G ∩ C A [ r ]) an expansion of [ r ] . Hence g = g ⊕ C A [ s ] for g representedby ( ˆ F = F ∩ i L C + A [ r ] , ˆ G = ˆ G ∩ i L C + A [ r ] , ˆ λ | ˆ F ) , which proves the result. This section and the next one are concerned with the proof of homological stability for the Higman–Thompson groups V n , r , the automorphism group of the free Cantor algebra C A [ r ] , with respect to the number r of generators, where A = { a , . . . , a n } as before. Given a family of groups satisfying a few properties, thepaper [R-WW17] yields a sequence of spaces whose high connectivity implies homological stability for thefamily of groups. In this section, we will show how the groups V n , r fit in the framework of [R-WW17] andconstruct spaces relevant to the proof of homological stability.For a fixed type A we collect the Higman–Thompson groups into a groupoid V A , which is a subgroupoid ofthe category Cantor A , or in fact of its groupoid of isomorphisms Cantor × A defined in the previous section.The objects of V A are the same as those of Cantor × A , namely the natural numbers r as placeholders for thefree Cantor algebras C A [ r ] , and the morphism sets are defined by setting V A ( C A [ r ] , C A [ s ]) = (cid:26) Cantor × A ( C A [ r ] , C A [ s ]) r = s Ø r (cid:54) = s In other words, V A ∼ = (cid:71) r (cid:62) V n , r is the groupoid of automorphisms in Cantor × A , with each group V n , r considered as a groupoid with one object.Recall that there are isomorphisms C A [ r ] ∼ = C A [ r + ( n − )] for any r (cid:62) n = | A | ). These isomorphismsare morphisms in the groupoid Cantor × A but we do not include these isomorphisms into the groupoid V A .Note that the symmetric monoidal structure of Cantor × A restricts to V A , making it a permutative groupoid. Recall from [R-WW17, Def. 1.3] that a monoidal category ( H , ⊕ , ) is called homogeneous if the monoidalunit 0 is initial and if the following two conditions are satisfied for every pair of objects X , Y in H : Theset H ( X , Y ) is a transitive Aut H ( Y ) –set under post-composition, and the homomorphismAut H ( X ) → Aut H ( X ⊕ Y ) that takes f to f ⊕ Y is injective with image { ϕ ∈ Aut H ( X ⊕ Y ) | ϕ ◦ ( ι X ⊕ Y ) = ι X ⊕ Y } . The main examples ofhomogeneous categories can be obtained by applying Quillen’s bracket construction to a groupoid. We recallthis construction here, and show that it yields a homogeneous category when applied to the groupoid V A .Let Q A = (cid:104) V A , V A (cid:105) denote the category obtained by applying Quillen’s bracket construction (see [Gra76,p. 219]) to the groupoid V A . The category Q A has the same objects as V A and there are no morphismsfrom C A [ r ] to C A [ s ] unless there exists a k such that C A [ k ] ⊕ C A [ r ] ∼ = C A [ s ] in V A , i.e. r (cid:54) s in our case,with k = s − r . If this is the case, morphisms are equivalence classes [ f ] of morphisms f in V n , s = Aut V A ( C A [ s ]) with f ∼ f (cid:48) if there exists an element g in V n , k such that f (cid:48) = f ◦ ( g ⊕ C A [ r ]) : C A [ s ] = C A [ k ] ⊕ C A [ r ] −→ C A [ s ] . = C A [ ] is now an initial object in the category Q A . We will write ι r : Ø → C A [ r ] for theunique morphism, which we can represent as the equivalence class [ C A [ r ]] of the identity in V n , r . Proposition 2.1.
The category Q A is a permutative and homogeneous category with maximal sub-groupoid V A .Proof. This is a direct application of three results in [R-WW17]: Because ( V A , ⊕ , σ , Ø ) is a symmetricmonoidal groupoid, [R-WW17, Prop. 1.7] gives that Q A (denoted U V A in that paper) inherits a symmetricmonoidal structure, with its unit Ø initial. And given that V A was actually permutative, so is Q A . We havethat Aut ( Ø ) = { id } and that there are no zero divisors in Q A : If there is an isomorphism C A [ r ] ⊕ C A [ s ] ∼ = Øin V A , then we must have that r = s =
0. Then [R-WW17, Prop. 1.6] gives that V A is the maximal sub-groupoid of Q A . Also, the groupoid V A satisfies cancellation (by construction): If there exists an isomor-phism C A [ r ] ⊕ C A [ s ] ∼ = C A [ r ] ⊕ C A [ s (cid:48) ] in V A , then C A [ s ] ∼ = C A [ s (cid:48) ] in V A ; they are in fact equal, because wehave C A [ r + s ] ∼ = C A [ r + s (cid:48) ] in the groupoid V A if and only if r + s = r + s (cid:48) . Finally, the groupoid V A satisfiesthat the map Aut V A ( C A [ r ]) → Aut V A ( C A [ r + s ]) adding the identity on C A [ s ] is injective by Proposition 1.15.It then follows from [R-WW17, Thm. 1.9] that Q A is homogenous, which completes the proof. Remark 2.2.
Explicitly, the monoidal structure of Q A is defined as follows. On objects, it is as in V A inducedby the sum of natural numbers. Given two morphisms [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) and [ g ] ∈ Q A ( C A [ r (cid:48) ] , C A [ s (cid:48) ]) ,the equivalence class of the compositionC A [ s + s (cid:48) ] C A [ s − r ] ⊕ C A [ s − r (cid:48) ] ⊕ C A [ r ] ⊕ C A [ r (cid:48) ] C A [ s − r ] ⊕ σ s − r (cid:48) , r ⊕ C A [ r (cid:48) ] (cid:15) (cid:15) C A [ s − r ] ⊕ C A [ r ] ⊕ C A [ s − r (cid:48) ] ⊕ C A [ r (cid:48) ] C A [ s ] ⊕ C A [ s (cid:48) ] f ⊕ g (cid:15) (cid:15) C A [ s ] ⊕ C A [ s (cid:48) ] C A [ s + s (cid:48) ] . defines [ f ] ⊕ [ g ] ∈ Q A ( C A [ r + r (cid:48) ] , C A [ s + s (cid:48) ]) , where σ s − r (cid:48) , r is the symmetry of V A . (See the proof of Proposi-tion 1.6 in [R-WW17].) A Recall from Proposition 1.11 that the isomorphisms of
Cantor A , and hence also the elements f of V n , s , admita unique minimal presentation ( E , F , λ ) where E , F are expansions of the standard generating set [ s ] of C A [ s ] ,and λ : E → F is a bijection. We will now define an analogous minimal representation for the morphismsin Q A ( C A [ r ] , C A [ s ]) for r < s . This description makes use of the posets E [ r ] , I [ r ] and I [ r ] of expansions,independent sets, and non-generating independent sets of [ r ] of Section 1.1.An element [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) is represented by some f ∈ V n , s , which itself admits a representa-tive ( E , F , λ ) , with E , F ∈ E [ s ] expansions of [ s ] and λ : E → F a bijection. Using the decomposition [ s ] = [ s − r ] ⊕ [ r ] , with the associated embedding i R : C + A [ r ] (cid:44) → C + A [ s ] , we can consider the intersection E : = E ∩ i R C + A [ r ] , whichis an expansion of [ r ] by Lemma 1.9. Then P : = λ ( E ) ⊂ λ ( E ) = F is an independent set of C A [ s ] . This waywe can associate to any morphism [ f ] ∈ Q A a triple ( E , P , λ ) with E ∈ E [ r ] and P ∈ I [ s ] and λ : E → P abijection. With this in mind, we extend the definition of Rep ( f ) from Section 1.2 to all morphisms of Q A : Definition 2.3.
For [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) , letRep [ f ] = { ( E , P , λ ) | E ∈ E [ r ] , P ∈ I [ s ] , λ = [ f ] | E : E ∼ = −→ P } , which we consider as a poset by setting ( E , P , λ ) (cid:54) ( E (cid:48) , P (cid:48) , λ (cid:48) ) if and only if E (cid:54) E (cid:48) in E [ r ] .12ote that [ f ] | E is a well-defined map as E ⊂ C A [ r ] ≡ i R C A [ r ] and any two representatives of [ f ] agree as mapson that subalgebra.If r = s , we have that Rep [ f ] = Rep ( f ) as defined earlier. In particular, if ( E , P , λ ) ∈ Rep [ f ] in this case,then P ∈ E [ s ] . On the other hand, if r < s and ( E , P , λ ) ∈ Rep [ f ] , we must have that P ∈ I [ s ] as P is non-generating in that case. Proposition 1.11 extends to the following result: Proposition 2.4.
For any morphism [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) , the set Rep [ f ] is non-empty and has a leastelement.Proof. We have already seen above that the set is non-empty. We will in this proof construct a leastelement. Pick a representative f ∈ V n , s of [ f ] , and let ( E , F , λ ) be the least element of Rep ( f ) givenby Proposition 1.11. As above, we consider the set [ s ] as the sum [ s − r ] ⊕ [ r ] , which gives embed-dings i L and i R of C A [ s − r ] and C A [ r ] inside C A [ s ] . Consider the triple ( E , P , λ | E ) with E = E ∩ i R C + A [ r ] and P = λ ( E ) . We have that E ∈ E [ r ] by Lemma 1.9 and P ∈ I [ s ] because P ⊂ λ ( E ) = F . Giventhat f | E = λ , we get that ( E , P , λ | E ) ∈ Rep [ f ] . We claim that ( E , P , λ | E ) is the least element of Rep [ f ] . Sosuppose ( E (cid:48) , P (cid:48) , λ (cid:48) ) ∈ Rep [ f ] is another element. Set E (cid:48) = ( E ∩ i L C + A [ s − r ]) ∪ E (cid:48) . By Lemma 1.9, we have E (cid:48) ∈ E [ s ] , and hence this is a basis for C A [ s ] . Now f ( E (cid:48) ) = f ( E ∩ C + A [ s − r ]) ∪ P (cid:48) isa basis, as f is an isomorphism, and it is included in C + A [ s ] as both f ( E ∩ C + A [ s − r ]) and P (cid:48) are. Hence it is anexpansion of [ s ] by Lemma 1.5 and ( E (cid:48) , f ( E (cid:48) ) , f | E (cid:48) ) is an element of Rep ( f ) . By the minimality of ( E , F , λ ) ,we must have that E (cid:54) E (cid:48) , which gives E (cid:54) E (cid:48) as requested.Define P A ( r , s ) = (cid:40) { ( E , P , λ ) | E ∈ E [ r ] , P ∈ I [ s ] , λ : E ∼ = −→ P } if r < s { ( E , P , λ ) | E ∈ E [ r ] , P ∈ E [ s ] , λ : E ∼ = −→ P } if r = s . We partially order P A ( r , s ) by setting ( E , P , λ ) (cid:54) ( E (cid:48) , P (cid:48) , λ (cid:48) ) if and only if E (cid:54) E (cid:48) and λ (cid:48) is the restriction to E (cid:48) of the map C A ( E ) → C A ( P ) → C A [ s ] induced by λ . As P (cid:48) = λ (cid:48) ( E (cid:48) ) , it follows that P (cid:48) is an expansion of P insuch a situation. Proposition 2.5.
For all (cid:54) r (cid:54) s, there is a poset isomorphism P A ( r , s ) ∼ = (cid:71) [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) Rep [ f ] . In particular, taking least representative in
Rep [ − ] defines an isomorphism between Q A ( C A [ r ] , C A [ s ]) and theset of minimal elements in P A ( r , s ) .Proof. Forgetting which [ f ] a triple represents defines a map α : (cid:71) [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) Rep [ f ] −→ P A ( r , s ) which is a map of posets as the order relation is defined in the same way in both cases. We want to show thatthis map is a poset bijection, that is a set bijection such that any two elements that are related in the target,are also related in the source, or equivalently, that any two elements that are related in P A ( r , s ) represent thesame morphism [ f ] .When r = s , any element ( E , P , λ ) ∈ P A ( s , s ) determines a unique automorphism f of C A [ s ] , as E and P are bases in this case. This proves that α is bijective in that case. It is also immediate that,if ( E , P , λ ) (cid:54) ( E (cid:48) , P (cid:48) , λ (cid:48) ) in P A ( s , s ) , then the triples represent the same morphism f , proving the result inthe case r = s . 13e now assume that r < s . To show that α is injective, suppose that [ f ] and [ f (cid:48) ] have the sameimage ( E , P , λ ) ∈ P A ( r , s ) . Consider the isomorphism g : = ( f (cid:48) ) − ◦ f : C A [ s ] = C A [ r − s ] ⊕ C A [ r ] −→ C A [ r − s ] ⊕ C A [ r ] = C A [ s ] . We have that g restricts to the identity on E as f | E = λ = f (cid:48) | E . By Proposition 1.17, it followsthat g = g ⊕ C A [ r ] for some g ∈ V n , s − r . Hence f = f (cid:48) ◦ ( g ⊕ C A [ r ]) , which proves that [ f ] = [ f (cid:48) ] .To show surjectivity in the case r < s , suppose that ( E , P , λ ) ∈ P A ( r , s ) . Let F be an expansionof [ s ] containing P . We have that | E | = r + a ( n − ) = | P | for some a (cid:62) | F | = s + b ( n − ) for some b (cid:62)
0, with | F | − | P | = s − r + ( b − a )( n − ) > ( E , P , λ ) ∈ P A ( r , s ) with r < s . If b − a <
0, replace F by an expansion of F still containing P by expanding an elementof F \ P (which is non-empty by the same assumption) at least ( b − a ) times. After doing this, we canassume moreover that b − a (cid:62)
0. Now let G be an expansion of [ s − r ] of cardinality s − r + ( b − a )( n − ) and pick a bijection µ : G → F \ P . Then G ∪ E is an expansion of [ s ] and ( G ∪ E , µ ( G ) ∪ P , µ ∪ λ ) rep-resents an element f ∈ V n , s . Let [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) be its equivalence class. By construction, wehave ( E , P , λ ) ∈ Rep [ f ] which gives surjectivity also in this case.Finally, still assuming r < s , we need to check that ( E , P , λ ) (cid:54) ( E (cid:48) , P (cid:48) , λ (cid:48) ) in P A ( r , s ) can only happen ifboth triples represent the morphism [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) , with λ and λ (cid:48) necessarily the restriction of arepresentative f of [ f ] to E and E (cid:48) respectively. By the surjectivity of α , we know that ( E , P , λ ) representssome [ f ] ∈ Q A ( C A [ r ] , C A [ s ]) . Let ( G ∪ E , µ ( G ) ∪ P , µ ∪ λ ) ∈ Rep ( f ) be the representative of such an f constructed above. Now ( G ∪ E , µ ( G ) ∪ P , µ ∪ λ ) (cid:54) ( G ∪ E (cid:48) , µ ( G ) ∪ P (cid:48) , µ ∪ λ (cid:48) ) , showing that both triplesnecessarily represent the same morphism f . Hence ( E , P , λ ) and ( E (cid:48) , P (cid:48) , λ (cid:48) ) are both in Rep [ f ] and werealready comparable there. A In the general context of the paper [R-WW17], given a pair ( B , X ) of objects in a homogeneous category, asequence of semi-simplicial sets W r ( B , X ) is defined, and the main theorem in that paper says that homo-logical stability holds for the automorphism groups of the objects B ⊕ X ⊕ r as long as the associated semi-simplicial sets are highly connected. In good cases, the connectivity of the semi-simplicial sets W r ( B , X ) canbe computed from the connectivity of closely related simplicial complexes S r ( B , X ) .We are here interested in the pair of objects ( B , X ) = ( Ø , C A [ ]) in the homogeneous category Q A . Indeed,the automorphism group of Ø ⊕ C A [ ] ⊕ r = C A [ r ] in the category Q A is the Higman–Thompson group V n , r .We will therefore begin by describing the semi-simplicial sets W r = W r ( Ø , C A [ ]) and the simplicial com-plexes S r = S r ( Ø , C A [ ]) from Definitions 2.1 and 2.8 in [R-WW17], and show that we are in a situationwhere we can use the connectivity of the latter to compute the connectivity of the former. In the followingSection 3 we will estimate that connectivity. Definition 2.6.
Given an integer r (cid:62)
1, let W r = W r ( Ø , C A [ ]) be the semi-simplicial set where the setof p –simplices is the set Q A ( C A [ p + ] , C A [ r ]) of morphisms C A [ p + ] → C A [ r ] in Q A and the i –th boundarymap Q A ( C A [ p + ] , C A [ r ]) → Q A ( C A [ p ] , C A [ r ]) is defined by precomposing with C A [ i ] ⊕ ι ⊕ C A [ p − i ] .To the semi-simplicial set W r we associate the following simplicial complex, of the same dimension r − Definition 2.7.
Given r (cid:62)
1, let S r = S r ( Ø , C A [ ]) be the simplicial complex with the same vertices as W r ,namely the set of maps Q A ( C A [ ] , C A [ r ]) , and where p + [ f ] , . . . , [ f p ] form a p –simplex ifthere exists a p –simplex of W r having them as vertices.Given a simplicial complex S , one can build a semi-simplicial set S ord that has a p –simplex for each orderingof the vertices of each p –simplex of S . 14 roposition 2.8. There is an isomorphism of semi-simplicial sets W r ∼ = S ord r . Moreover, if S r is ( r − ) –connected, then so is W r .Proof. This follows from [R-WW17, Prop. 2.9, Thm. 2.10], given that Q A is symmetric monoidal, once wehave checked that it is locally standard at ( Ø , C A [ ]) in the sense of [R-WW17, Def. 2.5]. This means twothings: Firstly, the morphisms C A [ ] ⊕ ι and ι ⊕ C A [ ] are distinct in Q A ( C A [ ] , C A [ ]) , and secondly, forall r (cid:62)
1, the map Q A ( C A [ ] , C A [ r − ]) → Q A ( C A [ ] , C A [ r ]) that takes a morphism [ f ] to [ f ] ⊕ ι is injective.For the first statement, we need to describe the two morphisms C A [ ] ⊕ ι and ι ⊕ C A [ ] precisely.Here C A [ ] ∈ Q A ( C A [ ] , C A [ ]) = V A ( C A [ ] , C A [ ]) is the identity, while ι = [ C A [ ]] ∈ Q A ( Ø , C A [ ]) isalso represented by the identity on C A [ ] , but this time only up to an automorphism of C A [ ] . Usingthe explicit definition of the composition given in Remark 2.2, we compute that C A [ ] ⊕ ι = [ σ , ] for σ , : C A [ ] = C A [ ] ⊕ C A [ ] → C A [ ] ⊕ C A [ ] = C A [ ] the symmetry, while ι ⊕ C A [ ] = C A [ ] is repre-sented by the identity (because the symmetry needed for that composition is σ , , which is the identity).To check that these morphisms are distinct in Q A ( C A [ ] , C A [ ]) , we use the minimal presentations (Proposi-tions 2.4 and 2.5). We have that [ σ , ] has minimal presentation ( { } , { } , λ ) while the second has presenta-tion ( { } , { } , µ ) , for λ , µ the unique maps, showing that they are indeed distinct.For the second statement, we have [ f ] ⊕ ι = [( f ⊕ C A [ ]) ◦ ( C A [ r − ] ⊕ σ , )] . If [ f ] is minimally presented by ( E , P , λ ) , then one can check that [ f ] ⊕ ι is minimally presentedby ( E , i L ( P ) , i L ◦ λ ) for i L : C + A [ r − ] → C + A [ r ] the left embedding. As i L is injective, the result follows.By [R-WW17, Prop. 2.9] we now know that W r “satisfies condition (A)”, which means that it is isomorphicto S ord r , and [R-WW17, Thm. 2.10] of the same paper gives the second part of the statement. Using the minimal representatives of morphisms of Q A given by Proposition 2.5, we can represent the ver-tices of W r and S r as triples ( E , P , λ ) where E ∈ E [ ] is an expansion of [ ] and P ∈ I [ r ] is an independentset (non-generating if r > r = E , and hence also P , necessarily has car-dinality 1 + a ( n − ) for some a (cid:62)
0. To study the connectivity of the simplicial complex S r , we will usevariants U r , U ∞ r and T ∞ r of this complex, where only the independent sets P are remembered.Recall that I [ r ] = I [ r ] \ E [ r ] denotes the set of non-generating independent sets. Definition 2.9.
Let U = E [ ] to be the simplicial complex of dimension 0 consisting of all the expansionsof the set [ ] . For r (cid:62)
2, let U r be the simplicial complex of dimension r − P ∈ I [ r ] of cardinality congruent to 1 modulo n −
1. A set of p + P , . . . , P q forms a q –simplex of U r if the sets P i are pairwise disjoint and • q < r − P (cid:116) · · · (cid:116) P q ∈ I [ r ] , or • q = r − P (cid:116) · · · (cid:116) P q ∈ E [ r ] .Recall from [HW10, Def. 3.2] that a complete join complex over a simplicial complex X is a simplicialcomplex Y with a projection π : Y → X , which is surjective, injective on each simplex, and with the propertythat, for each simplex σ = (cid:104) x , . . . , x q (cid:105) of X , the subcomplex π − ( σ ) of Y of simplices projecting down to X is the join complex π − ( x ) ∗ · · · ∗ π − ( x q ) . Proposition 2.10.
The simplicial complex S r is a complete join complex over U r .
15o prove the proposition, we will use the following lemma, which describes the vertices of any simplex of W r .We introduce first a notation: for any 1 (cid:54) j (cid:54) r , let i j : [ ] → [ r ] denote the map defined by i j ( ) = j . We will also write i j : C A [ ] −→ C A [ r ] for the induced map of Cantor algebras. Lemma 2.11.
Let [ f ] ∈ Q A ( C A [ q + ] , C A [ r ]) be a q–simplex of W r , and let ( E , P , λ ) ∈ Rep [ f ] be a triplerepresenting [ f ] . Then the vertices of [ f ] in W r are represented by the triples ( E , P , λ ) , . . . , ( E q , P q , λ q ) where E j = E ∩ i j C A [ ] , P j = λ ( E j ) and λ j = λ | E j . Moreover, ( E , P , λ ) is the minimal representative of [ f ] if and only if each ( E j , P j , λ j ) is likewise minimal.Proof. The j –th vertex of the simplex [ f ] is given by [ f ] ◦ ( ι j ⊕ C A [ ] ⊕ ι q − j ) ∈ Q A ( C A [ ] , C A [ r ]) . This com-position is represented by the following automorphism of C A [ r ] :C A [ r − q − ] ⊕ (cid:0) C A [ j ] ⊕ C A [ q − j ] ⊕ C A [ ] (cid:1) id ⊕ σ q − j , (cid:15) (cid:15) C A [ r − q − ] ⊕ (cid:0) C A [ j ] ⊕ C A [ ] ⊕ C A [ q − j ] (cid:1) id (cid:15) (cid:15) C A [ r − q − ] ⊕ C A [ q + ] f (cid:15) (cid:15) C A [ r ] (2.1)Now suppose [ f ] is represented by ( E , P , λ ) with f ∈ Aut ( C A [ r ]) a representative of [ f ] . This means that wehave E ∈ E [ q + ] , P ∈ I [ r ] and λ = f | E . Also, there exists F ∈ E [ r − q − ] such that f is represented by ( F ⊕ E , f ( F ) ⊕ P , f | F ⊕ λ ) = ( F ∪ i R ( E ) , f ( F ) ∪ i R ( P ) , f | F ∪ λ ) . On C + A [ r ] = C + A [ ] (cid:116) · · · (cid:116) C + A [ ] , we see that the composition (2.1) takes the last component C + A [ ] tothe ( r − j − ) –st position and then applies f . Thus it is represented by the triple ( F ⊕ E ⊕ · · · ⊕ E j − ⊕ E j + ⊕ · · · ⊕ E q ⊕ E j , f ( F ) ⊕ P , ( f | F ⊕ λ ) ◦ ( id ⊕ σ q − j , )) where E j = E ∩ i j C A [ ] ⊂ C A [ r ] . This proves the first part of the result, as the representative of the corre-sponding morphism in the category Q A ( C A [ ] , C A [ r ]) is obtained by restricting the source to i R C A [ ] .Finally, the statement about minimality follows as in the proof of Proposition 2.4 by showing that if the j –th vertex has a smaller representative ( E (cid:48) j , P (cid:48) j , λ (cid:48) j ) , we can use it to construct a smaller representative for [ f ] simply by replacing E ∩ i j C A [ ] with i j ( E (cid:48) j ) , and vice versa. Proof of Proposition 2.10.
Using Proposition 2.5, we identify the vertices of the simplicial complex S r with the minimal elements of P A ( , r ) . Using the lemma, we see that, under this identification, a collec-tion of vertices ( E , P , λ ) , . . . , ( E q , P q , λ q ) forms a q –simplex if and only if there exists a minimal ele-ment ( E , P , λ ) ∈ P A ( q , r ) such that E = E (cid:116) · · · (cid:116) E q for E j = E ∩ i j C + A [ ] , and for each j , λ j = λ | E j and P j = λ ( E j ) . It follows that P = P ∪ · · · ∪ P q with each P j pairwise disjoint in C + A [ r ] , and λ = λ ∪ · · · ∪ λ q .Note that by Lemma 1.9, each E j being an expansion of [ ] gives that E is an expansion of [ q + ] . It followsthat P ∪ · · · ∪ P q is an independent set that is non-generating if q + < r , and an expansion of [ r ] if q + = r .We define the projection π : S r → U r on vertices by setting π ( E , P , λ ) = P . From the above discussion, wesee that π is a map of simplicial complexes. The map is surjective as for any q –simplex (cid:104) P , . . . , P q (cid:105) of U r ,16e can choose expansions E j of [ ] with | E j | = | P j | , by the condition on the cardinality of the P j ’s, andbijections λ j : E j → P j . Then the collection ( E , P , λ ) , . . . , ( E q , P q , λ q ) will define a simplex of S r projectingdown to the given simplex. The map is injective on individual simplices as the P j ’s of distinct verticesin a simplex of S r are by definition distinct. Finally we check the join condition. For a vertex P of U r ,we have that π − ( P ) is the set of ( E , P , λ ) where E ∈ E [ ] has the same cardinality as P and λ : P → E is a bijection. As any choice of ( E j , λ j ) for each vertex P j of a simplex σ = (cid:104) P , . . . , P q (cid:105) of U r defines alift (cid:104) ( E , P , λ ) , . . . , ( E q , P q , λ q ) (cid:105) to S r , we see that π − ( σ ) is the join π − ( P ) ∗ · · · ∗ π − ( P q ) , finishing theproof of the result.In contrast to the simplicial complexes S r and U r , the variants U ∞ r and T ∞ r that we will now introduce are bothinfinite-dimensional (as the notation suggests). Definition 2.12.
For r (cid:62)
1, let U ∞ r be the simplicial complex whose vertices are the non-generating indepen-dent sets P ∈ I [ r ] of cardinality congruent to 1 modulo n −
1. Distinct vertices P , . . . , P p form a p -simplexin U ∞ r if the subsets are pairwise disjoint and their union P (cid:116) · · · (cid:116) P p is also an element of I [ r ] , that is theunion is a non-generating independent set.Note that U ∞ r and U r have the same set of vertices when r (cid:62) r =
1. In fact we have thefollowing:
Lemma 2.13.
The simplicial complexes U r and U ∞ r share the same ( r − ) –skeleton.Proof. This follows immediately from the definitions.
Definition 2.14.
For r (cid:62)
1, let T ∞ r denote the full subcomplex of U ∞ r on the vertices that have cardinality 1.The diagram U ∞ r T ∞ r ⊇ (cid:111) (cid:111) S r (cid:47) (cid:47) U r sk r − U r ⊆ (cid:79) (cid:79) sk r − U ∞ r ⊇ (cid:79) (cid:79) indicates the relations between the simplicial complexes that we have introduced so far, with the heightreflecting their dimension. In the next section, we will prove high connectivity of S r using this sequence ofmaps. This sequence of maps is chosen using the following ideas: • Going from S r to U r : only the sets P i are essential for determining the connectivity; • Going from U r to sk r − U r : removing the top simplices means that all simplices correspond now tonon-generating sets, and hence there is always something, and hence a lot, independent from any givensimplex—this will allow coning off; • Going from U r to U ∞ r : it is often easier to show that an infinite complex is contractible; • Going from U ∞ r to T ∞ r : we have more control over the vertices of the latter complex. In this section, we estimate the connectivity of the simplicial complexes defined in the previous section, anduse these to deduce homological stability for the canonical diagrams ( (cid:63) ) of the Higman–Thompson groups.As before, the chosen set A = { a , . . . , a n } is of cardinality n (cid:62) ∞ r and U ∞ r from the previous section. All the results of this section will bebased on the following two results. 17 roposition 3.1. Suppose | A | (cid:62) . Then for each integer r (cid:62) the simplicial complex T ∞ r is contractible. Proposition 3.2.
Suppose | A | (cid:62) . Then for each integer r (cid:62) the simplicial complex U ∞ r is contractible. Before we give proofs of these two propositions in Section 3.2, let us state and prove some of their conse-quences.Recall from [HW10, Def. 3.4] that a simplicial complex is called weakly Cohen–Macaulay of dimension n ifit is ( n − ) –connected and the link of every p –simplex of it is ( n − p − ) –connected. Corollary 3.3.
For all r (cid:62) the simplicial complexes U r are weakly Cohen–Macaulay of dimension r − .Proof. A simplicial complex is ( r − ) –connected if and only if its ( r − ) –skeleton is. Since the simplicialcomplexes U r and U ∞ r share the same ( r − ) –skeleton by Lemma 2.13, we see that the simplicial complex U r is ( r − ) –connected if the simplicial complex U ∞ r is, and the latter is even contractible by Proposition 3.2.Let now σ be a p –simplex of U r with vertices P , . . . , P p . We need to check that the link of σ is ( r − p − ) –connected. For p (cid:62) r −
2, there is nothing to check as ( r − ) − p − (cid:54) − ( − ) –connected is a non-condition. So assume p (cid:54) r −
3. Then P ∪· · ·∪ P p ∈ I [ r ] is a non-generating independent set. By Lemma 1.7,we can find a least Q ∈ I [ r ] (disjoint from the P i ’s) such that Q ∪ P ∪ · · · ∪ P p is an expansion of [ r ] . Notethat Q is non-empty as ∪ i P i is non-generating.To show the ( r − p − ) –connectivity of the link, it is enough to consider its ( r − p − ) -skeleton. We claimthat there is an isomorphism sk r − p − ( Link ( σ )) ∼ = sk r − p − ( U ∞ q ) for q = | Q | .To write a map sk r − p − ( U ∞ q ) → sk r − p − ( Link ( σ )) , we use the injection i : C + A [ q ] → C + A [ r ] induced by a cho-sen bijection [ q ] ∼ = Q and the inclusion Q ⊂ C + A [ r ] . (As we will see in Section 4, there is actually a canonicalsuch bijection as every Q has a canonical ordering induced by that of [ r ] and that of A , but we can use anybijection here.) This map induces a map i : I [ q ] → I [ r ] because Q was an independent set. Define α : sk r − p − ( U ∞ q ) −→ sk r − p − ( Link ( σ )) by setting α (cid:104) Q , . . . , Q q (cid:105) = (cid:104) i ( Q ) , . . . , i ( Q q ) (cid:105) . This is an injective map of simplicial complexes because i isinjective and preserves the independence condition. To show surjectivity, suppose that (cid:104) P p + , . . . , P s (cid:105) is a sim-plex of sk r − p − ( Link ( σ )) . Then (cid:104) P , . . . , P p , P p + , . . . , P s (cid:105) is a simplex of U r and there must exist Q (cid:48) disjointfrom the P i ’s such that Q (cid:48) ∪ P ∪ · · · ∪ P s is an expansion of [ r ] . Because Q was chosen so that Q ∪ P ∪ · · · ∪ P p is the least expansion containing P ∪ · · · ∪ P p , we must have that Q (cid:48) ∪ P p + ∪ · · · ∪ P s is an expansion of Q .Now s (cid:54) p + ( r − p − ) = r − (cid:104) P p + , . . . , P s (cid:105) is in the ( r − p − ) –skeleton of the link. Hencethe P i ’s together form a non-generating independent set and | Q (cid:48) | >
0. It follows that (cid:104) i − ( P p + ) , . . . , i − ( P s ) (cid:105) is a simplex of sk r − p − ( U ∞ q ) . This proves that the map α is an isomorphism.The required connectivity of the link follows from the contractibility of U ∞ q .Note that q = ∞ r is contractiblealso when r =
1, even though U ∞ has no relationship to U . Corollary 3.4.
For each r (cid:62) the spaces S r and W r are ( r − ) –connected.Proof. By Proposition 2.10, the simplicial complex S r is a complete join complex over U r , and U r is weaklyCohen–Macaulay of dimension r − r is alsoweakly Cohen–Macaulay of that dimension, and so in particular ( r − ) –connected. Then, by Proposition 2.8,it follows that the semi-simplicial sets W r are also ( r − ) –connected. Corollary 3.5.
The semi-simplicial set W r + is ( r − ) –connected for all r (cid:62) . roof. There is a morphism C A [ ] → C A [ r + ] in the category Q A as soon as r + (cid:62)
1, or equivalently r (cid:62) r + is non-empty for all r (cid:62)
0. That gives the cases r = , r (cid:62)
2, Corollary 3.4 gives that W r + is ( r − ) –connected, which, under the assumption on r , implies thatit is at least ( r − ) –connected. Define the stabilization homomorphism s r : V n , r → V n , r + as the map that takes an element f to f ⊕ C A [ ] .The maps s r for all r (cid:62) (cid:63) ) of groups of the introduction. Theorem 3.6.
Suppose n (cid:62) . The stabilization homomorphisms induce isomorphismss r : H d ( V n , r ; M ) −→ H d ( V n , r + ; M ) in homology in all dimensions d (cid:62) , for all r (cid:62) , and for all H ( V n , ∞ ) –modules M.Proof. First, we can apply the stability result of [R-WW17] to the category Q A with the chosen pair ofobjects ( C A [ ] , C A [ ]) : our Corollary 3.5 shows that the complexes W r ( C A [ ] , C A [ ]) ∼ = W r + ( Ø , C A [ ]) are ( r − ) –connected, which implies, by [R-WW17, Thm. 3.4], that s r is an isomorphism in the range ofdimensions d (cid:54) r − , that is a range that increases with the ‘rank’ r . Recall now that the group V n , r is iso-morphic to the group V n , r +( n − ) as soon as r (cid:62)
1. We choose an isomorphism between the two groups as fol-lows. Let h : C A [ ] → C A [ n ] be the isomorphism of Example 1.12 with minimal presentation ( { } × A , [ n ] , λ ) for λ : A → [ n ] a bijection (for example the canonical bijection coming from the picked ordering of A ).Then h r = h ⊕ C A [ r − ] : C A [ r ] → C A [ n + r − ] is also an isomorphism. Let γ ( h r ) denote conjugation with h r .Then we get a commutative diagram V n , +( r − ) s r (cid:47) (cid:47) γ ( h r ) (cid:15) (cid:15) V n , +( r − )+ γ ( h r + ) (cid:15) (cid:15) V n , n +( r − ) s r + (cid:47) (cid:47) V n , n +( r − )+ , as ( h ⊕ C A [ r ]) − ◦ ( f ⊕ C A [ ]) ◦ ( h ⊕ C A [ r ]) = (cid:0) ( h ⊕ C A [ r − ]) − ◦ f ◦ ( h ⊕ C A [ r − ]) (cid:1) ⊕ C A [ ] for all r (cid:62)
1. Given that the vertical maps are isomorphisms and increase the rank of the group (under theassumption n (cid:62) Remark 3.7.
For the purposes of the present paper, we will only need homological stability with respectto trivial, or potentially abelian, coefficients, in the form stated. Applying the more general Theorem Aof [R-WW17] instead of Theorem 3.4 in that paper, one obtains that stability also holds with finite degreecoefficient systems. We refer the interested reader to [R-WW17] for the definitions.
In the rest of this section we give proofs of Propositions 3.1 and 3.2.Recall that a vertex of T ∞ r is a non-generating independent set of cardinality 1 in C + A [ r ] . The setC + A [ r ] = (cid:71) h (cid:62) [ r ] × A h
19s canonically graded by the height h of its elements. If r (cid:54) =
1, any element of C + A [ r ] is non-generating andindependent, so the set of vertices of T ∞ r is the set of elements of C + A [ r ] , whereas for r = y ∈ C + A [ r ] that are independent of any givennon-generating independent set P , as long as we go high enough in C + A [ r ] . Lemma 3.8.
For any r , k , N (cid:62) , there exists a height h r , k , N (cid:62) , such that for any non-generating independentset P ∈ I [ r ] of cardinality k, and any h (cid:62) h r , k , N , there are at least N elements y of height h in C + A [ r ] \ P suchthat P ∪ { y } is still independent.Proof. If we were only concerned with one particular non-generating independent set P , we could pro-duce such a height simply by picking one element y ∈ C + A [ r ] independent of P , which exists by the non-generating assumption, and considering the A ∗ –subset C + A ( y ) ⊂ C + A [ r ] generated by y . If y has height h , thenat height h = h + (cid:96) , this A ∗ –subset has n (cid:96) elements, all independent of P . And the number n (cid:96) can be chosenas big as we like by increasing (cid:96) appropriately. To prove the lemma, we need to show that we can find aheight that works for any given P of some fixed cardinality k .Note that it is enough to find a height H such that for any independent set P of cardinality k , there is at least 1element y of height H independent of P . (Here y will depend on P .) Indeed, given such an H , just like in thecase of a fixed set P , for all h (cid:62) H , there will then be at least at least n h − H elements independent of P , namelythe descendents of y of height H . Setting h r , k , = H and more generally h r , k , N = H + (cid:96) for (cid:96) such that n (cid:96) (cid:62) N ,will then prove the result.We claim that we can take H = (cid:24) k − r + n − (cid:25) . Indeed, an element y ∈ C + A [ r ] of height H is dependent of P if there is a p ∈ P such that either y ∈ C + A ( p ) or p ∈ C + A ( y ) . There are rn H elements of height H in C + A [ r ] . Suppose P = { p , . . . , p k } with p i of height h i ,where we have ordered the elements of P so that h (cid:54) . . . (cid:54) h i < H (cid:54) h i + (cid:54) . . . (cid:54) h k . Then thereare rn H − ( n H − h + · · · + n H − h i + k − i ) elements of height H that are independent of P . This number islowest when the heights of the elements of P are lowest, where lowest means as many elements as possibleof height 0, then as many as possible of height one, and so on. It therefore remains to be shown that, if wechoose a set P with lowest heights, then there is still an element of height H that is independent of P . As P isassumed to be a non-generating independent set, it has at most ( r − ) elements of height 0, and if P indeedhas ( r − ) elements of height 0, then it can have at most ( n − ) elements of height 1, etc. (A set P withlowest height is displayed in Figure 3.1.) This shows that there will always be an element left in height H when we have k (cid:54) ( r − ) + ( n − ) H . By definition of H , this inequality is satisfied, and we are done.Figure 3.1: Non-generating independent set P ⊂ C + A [ ] with | A | = | P | = h , , = h , , = h r , k , N as the height such that there are at least N elements y of that heightwith P ∪ { y } still an independent set of cardinality k +
1. If we want P ∪ { y } to be non-generating , and henceagain defining a simplex of T ∞ r , we need N (cid:62)
2. 20 roof of Proposition 3.1.
We will show that, for all k (cid:62)
0, all maps S k → T ∞ r from spheres into the space T ∞ r are null-homotopic. Let us be given a map f : S k → T ∞ r . From [Zee64], we can assume that the sphere S k comes with a triangulation such that the map f is simplicial. Let v i be the number of simplices of dimension i is this triangulation of the sphere, and let v = v + · · · + v k be the total number of simplices of all dimensionsof that triangulation. In particular, this triangulation has v (cid:54) v vertices. Let N = v + h = h r , k , N be thecorresponding height obtained in Lemma 3.8. We will start by showing that the map f is homotopic to a mapwhose image only contains vertices of height at least h in T ∞ r , and at most v of them: we will progressivelyretriangulate the sphere, and the new triangulation at all time will still have at most v vertices.We call a simplex of the sphere S k bad for f if all of its vertices are mapped to vertices in T ∞ r that haveheight less than h . We will modify f by removing the bad simplices inductively starting by those of highestdimension. So let σ be a bad simplex of maximal dimension p among all bad simplices. We will modify f and the triangulation of the sphere in the star of that simplex in a way that does not add any new bad simplex.In the process, we will increase the number of vertices by at most 1, and not at all if σ was a vertex. Thisimplies that, after doing this for all bad simplices, we will have increased the number of vertices of thetriangulation of the sphere by at most v + · · · + v k . As the sphere originally had v vertices, at the end of theprocess its new triangulation will have at most v = v + v + · · · + v k vertices (as we never introduced any newbad simplices, so only bad simplices of the original triangulation affect that number). There are two cases: Case p = k. If the bad simplex σ is of the dimension k of the sphere S k , then its image f ( σ ) is a non-generating independent subset of C + A [ r ] . Because it is non-generating, we can choose y ∈ C + A [ r ] that hasheight at least h and that is not a descendant of any vertex of f ( σ ) and still, together with f ( σ ) , gives anon-generating independent subset. As the union f ( σ ) ∪ { y } is again a simplex of T ∞ r , we can add a vertex a in the center of σ , replacing σ by ∂ σ ∗ a and replace f by the map ( f | ∂σ ) ∗ ( a (cid:55)→ y ) on ∂ σ ∗ a . This map ishomotopic to f through the simplex f ( σ ) ∪{ y } . We have added a single vertex to the triangulation. Because y has height h , we have not added any new bad simplex, and we have removed one bad simplex, namely σ . Case p < k. If the bad simplex σ is a p –simplex for some p < k , by maximality of its dimension, the linkof σ is mapped to vertices of height at least h in the complement of the free A ∗ –set C + A ( f ( σ )) generatedby f ( σ ) . The simplex σ has p + + A [ r ] of cardinalityat most p + (cid:54) k . By our choice of h , there are at least N = v + y , . . . , y N of height h such thateach f ( σ ) ∪ { y i } still forms an independent set. Consider the A ∗ –setC + A ( y ) (cid:116) · · · (cid:116) C + A ( y N ) ⊂ C + A [ r ] . As there are fewer vertices in the link than in the whole sphere, and the whole sphere has at most v vertices, bythe pigeonhole principle, the vertices of Link ( σ ) are mapped to at most v of the subsets C + A ( y i ) . As N = v + y i and y j of height h such that no vertex of the link is mapped toany of their descendants C + A ( y i ) (cid:116) C + A ( y j ) . As these vertices are to start with independent of f ( σ ) , it followsthat for any simplex τ of Link ( σ ) , the union f ( τ ) ∪ f ( σ ) ∪ { y i } is an independent set. Moreover this setis also non-generating because y j is still independent of it. Hence each f ( τ ) ∪ f ( σ ) ∪ { y i } forms a simplexof T ∞ r . We can then replace f inside the starStar ( σ ) = Link ( σ ) ∗ σ (cid:39) S k − p − ∗ D p by the map ( f | Link ( σ ) ) ∗ ( a (cid:55)→ y i ) ∗ ( f | ∂σ ) onLink ( σ ) ∗ a ∗ ∂ σ (cid:39) S k − p − ∗ D ∗ S p − . which agrees with f on the boundary Link ( σ ) ∗ ∂ σ of the star, and is homotopic to f through themap ( f | Link ( σ ) ) ∗ ( a (cid:55)→ y i ) ∗ ( f | σ ) defined onLink ( σ ) ∗ a ∗ σ (cid:39) S k − p − ∗ D ∗ D p . Now Link ( σ ) ∗ a ∗ ∂ ( σ ) has exactly one extra vertex compared to the star of σ , unless σ was just a vertex,in which case its boundary is empty, and it has the same number of vertices. As y i has height h , we have not21dded any new bad simplices. Hence we have reduced the number of bad simplices by one because σ wasremoved.By induction, we can now assume that there are no bad simplices for f with respect to a triangulation with atmost v vertices. With this assumption, we can cone off f just as we coned off the links in the above argument:We have more than N = v + h in T ∞ r , and at most v vertices in the sphere. These verticesare mapped to vertices of height at least h , that is to descendants of the vertices of height h . By the pigeonholeprinciple, we know that there are at least two vertices y i and y j , of height h such that no vertex of the sphere ismapped to any of their descendants. Hence we can cone off the sphere using { y i } . Indeed, this { y i } is disjointand independent from the set f ( σ ) for every simplex σ of the sphere, ensuring that the union f ( σ ) ∪ { y i } still forms an independent set, and any such set is non-generating because y j is independent of it. Henceevery f ( σ ) ∪ { y i } defines a simplex of T ∞ r and we can cone off the sphere by adding a single vertex mappedto y i . Proof of Proposition 3.2.
Consider a map f : S k → U ∞ r . Again we can assume that it is simplicial for sometriangulation of the sphere S k . We will show that there is a homotopy from f to a map that lands inside T ∞ r .This will prove the result by Proposition 3.1.We will, just like in the previous proof, modify f by a homotopy on the stars of the bad simplices in S k ,namely those whose vertices are all mapped to U ∞ r \ T ∞ r . We will show how to reduce their number one byone, so that the result follows by induction.Let σ be a bad simplex of maximal dimension, say p , in the sphere S k . By maximality, the link of σ ismapped to T ∞ r ∩ Link ( f ( σ )) ⊂ U ∞ r . We now argue that the intersection T ∞ r ∩ Link ( f ( σ )) is isomorphic to T ∞ q for some q (cid:62) f ( σ ) is a collection of disjoint subsets of C + A [ r ] that together form a non-generating independentsubset P ∈ I [ r ] . By Lemma 1.7, there exists a least expansion E of [ r ] containing P , and because P is non-generating, the set E \ P = Q has cardinality q (cid:62)
1. We claim that T ∞ r ∩ Link ( f ( σ )) is isomorphic to T ∞ q . Towrite a map, we first pick a bijection λ : [ q ] → Q . Then λ , together with the inclusion Q ⊂ C + A [ r ] induces amap ˆ λ : C + A [ q ] −→ C + A [ r ] which in turn induces a map T ( ˆ λ ) : T ∞ q −→ T ∞ r defined on vertices by T ( ˆ λ )( y ) = ˆ λ ( y ) . Indeed, the map ˆ λ respects independency of subsets, which showsthat T ( ˆ λ ) is simplicial. Now note that any simplex of T ∞ r in the image of T ( ˆ λ ) lies in the link of f ( σ ) because it necessarily is an independent strict subset of some expansion of Q , and hence is independentof f ( σ ) , and, together with f ( σ ) , non-generating. On the other hand, any simplex τ of T ∞ r ∩ Link ( f ( σ )) isdefined by an independent set R such that R ∪ P is (non-generating) independent. Let E (cid:48) be some expansionof [ r ] containing R ∪ P . By minimality of E = Q ∪ P , we must have that E (cid:48) is an expansion of E and hencethat R lies in some expansion of Q . It follows that τ was actually in the image of T ( ˆ λ ) . Hence T ( ˆ λ ) definesan isomorphism T ∞ q ∼ = T ∞ r ∩ Link ( f ( σ )) .Let us consider the restriction of the map f to the star of the simplex σ . Since we are working inside a k -sphere, we have Star ( σ ) ∼ = Link ( σ ) ∗ σ ∼ = S k − p − ∗ σ . The simplicial complex T ∞ q is contractible by Proposition 3.1. As the link of f ( σ ) is mapped into theintersection T ∞ r ∩ Link ( f ( σ )) , which is isomorphic to T ∞ q , we can extend the map from the link to amap g : D k − p → T ∞ r ∩ Link ( f ( σ )) . Using such a map, the restriction of the map f to the star of the sim-plex σ can be extended to get a map ˆ f = g ∗ f : D k − p ∗ σ → U ∞ r on the ball D k − p ∗ σ . This extension definesa homotopy on the star of σ in S k , relative to the boundary link, from the restriction of f to a map with strictlyfewer bad simplices: The new map g ∗ f : D k − p ∗ ∂ σ → U ∞ r has fewer old bad simplices, because σ has been22emoved, and there are no new bad simplices, because new simplices are joins of faces of ∂ σ and simplicesin the disc D k − p , but the latter is mapped to good vertices. The result follows by induction. As before, we fix a finite set A = { a , . . . , a n } of cardinality n (cid:62)
2. Recall from Section 1.3 the groupoids
Set × of finite sets [ r ] = { , . . . , r } , with r (cid:62)
0, and their bijections, and the groupoid
Cantor × A of free Cantoralgebras C A [ r ] of type A , with r (cid:62)
0, and morphisms their isomorphisms. The groupoids
Set × and Cantor × A are both permutative categories, and taking the free Cantor algebra on a set defines a symmetric monoidalfunctor C A : Set × → Cantor × A . The unit of the permutative category
Set × is the empty set [ ] = Ø and that of
Cantor × A is the empty Cantoralgebra C A [ ] . Both have no non-trivial automorphisms and behave like disjoint basepoints: Set × = Set × (cid:116) { [ ] } and Cantor × A = Cantor × A (cid:116) { C A [ ] } for Set × the full subgroupoid of Set × on the objects [ r ] with r >
0, and
Cantor × A the full subgroupoidof Cantor × A on the objects C A [ r ] with r >
0. Following Thomason [Tho82], we will in this section discardthese units and work with non-unital permutative categories like
Set × and Cantor × A , inserting units backagain as disjoint basepoints only at the very end, just before taking spectra, as we explain now.There are algebraic K-theory “machines” that produce for every unital symmetric monoidal category C aspectrum K ( C ) whose underlying infinite loop space Ω ∞ K ( C ) is a group completion of the classifyingspace | C | of C . (See [Tho82, App C] for the particular machine that we will be using here.) If C is anon-unital symmetric monoidal category, we will, following Thomason [Tho82, App C], define K ( C ) : = K ( C + ) for C + the category build from C by adding a disjoint unit. In particular, when C = Set × , we havethat C + ∼ = Set × and similarly for Cantor × A and Cantor × A . The Barratt–Priddy–Quillen theorem says thatthe spectrum K ( Set × ) = K ( Set × ) (cid:39) S is the sphere spectrum, so that there is a group completion | Set × | → Ω ∞ S (see [BP72]). Our goal here is tocompute K ( Cantor × A ) . We will do this using a permutative category Tho A , build from Set × and the set A ,using a homotopy colimit construction of Thomason. Our main result for the section is the following: Theorem 4.1.
There is a strict symmetric monoidal functor
Tho A −→ Cantor × A of non-unital permutativecategories that induces a homotopy equivalence | Tho A | (cid:39) −→ | Cantor × A | between the classifying spaces. The category
Tho A will be build as a non-unital permutative category, and can be made unital by addinga disjoint basepoint. By a strict symmetric monoidal functor between non-unital permutative categories C and D , we will mean a functor F : C → D such that F ( X ⊕ Y ) = F ( X ) ⊕ F ( Y ) for every objects X , Y of C and likewise for morphisms. Adding disjoint units will give an associated strict symmetric monoidal func-tor F + : C + → D + in the usual sense.Using [Tho82, Lem. 2.3], we get the following corollary: Corollary 4.2.
There is an equivalence K ( Tho A ) (cid:39) −→ K ( Cantor × A ) = K ( Cantor × A ) of spectra. The idea behind the permutative category
Tho A is as follows. There is a symmetric monoidal endofunctor Σ A : Set × −→ Set × A ” (see Section 4.3 for a precise definition). Now the functor C A : Set × → Cantor × A hasthe property that it is insensitive to pre-composition with Σ A : as seen in Example 1.12, there are isomor-phisms C A ( X × A ) ∼ = C A ( X ) , and these isomorphisms are essentially the defining property of Cantor algebrasof type A . There is in fact a natural isomorphism C A ◦ Σ A ∼ = C A of functors. This suggests that the functor C A extends over a “mapping torus of Σ A ” and the category Tho A , which we define below, will be precisely sucha device.Given a diagram C : λ (cid:55)→ C λ of non-unital permutative categories, indexed on a small category Λ , Thoma-son [Tho82, Const. 3.22] defines a new non-unital permutative category, denoted here by hocolim Λ C λ , withthe property that K ( hocolim Λ C λ ) (cid:39) hocolim Λ K ( C λ ) . (4.1)(See [Tho82, Thm. 4.1].) In order to construct Tho A , we take the diagram that is indexed by the monoid N of natural numbers, thought of as a category with one object. Given a pair ( C , F ) consisting of a non-unitalpermutative category C together with a symmetric monoidal endo-functor F , we get a diagram on N thatassociates to the unique object the category C and to the morphism k ∈ N the functor F k . We define Tho A = hocolim N ( Set × , Σ A ) (4.2)for ( Set × , Σ A ) the corresponding diagram on N . By the universal property of Thomason’s construction(see [Tho82, Prop 3.21]), symmetric monoidal functors from Tho A to Cantor × A can be defined by the fol-lowing data: a symmetric monoidal functor F : Set × → Cantor × A and a symmetric monoidal natural transfor-mation F ◦ Σ A → F . Taking F = C A will thus give rise to a symmetric monoidal functor Tho A → Cantor × A .We will construct this functor explicitly, and show that it induces an equivalence on the level of classifyingspaces by showing that it fits inside a diagram of categories and functors Tho A (cid:47) (cid:47) Cantor × A Lev A ∼ (cid:100) (cid:100) ∼ (cid:47) (cid:47) Exp A ∼ (cid:57) (cid:57) where all the other arrows induce homotopy equivalences on classifying spaces, and such that the diagramcommutes up to homotopy. The idea behind the intermediate categories Exp A and Lev A is as follows. Con-sidering the functor Tho A → Cantor × A , one sees that morphisms of Cantor × A differ from morphisms of Tho A in two ways. Firstly, the morphisms of Tho A that map to the canonical isomorphism C A [ ] → C A [ n ] are notinvertible in Tho A . Secondly, only certain simple types of expansions occur directly as morphisms of Tho A ,those that we will call “level expansions”. We will take care of these two issues one at a time, writing thehomotopy equivalence in three steps, studying the homotopy fibers each time.The section is organized as follows: In Section 4.1, we define the category Exp A and show that it is equivalentto Cantor × A . In Section 4.2, we define the category Lev A and show that it is equivalent to Exp A . In Section 4.3we defined the category Tho A and show that it is equivalent to Lev A . Finally, in Section 4.4 we define thefunctor Tho A → Cantor × A and show that the resulting diagram commutes. We will define the non-unital permutative category
Exp A as a subcategory of Cantor × A , and show that passingfrom the groupoid Cantor × A to its subcategory Exp A preserves the homotopy type of the classifying space.Here Cantor × A denotes, as above, the category of free Cantor algebras C A [ r ] with r > A [ r ] to C A [ s ] in Cantor × A can be described by a triple ( E , F , λ ) ,with the set E an expansion of [ r ] , the set F an expansion of [ s ] , and λ : E → F a bijection. Recall also thatthere is a minimal such representative, where the minimality is defined using the poset structure of the set E [ r ] of expansions of [ r ] . Note that if either E = [ r ] or if F = [ s ] , the representative is necessarily the minimal one.This allows for the following definition: Definition 4.3.
The expansion category
Exp A is the subcategory of Cantor × A , with the same objects, and withmorphisms C A [ r ] → C A [ s ] the morphisms of Cantor × A that can be represented by a triple ( E , F , λ ) with F = [ s ] .We will write ( E , λ ) for such a morphism. The symmetric monoidal structure of Cantor × A restricts to oneon Exp A , making it a non-unital permutative category.Note that Exp A is indeed a subcategory: if f : C A [ r ] → C A [ s ] and g : C A [ s ] → C A [ t ] in Cantor × A are repre-sented by ( E , [ s ] , λ ) and ( F , [ t ] , µ ) respectively, then their composition g ◦ f is represented by ( ˆ E , [ t ] , µ ◦ ˆ λ ) for ˆ E = ˆ λ − ( F ) the expansion of E corresponding to F under λ . Remark 4.4.
Recall from Section 1.1 that an expansion E of [ r ] can be thought of as a planar n –ary foreston r roots corresponding to the elements of [ r ] , whose leaves identify with the elements of E . A bijec-tion λ : E → [ s ] can then be interpreted as a labeling of the leaves of this forest. Identifying the objectsof Exp A with the natural numbers and the the morphisms with labeled planar forests in this way, we thus seethat Exp A can be thought of as a certain “cobordism category” of sets where the cobordisms are planar n –aryforests.Our main result in this section is the following: Proposition 4.5.
The inclusion I : Exp A → Cantor × A is a symmetric monoidal functor of non-unital permu-tative categories that induces an equivalence | Exp A | (cid:39) | Cantor × A | on classifying spaces. Remark 4.6.
The above result can be seen as a special case of [Thu17, Prop 2.13] by interpreting theHigman–Thompson groups as operad groups and applying the results in Section 3 of that paper. As thepreparatory work for our proof will be useful in the following sections, we decided to keep the proof as israther than explaining this alternative approach.
Notation 4.7 (Induced maps ˆ λ ) . Given a bijection λ : X ∼ = → Y between finite sets, we get an induced isomor-phism C A ( λ ) : C A ( X ) ∼ = −→ C A ( X ) of Cantor algebras, which in turn induces an isomorphism of posets E ( λ ) : E ( X ) ∼ = −→ E ( Y ) as C A ( λ ) takes expansions to expansions. And if E is an expansion of X and F = E ( λ )( E ) is the expansionof Y corresponding to E under λ , then restricting C A ( λ ) to E induces a bijection E → F . In what follows, toease notations, we will write ˆ λ for these three types of maps induced by λ : the map of Cantor algebra C A ( λ ) ,its restriction to subsets of C A ( X ) , and the poset map E ( λ ) .Recall that A = { a , . . . , a n } is an ordered set and that any expansion E of [ r ] is a subset E ⊂ (cid:71) n (cid:62) [ r ] × A n = C + A [ r ] . Hence E is also canonically ordered, using the lexicographic ordering on the words (cid:70) n (cid:62) [ r ] × A n . (In termsof forests, this corresponds to the naturally induced planar ordering.) In the proof of Proposition 4.5, as well25s later in Section 4, we will use this lexicographic ordering to identify any expansion E with the set [ e ] of thesame cardinality as E . The following result shows that this chosen identification is compatible with takingfurther expansions. Lemma 4.8.
Let A = { a , . . . , a n } . For any expansion E of [ r ] of cardinality e, denote by λ E : E ∼ = −→ [ e ] thebijection defined by the lexicographic ordering of E. Let F be an expansion of [ e ] , with E (cid:48) = ˆ λ − E ( F ) thecorresponding expansion of E under λ E . Then λ F ◦ ˆ λ E = λ E (cid:48) , i.e. the upper triangle in the following diagramcommutes: E (cid:48) (cid:54) ∼ = ˆ λ E (cid:47) (cid:47) ∼ = λ E (cid:48) (cid:39) (cid:39) ˆ λ E ( E (cid:48) ) = F (cid:54) ∼ = λ F (cid:47) (cid:47) [ e (cid:48) ] E (cid:54) ∼ = λ E (cid:47) (cid:47) [ e ][ r ] (4.3) Proof.
The compatibility property follows from the fact that the map λ E also induces a bijection of theset C + A ( E ) ⊂ C + A [ r ] with C + A [ e ] , and that this map respects the lexicographic order: this is true by definitionof λ E on the generating set E , and by definition of the lexicographic order on the remaining elements. So thelexicographic order of E (cid:48) ⊂ C + A ( E ) ⊂ C + A [ r ] agrees with that of ˆ λ E ( E (cid:48) ) ⊂ C + A [ e ] . Proof of Proposition 4.5.
We have already seen that the categories are (non-unital) permutative and that theinclusion respects this structure. The result will follow Quillen’s Theorem A [Qui73, §
1] if we show that,for any r >
0, the fiber of I under C A [ r ] is equivalent to the poset E [ r ] . Indeed, the poset E [ r ] has [ r ] as leastelement, and hence is contractible.The fiber C A [ r ] \ I of the functor I under an object C A [ r ] of Cantor × A has objects the pairs ( C A [ s ] , f ) with f : C A [ r ] → C A [ s ] a morphism in Cantor × A , and morphisms ( C A [ s ] , f ) → ( C A [ s (cid:48) ] , f (cid:48) ) given by mor-phisms ( F , λ ) : C A [ s ] → C A [ s (cid:48) ] in Exp A such that the diagramC A [ s ] I ( F , λ ) (cid:47) (cid:47) C A [ s (cid:48) ] C A [ r ] f (cid:98) (cid:98) f (cid:48) (cid:59) (cid:59) commutes in Cantor × A .We define a functor Λ : E [ r ] → C A [ r ] \ I on objects by Λ ( E ) = ( C A [ e ] , f E ) , where e = | E | is the cardinalityof E , and the map f E : C A [ r ] → C A [ e ] is represented by the triple ( E , [ e ] , λ E ) with λ E as in Lemma 4.8.If E (cid:48) (cid:62) E is an expansion of E , we have a diagram E (cid:48) (cid:54) ∼ = ˆ λ E (cid:47) (cid:47) ∼ = λ E (cid:48) (cid:38) (cid:38) F (cid:54) ∼ = λ F (cid:47) (cid:47) [ e (cid:48) ] E ∼ = λ E (cid:47) (cid:47) [ e ] (4.4)26here triangle commutes by Lemma 4.8. We define Λ on the inequality E (cid:54) E (cid:48) in E [ r ] to be the morphismof Exp A defined by the pair ( F , λ F ) : C A [ e ] −→ C A [ e (cid:48) ] for F = ˆ λ E ( E (cid:48) ) as in (4.4). The fact that the diagramC A [ e ] I ( F , λ F ) (cid:47) (cid:47) C A [ e (cid:48) ] C A [ r ] ( E , [ e ] , λ E ) (cid:99) (cid:99) ( E (cid:48) , [ e (cid:48) ] , λ E (cid:48) ) (cid:59) (cid:59) commutes is the commutativity of the triangle in (4.4).We now define a functor Π : C A [ r ] \ I → E [ r ] in the other direction: Given an object ( C A [ s ] , f ) in the fiber,with f : C A [ r ] → C A [ s ] an isomorphism of Cantor algebras, let ( E , G , µ ) be the minimal representativeof f . We set Π ( C A [ s ] , f ) = E . On morphisms, we have no choice as the target category is a poset,but we have to check that, given a morphism ( F , λ ) : ( C A [ s ] , f ) → ( C A [ s (cid:48) ] , f (cid:48) ) in the fiber, the expan-sion Π ( C A [ s (cid:48) ] , f (cid:48) ) of [ r ] is an expansion of Π ( C A [ s ] , f ) . This follows from the fact that f (cid:48) = I ( F , λ ) ◦ f ,implying that also f − = f (cid:48)− ◦ I ( F , λ ) and the fact that I ( F , λ ) comes from Exp A : if f (cid:48) has minimal presen-tations ( E (cid:48) , G (cid:48) , µ ) , then f (cid:48)− can be represented by ( G (cid:48) , E (cid:48) , µ (cid:48)− ) , and the composition f (cid:48)− ◦ I ( F , λ ) can becomputed using the diagram: ˆ F (cid:54) ˆ λ (cid:47) (cid:47) G (cid:48) (cid:54) µ (cid:48)− (cid:47) (cid:47) E (cid:48) (cid:54) F (cid:54) λ (cid:47) (cid:47) [ s (cid:48) ] [ r ][ s ] with the triple ( ˆ F , E (cid:48) , µ (cid:48)− ◦ ˆ λ ) representing the composition. Given that this composition is equal to f − ,which has minimal presentation ( G , E , µ − ) , we must have that E (cid:48) (cid:62) E , as required.The composition ΠΛ is the identity: it is enough to check this on objects as E [ r ] is a poset, and Λ takes anexpansion E to the object ( C A [ e ] , ( E , [ e ] , λ E )) , itself taken back to E by Π . On the other hand, the composi-tion ΛΠ takes an object ( C A [ s ] , f ) with f minimally represented by ( E , G , µ ) to ( C A [ e ] , f E ) for e = | E | and f E represented par ( E , [ e ] , λ E ) . Now note that the diagram in Cantor × A C A [ s ] ( G , [ e ] , λ E ◦ µ − ) (cid:47) (cid:47) C A [ e ] C A [ r ] ( E , G , µ ) (cid:99) (cid:99) ( E , [ e ] , λ E ) (cid:59) (cid:59) commutes, where we have written the morphisms in terms of representatives. Thus ( G , λ E ◦ µ − ) defines amorphism in C A [ r ] \ I from ( C A [ s ] , f ) to ΛΠ ( C A [ s ] , f ) . We check now that these morphisms assemble to anatural transformation between the identity functor on C A [ r ] \ I and the composition ΛΠ . Indeed, consider amorphism ( F , λ ) : ( C A [ s ] , f ) → ( C A [ s (cid:48) ] , f (cid:48) ) where f has minimal representative ( E , G , µ ) and f (cid:48) has minimalrepresentative ( E (cid:48) , G (cid:48) , µ (cid:48) ) . We need to check that the following square commutes in C A [ r ] \ I : ( C A [ s ] , f ) (cid:104) (cid:104) f ( F , λ ) (cid:15) (cid:15) ( G , λ E ◦ µ − ) (cid:47) (cid:47) ( C A [ e ] , f E ) = ΛΠ ( C A [ s ] , f ) ΛΠ ( F , λ ) (cid:15) (cid:15) (cid:52) (cid:52) f E C A [ r ]( C A [ s (cid:48) ] , f (cid:48) ) (cid:118) (cid:118) f (cid:48) ( G (cid:48) , λ E (cid:48) ◦ µ (cid:48)− ) (cid:47) (cid:47) ( C A [ e (cid:48) ] , f E (cid:48) ) = ΛΠ ( C A [ s ] , f ) (cid:42) (cid:42) f E (cid:48) Exp A if and only if its image in Cantor × A commutes, as Exp A is a subcategoryof Cantor × A . But in Cantor × A , each triangle in the diagram commutes by the fact that the maps in the squareare maps in the fiber C A [ r ] \ I . As all the maps are isomorphisms of Cantor algebras, it follows that the outsidesquare commutes, as needed. The category
Exp A has morphisms given by pairs consisting of an expansion and a bijection. We will nowdecompose the expansions into simpler types of expansions, which we call level expansions , that can bedescribed in terms of subsets. We will construct a category Lev A , which we will show is equivalent to Exp A ,where the morphisms will now be given by level expansions and bijections. Definition 4.9.
An expansion E of X is called a level expansion if there exists a subset P of X suchthat E = ( P × A ) ∪ Q as a subset of C + A ( X ) , where Q = X \ P is the complement of P in X .Note that level expansions do not “compose” in the sense that if E is a level expansion of X and F a levelexpansion of E , then F , while still an expansion of X , need not be a level expansion of X . For that reason,we do not get a category of level expansions analogous to Exp A right away. To define the category Lev A , wewill start with a semi-simplicial set of level expansions, and then pass to its poset of simplices. Definition 4.10.
Given a finite set X , we define L ( X ) to be the semi-simplicial set with 0–simplices theexpansions of X and with p –simplices the sequences E < E < E < · · · < E p in the poset E ( X ) satisfying that each E i is a level expansion of the smallest set E . The i –th face mapforgets E i : d i ( E < · · · < E p ) = E < · · · < (cid:98) E i < · · · < E p . Note that also the differential d makes sense. In fact, if E i = ( Q i × A ) ∪ E \ Q i and E j = ( Q j × A ) ∪ E \ Q j are both level expansions of E with E j > E i , then one necessarily has that Q i ⊂ Q j and E j is the levelexpansion of E i as well. This leads to the following alternative description of L ( X ) : Definition 4.11.
Given a finite set X , we define L (cid:48) ( X ) to be the semi-simplicial set with 0–simplices theexpansions of X and with p –simplices the tuples ( E , ( P , . . . , P p )) with E an expansion of X and ( P , . . . , P p ) a collection of disjoint non-empty subsets of E . The face maps aredefined by d i ( E , ( P , . . . , P p )) = (cid:0) ( P × A ) ∪ ( E \ P ) , ( P , . . . , P p ) (cid:1) i = ( E , ( P , . . . , P i ∪ P i + , . . . , P p )) < i < p ( E , ( P , . . . , P p − )) i = p . Lemma 4.12.
The map Φ : L (cid:48) ( X ) → L ( X ) that takes ( E , ( P , . . . , P p )) to E < E < · · · < E p with the sets E i defined by E i = ( P [ , i ] × A ) ∪ ( E \ P [ , i ] ) for P [ , i ] = P ∪ · · · ∪ P i , is an isomorphism of semi-simplicial sets.Proof. The map Φ is well-defined since each E i is by definition a level expansion of E = E , and thefact that E i < E i + follows from the fact that P [ , i ] ⊂ P [ , i + ] ; in fact, E i + is the level expansion of E i along P i + = P [ , i + ] \ P [ , i ] . The face maps in L (cid:48) ( X ) were defined to make this map simplicial: d corre-sponds to replacing E by E , forgetting E = E , and when 0 < i < p , the map d i corresponds to forgetting E i ,and d p forgets E p . Injectivity is immediate. For surjectivity, note that if E < · · · < E p is a simplex of L ( X ) ,then we must have that E i = ( Q i × A ) ∪ ( E \ Q i ) for each i , as E i is a level expansion of E , and the factthat E i < E i + imposes that Q i ⊂ Q i + as we have seen above. Setting P i = Q i \ Q i − for each i gives asimplex ( E , ( P , . . . , P p )) of L (cid:48) ( X ) mapping to E < · · · < E p , showing that Φ is also surjective.28his lemma is in some way crucial, as it is the point where we switch from talking about expansions andmorphisms of Cantor algebras, to solely talking about sets and subsets. Both ways of thinking of the simplicesof L ( X ) will be useful throughout the rest of the section.Note that any expansion E of a set X can be factored as a “composition” of level expansions, in the sense thatone can always find expansions E , . . . , E k of X such that X = E < E < · · · < E k < E k + = E in the poset E ( X ) of expansions of X , in such a way that each E i is a level expansion of E i − . (There is evena canonical such factorization using the height filtration of C + A ( X ) , but we will not use it.) Given that E ( X ) is contractible, we get that L ( X ) is connected for every finite set X . The following result shows that therealization of L ( X ) ∼ = L (cid:48) ( X ) , which is a subspace of the nerve of E ( X ) , is in fact, like E ( X ) , contractible. Remark 4.13.
Semi-simplicial sets have a realization, defined just like the “thick” realization of sim-plicial sets (see eg. [ER-W17]). Given a poset P , its nerve is a simplicial set, which can be consid-ered as a semi-simplicial set by forgetting the degeneracies. This semi-simplicial set has q –simplices thesequences p (cid:54) . . . (cid:54) p q in the poset P , with the face map d i forgetting the i –th element. Alternatively, wecan associate a smaller semi-simplicial set to P , that has q –simplices the sequences p < · · · < p q in theposet P . Now the classical nerve of P can be recovered from this smaller nerve by freely adding all thedegeneracies. The classical nerve as simplicial set or as semi-simplicial set (forgetting the degeneracies), andthe smaller nerve using only strict inequalities, all have homotopy equivalent realizations (see eg. [ER-W17,Lem. 1.7, 1.8]). In particular, if a poset has a least or greatest element, its realization is contractible, usingwhichever of these three possible realizations. Proposition 4.14.
For all finite sets X, the semi-simplicial set L ( X ) ∼ = L (cid:48) ( X ) is contractible.Proof. If Y is an expansion of X , we define L ( X , Y ) to be the full subcomplex of L ( X ) whose verticesare expansions E of X admitting Y as an expansion. As any finite collection of expansions of X admits acommon expansion (by repeated use of Lemma 1.8), compactness of the spheres implies that every homotopyclass can be represented by a map into some L ( X , Y ) ⊆ L ( X ) . It is therefore sufficient to show that thecomplexes L ( X , Y ) are contractible.For an expansion E of X , we define its rank to be rk ( E ) = ( | E | − | X | ) / ( n − ) . This is the number of simpleexpansions (expanding a single element x once) needed to obtain E from X , and thus also the rank of E in theposet E ( X ) . Suppose that the expansion Y has rank r . Let F i L ( X , Y ) denote the full subcomplex of L ( X , Y ) on the vertices of rank at least i . This defines a descending filtration of L ( X , Y ) .L ( X , Y ) = F L ( X , Y ) ⊇ F L ( X , Y ) ⊇ F L ( X , Y ) ⊇ · · · ⊇ F r L ( X , Y ) = { Y } For each i < r , the complex F i L ( X , Y ) is obtained from F i + L ( X , Y ) by attaching cones on the vertices E ofrank i along their links, because no two such vertices are part of the same simplex. Now E < · · · < E p isin Link ( E ) ∩ F i + ( X , Y ) if and only if E < E < · · · < E p (cid:54) Y , that is if and only if E < · · · < E p is a sequence of non-trivial level expansions of E admitting Y as anexpansion. Let E L ev ( E , Y ) denote the set of non-trivial level expansions of E admitting Y as an expansion.Consider E L ev ( E , Y ) as a subposet of E ( E ) . We claim that Link ( E ) ∩ F i + ( X , Y ) is isomorphic to its associated(small) semi-simplicial set (in the sense of in Remark 4.13). Indeed, if E , . . . , E p ∈ E L ev ( E , Y ) are levelexpansions of E such that E < · · · < E p is a simplex in the nerve of the poset E , and thus also by definition inthe nerve of E L ev ( E , Y ) , then E < E < · · · < E p (cid:54) Y and we exactly have a simplex in Link ( E ) ∩ F i + ( X , Y ) .Our last step in the proof is to show that E L ev ( E , Y ) has a greatest element, and is hence contractible.Under the isomorphism of Lemma 4.12, E L ev ( E , Y ) is isomorphic to the poset P ( E , Y ) of Ø (cid:54) = P ⊂ E such that E ( P ) = ( P × A ) ∪ ( E \ P ) admits Y as an expansion, where the poset structure is now inclusion.29et ˆ P = (cid:83) P ∈ P ( E , Y ) P be the union of all such P ’s. The check that ˆ P ∈ P ( E , Y ) , we need to check that theexpansion E ( ˆ P ) = ( ˆ P × A ) ∪ ( E \ ˆ P ) still admits Y as an expansion. This follows from the fact that we cancheck this component-wise: writing Y = ∪ e ∈ E Y e for Y e = Y ∩ C + A ( e ) ⊂ C + A [ E ] , the condition that Y (cid:62) E ( ˆ P ) is equivalent to Y e ∈ C + A ( e × A ) ⊂ C + A ( e ) for every e ∈ ˆ P , which holds by the definition of ˆ P as every such e lies in an element P of P ( E , Y ) . Hence ˆ P is a greatest element for P ( E , Y ) , showing that P ( E , Y ) , and thusalso E L ev ( E , Y ) , is contractible, as needed. The result follows by induction.Let L ( X ) denote the poset of simplices of L (cid:48) ( X ) , so the elements of L ( X ) are the simplices of L (cid:48) ( X ) , and themorphisms are the inclusions among them. Explicitly, the elements of L ( X ) are the tuples ( E , ( P , . . . , P p )) with E an expansion of X and ( P , . . . , P p ) , for some p (cid:62)
0, a sequence of disjoint non-empty subsets of E . Todescribe the poset structure, recall from Definition 4.11 that the face map d in L (cid:48) ( X ) takes a level expansionalong P , then for 0 < i < p , the face map d i takes the union of the neighbouring subsets P i and P i + , andfinally the last face map d p forgets the last subset P p . Writing P [ i , j ] : = P i ∪ · · · ∪ P j for i (cid:54) j , we thus have that ( F , ( Q , . . . , Q q )) (cid:54) ( E , ( P , . . . , P p )) if there exists 0 (cid:54) p , p (cid:54) p and κ , . . . , κ q (cid:62) p + ∑ qj = κ j + p = p such that F = ( P [ , p ] × A ) ∪ ( E \ P [ , p ] ) is the level expansion of E along the first p subsets and each Q j is a union Q j = P [ p + κ + ··· + κ j − + , p + κ + ··· + κ j ] . In particular Q ∪ · · · ∪ Q q = P [ p + , p − p ] . (The numbers p and p count the number of 0–th and last facemaps that have been applied.)Note that the order relation in the poset L [ r ] is opposite to the order relation in the poset E [ r ] in the sense thatif ([ r ] , ( P , . . . , P p )) (cid:62) ( F , ( Q , . . . , Q q ) in L [ r ] , then F (cid:62) [ r ] in E [ r ] . This is why the order relation of L [ r ] willoccur in the reversed direction in the definition of Lev A below.We have already seen that a bijection λ : X → Y induces an isomorphism ˆ λ : E ( X ) → E ( Y ) of posets (seeNotation 4.7). This in turn induces a map ˆ λ : L ( X ) → L ( Y ) of semi-simplicial sets, and hence also amap ˆ λ : L ( X ) → L ( Y ) of posets of simplices, which we will denote by ˆ λ again. Explicitly, this last maptakes ( E , ( P , . . . , P p )) to ( ˆ λ E , ( ˆ λ P , . . . , ˆ λ P p )) . Note also that for any expansion E of X , the poset L ( E ) identifies with the subposet of L ( X ) with objects the tuples ( F , ( P , . . . , P p )) such that F is an expansion of E .We are now ready to construct the category Lev A from the posets L ( X ) in a way similar to the one in whichwe constructed Exp A from the posets E ( X ) . As in the case of Cantor × A and Exp A , we will restrict to objectsbuild from the non-zero natural numbers. Definition 4.15.
The category
Lev A has as objects the sequences ([ r ] , ( P , . . . , P p )) for r > ( P , . . . , P p ) a p –tuple of p (cid:62) [ r ] . (In particular, ([ r ] , ( P , . . . , P p )) ∈ L [ r ] .) The morphisms ϕ : ([ r ] , ( P , . . . , P p )) −→ ([ s ] , ( Q , . . . , Q q )) are given by tuples ϕ = ( F , ( Q (cid:48) , . . . , Q (cid:48) q ) , λ ) where the element ( F , ( Q (cid:48) , . . . , Q (cid:48) q )) ∈ L [ r ] is such that the rela-tion ([ r ] , ( P , . . . , P p )) (cid:62) ( F , ( Q (cid:48) , . . . , Q (cid:48) q )) holds in L ([ r ]) and λ : F → [ s ] is a bijection such that λ ( Q (cid:48) i ) = Q i for each i . (In particular, F is a level expansion of [ r ] along some P i ’s and each Q (cid:48) i is a union of P i ’s.)The composition of ϕ with a morphism ψ = ( G , ( R (cid:48) , . . . , R (cid:48) u ) , µ ) : ([ s ] , ( Q , . . . , Q q )) → ([ t ] , ( R , . . . , R u )) isdefined by ( G , ( R (cid:48) , . . . , R (cid:48) u ) , µ ) ◦ ( F , ( Q (cid:48) , . . . , Q (cid:48) q ) , λ ) = ( ˆ λ − G , ( ˆ λ − R (cid:48) , . . . , ˆ λ − R (cid:48) q ) , µ ◦ ˆ λ ) λ − ( G ) ˆ λ (cid:47) (cid:47) (cid:54) G µ (cid:47) (cid:47) (cid:54) [ t ] F λ (cid:47) (cid:47) (cid:54) [ s ][ r ] and where we use that λ induces an isomorphism ˆ λ : L ( F ) → L [ s ] and that L ( F ) is naturally a subposetof L [ r ] , so that ([ s ] , ( Q , . . . , Q q )) (cid:62) ( G , ( R (cid:48) , . . . , R (cid:48) u ) in L [ s ] gives ([ r ] , ( P , . . . , P p )) (cid:62) ( F , ( Q (cid:48) , . . . , Q (cid:48) q ) (cid:62) ( ˆ λ − G , ( ˆ λ − R (cid:48) , . . . , ˆ λ − R (cid:48) q )) in L [ r ] . (In particular, ˆ λ − ( G ) is necessarily again a level expansion of [ r ] along some P i ’s and each ˆ λ − ( R (cid:48) i ) is a union of P i ’s.)We see here that it is crucial for the composition that we use the poset structure of L [ r ] and not the simplernotion of level expansion to define the morphisms in Lev A , as level expansions in general do not compose tolevel expansions.Note that part of the data of a morphism ϕ = ( F , ( Q (cid:48) , . . . , Q (cid:48) q ) , λ ) from ([ r ] , ( P . . . , P p )) to ([ s ] , ( Q , . . . , Q q )) in Lev A is an expansion F of [ r ] and a bijection λ : F ∼ = −→ [ s ] . In fact, there is a functor J : Lev A → Exp A defined on objects and morphisms as a forgetful map: J ([ r ] , ( P . . . , P p )) : = C A [ r ] and J ( F , ( Q (cid:48) , . . . , Q (cid:48) q ) , λ ) = ( F , λ ) . This is compatible with composition as can be seen directly from the definition of composition in
Lev A above. Proposition 4.16.
The functor J : Lev A → Exp A defined above induces an equivalence | Lev A | (cid:39) | Exp A | on classifying spaces.Proof. We will show that the fiber of J under C A [ r ] is homotopy equivalent to L [ r ] . As L [ r ] is the poset ofsimplices of L (cid:48) [ r ] , the result will then follow from Proposition 4.14, which says that L (cid:48) [ r ] is contractible. Description of the fibers.
The objects of the fiber C A [ r ] \ J can be written as tuples ([ s ] , ( P , . . . , P p ) , [ s ] λ ← E ) for ([ s ] , ( P , . . . , P p )) an object of the category Lev A , the set E an expansion of [ r ] and λ : E ∼ = −→ [ s ] a bijectiondefining a morphism ( E , λ ) : C A [ r ] → C A [ s ] = J ([ s ] , ( P , . . . , P p )) in Exp A . A morphism ϕ : ([ s ] , ( P , . . . , P p ) , [ s ] λ ← E ) −→ ([ t ] , ( Q . . . , Q q ) , [ t ] µ ← F ) in the fiber is given by a morphism ϕ = ( G , ( Q (cid:48) , . . . , Q (cid:48) q ) , G κ → [ t ]) : ([ s ] , ( P , . . . , P p )) −→ ([ t ] , ( Q , . . . , Q q )) in the category Lev A , with in particular G an expansion of [ s ] and κ a bijection (where we have spelled outthe source and target of κ for clarity), such that the diagram J ([ s ] , ( P , . . . , P p )) = C A [ s ] J ( ϕ )=( G , κ ) (cid:47) (cid:47) C A [ t ] = J ([ t ] , ( Q . . . , Q q )) C A [ r ] ( E , λ ) (cid:101) (cid:101) ( F , µ ) (cid:57) (cid:57) Exp A .Just as in the proof of Proposition 4.5, we define a pair of functors Λ : L [ r ] op ←→ C A [ r ] \ J : Π , where theorder of L [ r ] is reversed. Definition of Λ : L [ r ] op −→ C A [ r ] \ J. Define Λ on objects by Λ ( E , ( P , . . . , P p )) = ([ e ] , ( λ E P , . . . , λ E P p ) , [ e ] λ E ← E ) for e = | E | and λ E : E → [ e ] the map of Lemma 4.8 given by the lexicographic ordering of E . The functor Λ is given on morphisms as follows: if ( E , ( P . . . , P p )) (cid:62) ( E (cid:48) , ( Q , . . . , Q q )) in L [ r ] , we have that E (cid:48) is anexpansion of E along P ∪ · · · ∪ P i for some i and we get a diagram of expansions and isomorphisms [ e (cid:48) ] E (cid:48) λ E (cid:48) (cid:111) (cid:111) (cid:54) ˆ λ E (cid:47) (cid:47) ˆ λ E E (cid:48) (cid:54) E λ E (cid:47) (cid:47) [ e ] . We define a morphism ϕ : ([ e ] , ( λ E P , . . . , λ E P p ) , [ e ] λ E ← E ) −→ ([ e (cid:48) ] , ( λ E (cid:48) Q , . . . , λ E (cid:48) Q q ) , [ e (cid:48) ] λ E (cid:48) ← E (cid:48) ) in the fiber by setting ϕ = ( ˆ λ E E (cid:48) , ˆ λ E Q , . . . , ˆ λ E Q q , ˆ λ E E (cid:48) λ E (cid:48) ˆ λ − E −−−−→ [ e (cid:48) ]) . This makes sense because ([ e ] , ( λ E P , . . . , λ E P p )) (cid:62) ( ˆ λ E E (cid:48) , ( ˆ λ E Q , . . . , ˆ λ E Q q )) in L [ e ] , as it is the image under ˆ λ E : L ( E ) ⊂ L [ r ] → L [ e ] of ( E , ( P . . . , P p )) (cid:62) ( E (cid:48) , ( Q , . . . , Q q )) in L [ r ] . Also,we have λ E (cid:48) ˆ λ − E ( ˆ λ E Q i ) = λ E (cid:48) Q i for each i , so ϕ defines a morphism in Lev A from ([ e ] , ( λ E P , . . . , λ E P p )) to ([ e (cid:48) ] , ( λ E (cid:48) Q , . . . , λ E (cid:48) Q q )) . Finally, we see that the diagramC A [ e ] ( ˆ λ E E (cid:48) , λ E (cid:48) ˆ λ − E ) (cid:47) (cid:47) C A [ e (cid:48) ] C A [ r ] , ( E , λ E ) (cid:99) (cid:99) ( E (cid:48) , λ E (cid:48) ) (cid:59) (cid:59) commutes in Exp A , showing that ϕ indeed defines a morphism in the fiber. Functoriality follows from thefollowing computation: if ( E , ( P . . . , P p )) (cid:62) ( E (cid:48) , ( Q , . . . , Q q )) (cid:62) ( E (cid:48)(cid:48) , ( R , . . . , R u )) in L [ r ] , the compositionof the images of these morphisms under Λ is (cid:0) ˆ λ E (cid:48) E (cid:48)(cid:48) , ( ˆ λ E (cid:48) R , . . . , ˆ λ E (cid:48) R u ) , λ E (cid:48)(cid:48) ˆ λ − E (cid:48) (cid:1) ◦ (cid:0) ˆ λ E E (cid:48) , ( ˆ λ E Q , . . . , ˆ λ E Q q ) , λ E (cid:48) ˆ λ − E (cid:1) = (cid:0) ( ˆ λ E (cid:48) ˆ λ − E ) − ˆ λ E (cid:48) E (cid:48)(cid:48) , (( ˆ λ E (cid:48) ˆ λ − E ) − ˆ λ E (cid:48) R , . . . , ( ˆ λ E (cid:48) ˆ λ − E ) − ˆ λ E (cid:48) R u ) , ( λ E (cid:48)(cid:48) ˆ λ − E (cid:48) ) ◦ ( ˆ λ E (cid:48) ˆ λ − E ) (cid:1) = (cid:0) ˆ λ E E (cid:48)(cid:48) , ( ˆ λ E R , . . . , ˆ λ E R u ) , λ E (cid:48)(cid:48) ˆ λ − E (cid:1) , which is equal to image under Λ of the composition. Definition of Π : C A [ r ] \ J −→ L [ r ] op . Define the functor Π on objects by Π ([ s ] , ( P , . . . , P p ) , [ s ] λ ← E ) = ( E , ( λ − P , . . . , λ − P p )) which makes sense as E is an expansion of [ r ] and λ a bijection. As the target of Π is a poset, we are left tocheck that the existence of a morphism in the source gives the appropriate inequality in the target. Given amorphism ϕ = ( G , ( Q (cid:48) , . . . , Q (cid:48) q ) , G κ → [ t ]) : ([ s ] , ( P , . . . . P p ) , [ s ] λ ← E ) −→ ([ t ] , ( Q , . . . , Q q ) , [ t ] µ ← F )
32n the fiber C A [ r ] \ J , we have ([ s ] , ( P , . . . . P p )) (cid:62) ( G , ( Q (cid:48) , . . . , Q (cid:48) q )) in L [ s ] and κ : G → [ t ] an isomorphismtaking Q (cid:48) i to Q i for each i . Pulling back along the bijection λ : E → [ s ] we get that ( E , ( λ − P , . . . , λ − P p )) (cid:62) ( ˆ λ − G , ( ˆ λ − Q (cid:48) , . . . , ˆ λ − Q (cid:48) q )) in the poset L ( E ) , which can be identified with a subposet of L [ r ] . We claim that the right hand side is equalto ( F , ( µ − Q , . . . , µ − Q q )) , which will give the required inequality. Indeed, because ϕ is a morphism in thefiber, we have ( G , κ ) ◦ ( E , λ ) = ( F , µ ) . Computing the left hand side inˆ λ − G ˆ λ (cid:47) (cid:47) (cid:54) G κ (cid:47) (cid:47) (cid:54) [ t ] E λ (cid:47) (cid:47) (cid:54) [ s ][ r ] , shows that ˆ λ − G = F and κ ◦ ˆ λ = µ . The claim then follows from the fact that Q (cid:48) i = κ − Q i . The functors Λ and Π define an equivalence. The composition ΠΛ is the identity on L op [ r ] : it isenough to check this on objects as L op [ r ] is a poset. An object ( E , ( P , . . . , P p )) ∈ L op [ r ] is mapped by Λ to ([ e ] , ( λ E P , . . . , λ E P p ) , [ e ] λ E ← E ) , which is mapped back to the original tuple by Π . On the other hand, wehave ΛΠ ([ s ] , ( P , . . . , P p ) , [ s ] λ ← E ) = ([ s ] , ( λ E λ − P , . . . , λ E λ − P p ) , [ s ] λ E ← E ) where we have used that e = s as E and [ s ] have the same cardinality. These objects of C A [ r ] \ J are not equalin general, but η = ([ s ] , ( P , . . . , P p ) , [ s ] λ E λ − −−−−→ [ s ]) is a morphism in Lev A from ([ s ] , ( P , . . . , P p )) to ([ s ] , ( λ E λ − P , . . . , λ E λ − P p ) , which induces a morphism inthe fiber from ([ s ] , ( P , . . . , P p ) , [ s ] λ ← E ) to its image by ΛΠ by the commutativity of the diagramC A [ s ] ([ s ] , λ E λ − ) (cid:47) (cid:47) C A [ s ] C A [ r ] ( E , λ E ) (cid:59) (cid:59) ( E , λ ) (cid:99) (cid:99) in Exp A . We are left to check that these morphisms η fit together to define a natural transformation from theidentity to ΛΠ . Consider a morphism ϕ = ( G , ( Q (cid:48) , . . . , Q (cid:48) q ) , G κ → [ t ]) : ([ s ] , ( P , . . . . P p ) , [ s ] λ ← E ) −→ ([ t ] , ( Q , . . . , Q q ) , [ t ] µ ← F ) in the fiber. Then Π ( ϕ ) just remembers that the images under Π of the source and target are comparablein L [ r ] , and ΛΠ ( ϕ ) = ( ˆ λ E F , ( ˆ λ E µ − Q , . . . , ˆ λ E µ − Q q ) , ˆ λ E F λ F ˆ λ − E −−−−→ [ t ]) . We need to check that ([ s ] , ( P , . . . . P p )) ([ s ] , ( P ,..., P p ) , [ s ] λ E λ − −−−−→ [ s ]) (cid:15) (cid:15) ϕ =( G , ( Q (cid:48) ,..., Q (cid:48) q ) , G κ → [ t ]) (cid:47) (cid:47) ([ t ] , ( Q , . . . , Q q )) ([ t ] , ( Q ,..., Q q ) , [ t ] λ F µ − −−−−→ [ t ]) (cid:15) (cid:15) ([ s ] , ( λ E λ − P , . . . , λ E λ − P p )) ΛΠ ( ϕ ) (cid:47) (cid:47) ([ t ] , ( λ F µ − Q , . . . , λ F µ − Q q ) (4.5)33ommutes in Lev A . Now because all the morphisms in the diagram define morphisms in the fiber C A [ r ] \ J ,we get a diagram C A [ s ] ([ s ] , λ E λ − ) (cid:15) (cid:15) ( G , κ ) (cid:47) (cid:47) C A [ t ] ([ t ] , λ F µ − ) (cid:15) (cid:15) C A [ r ] (cid:102) (cid:102) (cid:56) (cid:56) (cid:120) (cid:120) (cid:38) (cid:38) C A [ s ] ( ˆ λ E F , λ F ˆ λ − E ) (cid:47) (cid:47) C A [ t ] of commuting triangles in Exp A , from which it follows that the outer square commutes as Exp A is a sub-category of the groupoid Cantor × A . From this, we get that G = ˆ λ ( F ) and λ F µ − κ = λ F ˆ λ − . Now goingalong the top of diagram (4.5), the maps compose to ( G , ( Q (cid:48) , . . . , Q (cid:48) q ) , G λ F µ − κ −−−−−→ [ t ]) while going along thebottom they compose to (( ˆ λ F , ( λ µ − Q , . . . , λ µ − Q q ) , ˆ λ F λ F ˆ λ − −−−−→ [ t ]) . And these are equal by the previouscomputation, using also that Q i = κ ( Q (cid:48) i ) for each i . This completes the proof. We will now recall Thomason’s explicit model for the homotopy colimit (4.2). Let Σ A : Set × → Set × be the functor defined on objects by Σ A [ r ] = [ rn ] and on morphisms λ : [ r ] → [ r ] by the diagram [ rn ] Σ A ( λ ) (cid:47) (cid:47) [ rn ][ r ] × A = E ˆ λ = λ × A (cid:47) (cid:47) (cid:54) λ E (cid:79) (cid:79) E = [ r ] × A (cid:54) λ E (cid:79) (cid:79) [ r ] λ (cid:47) (cid:47) [ r ] , (4.6)that is setting Σ A ( λ ) = λ E ◦ ˆ λ ◦ λ − E , where λ E is the map given by lexicographic ordering of Lemma 4.8.Explicitly λ E orders the elements of [ r ] × A as ( , a ) , . . . , ( , a n ) , . . . , ( r , a ) , . . . , ( r , a n ) , so Σ A ( λ ) is the well-known “block” version of λ , permuting blocks of size n . Using this block permutationdescription, one sees that Σ A is a strict symmetric monoidal functor.Recall that Tho A = hocolim N ( Set × , Σ A ) is obtained as a homotopy colimit in the category of non-unital permutative categories for N here consideredas the category with one object ∗ associated to the monoid ( N , +) and ( Set × , Σ A ) : N −→ Perm denoting the functor from N to the category of non-unital permutative categories taking the unique objectto Set × and a morphism k ∈ N to Σ kA , the k –th iterate of the functor Σ A . Here we use the explicit modelgiven by Thomason in [Tho82, Construction 3.22] for this homotopy colimit. The objects of Tho A are thetuples ([ r ] , . . . , [ r m ]) with m > r i >
0. Morphisms are given as tuples ( ψ , µ , f , . . . , f n ) : ([ r ] , . . . , [ r m ]) −→ ([ s ] , . . . , [ s n ]) ψ : [ m ] → [ n ] is a surjection, µ : [ m ] → N is a function, and for each j = , . . . , n , f j : [ ∑ i ∈ ψ − ( j ) r i n µ ( i ) ] −→ [ s j ] is a bijection. The composition of such morphisms is defined by composition of the surjections, correspond-ing “multi-additions” of the values µ ( i ) and composition of the bijections f j , appropriately multiplied withpowers of A : if ( ψ , µ , f , . . . , f n ) is as above and ( ψ (cid:48) , µ (cid:48) , f (cid:48) , . . . , f (cid:48) p ) : ([ s ] , . . . , [ s n ]) −→ ([ t ] , . . . , [ t p ]) , thenthe composition ( ψ (cid:48) , µ (cid:48) , f (cid:48) , . . . , f (cid:48) p ) ◦ ( ψ , µ , f , . . . , f n ) is given by ( ψ (cid:48) ◦ ψ , µ + ( µ (cid:48) ◦ ψ ) , f (cid:48) ◦ (cid:77) j ∈ ψ (cid:48)− ( ) Σ µ (cid:48) ( j ) A f j ◦ σ , . . . , f (cid:48) p ◦ (cid:77) j ∈ ψ (cid:48)− ( p ) Σ µ (cid:48) ( j ) A f j ◦ σ p ) for σ k the block permutation induced by the permutation reordering ( ψ (cid:48) ψ ) − ( k ) as ψ − ( j ) , . . . , ψ − ( j q ) for { j < · · · < j q } = ( ψ (cid:48) ) − ( k ) . The monoidal structure is given on objects by juxtaposition: ([ r ] , . . . , [ r m ]) ⊕ ([ s ] , . . . , [ s n ]) = ([ r ] , . . . , [ r m ] , [ s ] , . . . , [ s n ]) and likewise on morphisms, and the symmetry is the map ( σ m , n , , id , . . . , id ) is induced by the symmetryin Set × , with : [ m + n ] → N denoting the zero map.The goal of this section is to prove the following Theorem 4.17.
There is a functor H : Lev A −→ Tho A which induces a homotopy equivalence on classifying spaces.Part 1 of the proof of Theorem 4.17: Definition of the functor. The functor H : Lev A −→ Tho A is defined onobjects ([ r ] , ( P , . . . , P p )) of Lev A with r i = | P i | by setting H ([ r ] , ( P , . . . , P p )) = (cid:26) ([ r ] , . . . , [ r p ] , [ r − ∑ i r i ]) if ∑ i r i (cid:54) = r ([ r ] , . . . , [ r p ]) if ∑ i r i = r . Consider now a morphism ϕ : ([ r ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ −→ ([ s ] , ( Q , . . . , Q q )) in Lev A . The relation ([ r ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) in L [ r ] gives that there exists 0 (cid:54) p , p (cid:54) p and κ , . . . , κ q (cid:62) p + ∑ j κ j + p = p , with E the level expansion of [ r ] along P ∪ · · · ∪ P p and Q (cid:48) j = P p + κ ( )+ ··· + κ ( j − )+ (cid:116) · · · (cid:116) P p + κ ( )+ ··· + κ ( j ) . The object H ([ r ] , ( P , . . . , P p )) will be a sequence of p (cid:48) = p or p + ∪ (cid:96) P (cid:96) = [ r ] or not. Likewise, the image H ([ s ] , ( Q , . . . , Q q )) will be a sequence of q (cid:48) = q or q + H ( ϕ ) = ( ψ , µ , f , . . . , f q (cid:48) ) for ψ , µ , and the functions f j are defined as follows. Let ψ ( i ) = (cid:40) j if P i ⊂ Q (cid:48) j q + P i (cid:54)⊂ ∪ j Q (cid:48) j or i = p + . This gives a well-defined surjective map ψ : [ p (cid:48) ] → [ q (cid:48) ] : any j ∈ [ q ] with j (cid:54) q is in the image of ψ becauseany Q j is a union of P i ’s, and if q (cid:48) = q +
1, then we must either have p (cid:48) = p + ψ ( p + ) = q + p (cid:48) = p with p (cid:62) p (cid:62)
1, in which case either ψ ( ) = q + ψ ( p ) = q +
1. We define µ : [ p (cid:48) ] → N by µ ( i ) = (cid:26) (cid:54) i (cid:54) p . (cid:54) j (cid:54) q , let f j : [ ∑ i ∈ ψ − ( j ) r i ] → [ ∑ i ∈ ψ − ( j ) r i ] = [ s j ] for s j = | Q j | , be the bijection making thefollowing diagram commute: [ ∑ i ∈ ψ − ( j ) r i ] f j (cid:47) (cid:47) [ ∑ i ∈ ψ − ( j ) r i ] = [ s j ] (cid:70) i ∈ ψ − ( j ) [ r i ] ⊕ (cid:51) (cid:51) (cid:70) i ∈ ψ − ( j ) P i (cid:116) i λ Pi (cid:107) (cid:107) (cid:83) i ∈ ψ − ( j ) P i = Q (cid:48) j (cid:111) (cid:111) λ | Q (cid:48) j (cid:47) (cid:47) Q j λ Qj (cid:79) (cid:79) (4.7)for λ P i and λ Q j the lexicographic maps of Lemma 4.8. (In this case, each µ ( i ) = q (cid:48) = q +
1, we set f q + to be the bijection making the following diagram commute: [ ∑ i ∈ ψ − ( q + ) r i n µ ( i ) ] f q + (cid:47) (cid:47) [ ∑ i ∈ ψ − ( q + ) r i n µ ( i ) ] (cid:0) (cid:70) (cid:54) i (cid:54) p P i × A (cid:1) (cid:116) (cid:0) (cid:70) p − p < i (cid:54) p P i (cid:1) (cid:116) (cid:0) [ r ] \ ∪ i P i (cid:1) (cid:116) ( lexi ) (cid:79) (cid:79) E \ ( ∪ j Q (cid:48) j ) (cid:111) (cid:111) λ (cid:47) (cid:47) [ s ] \ ( ∪ j Q j ) ( lexi ) (cid:79) (cid:79) with the vertical maps defined as in the previous case.We check that H respects composition. Consider a composition in Lev A : ([ r ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ (cid:47) (cid:47) ([ s ] , ( Q , . . . , Q q )) (cid:62) ( G , ( R (cid:48) , . . . , R (cid:48) u )) κ (cid:47) (cid:47) ([ t ] , ( R , . . . , R u )) with image under H the composition ([ r ] , . . . , [ r p (cid:48) ]) ( ψ , µ , f ) (cid:47) (cid:47) ([ s ] , . . . , [ s q (cid:48) ]) ( ψ (cid:48) , µ (cid:48) , f (cid:48) ) (cid:47) (cid:47) ([ t ] , . . . , [ t u (cid:48) ]) . By definition of the composition in
Tho A , these maps compose to (cid:0) ψ (cid:48) ◦ ψ , µ + ( µ (cid:48) ◦ ψ ) , f (cid:48) ◦ ( ⊕ i ∈ ψ (cid:48)− ( ) Σ µ (cid:48) ( i ) A f i ) ◦ σ , . . . , f (cid:48) u (cid:48) ◦ ( ⊕ i ∈ ψ (cid:48)− ( u (cid:48) ) Σ µ (cid:48) ( i ) A f i ) ◦ σ u (cid:48) (cid:1) . The fact that the value of the functor H on the composed morphism ([ r ] , ( P , . . . , P p )) (cid:62) ( ˆ λ − G , ( ˆ λ − R (cid:48) , . . . , ˆ λ − R (cid:48) u )) κ ◦ ˆ λ (cid:47) (cid:47) ([ t ] , ( R , . . . , R u )) agrees on the first component follows from the fact that P i ⊂ ˆ λ − ( R k ) if and only if P i ⊂ Q (cid:48) j and λ ( Q (cid:48) j ) = Q j ⊂ R k . On the second component, it is because µ records the P i ’s that are expanded by the first mapand µ (cid:48) records those that are expanded by the second map. For the last component, assuming first 1 (cid:54) k (cid:54) u ,with ( ψ (cid:48) ) − ( k ) = { j < · · · < j m } , we have µ + ( µ (cid:48) ◦ ψ )( i ) = i ∈ ( ψ (cid:48) ψ ) − ( k ) , and the diagram [ ∑ i ∈ ( ψ (cid:48) ψ ) − ( k ) r i ] σ k ∼ = [ ∑ m (cid:96) = ( ∑ i ∈ ( ψ ) − ( j (cid:96) ) r i )] ⊕ (cid:96) f (cid:96) (cid:47) (cid:47) [ ∑ m (cid:96) = ( ∑ i ∈ ( ψ ) − ( j (cid:96) ) r i )] f (cid:48) k (cid:47) (cid:47) [ ∑ i ∈ ( ψ (cid:48) ψ ) − ( k ) r i ] (cid:70) i ∈ ( ψ (cid:48) ψ ) − ( k ) P i σ k ∼ = (cid:70) m (cid:96) = ( (cid:70) i ∈ ( ψ ) − ( j (cid:96) ) P i ) ← (cid:70) m (cid:96) = Q (cid:48) j (cid:96) ˆ λ (cid:47) (cid:47) (cid:116) ( lexi ) (cid:79) (cid:79) (cid:116) ( lexi ) (cid:79) (cid:79) (cid:70) m (cid:96) = Q j (cid:96) ← R (cid:48) k (cid:116) ( lexi ) (cid:79) (cid:79) κ (cid:47) (cid:47) R k ( lexi ) (cid:79) (cid:79) (cid:83) i ∈ ( ψ (cid:48) ψ ) − ( k ) P i = ˆ λ − R (cid:48) k (cid:107) (cid:107) (cid:109) (cid:109) (cid:105) (cid:105) ˆ λ (cid:69) (cid:69) κ ◦ ˆ λ (cid:54) (cid:54) commutes by definition of H on morphisms for the squares, and with the first three diagonal maps at thebottom of the diagram the canonical maps coming from the fact that the target is in each case a decompositionof the source into disjoint subsets of it. The top row is the map f (cid:48) u (cid:48) ◦ ( ⊕ i ∈ ψ (cid:48)− ( u (cid:48) ) Σ µ (cid:48) ( i ) A f i ) ◦ σ u (cid:48) , but is also equalto corresponding component of the value of H on the composition by the commutativity of the outer diagram.The case k = u +
1, when relevant, is similar. 36he proof that the functor H induces a homotopy equivalence on classifying spaces will be analogous to thatof the functor Lev A → Exp A : we will compare the homotopy fibers to a poset of simplices of a semi-simplicialset analogous to L ( X ) . We start by introducing the relevant semi-simplicial set.Let N = { , , , . . . } denote the set of natural numbers and ( N k , (cid:54) ) the poset of k –tuples of natural numbers,with the poset structure defined by ( m , . . . , m k ) (cid:54) ( n , . . . , n k ) if m i (cid:54) n i for each i . Definition 4.18.
For k >
0, let M ( k ) denote the semi-simplicial subset of the nerve of ( N k , (cid:54) ) with the samevertices and where a p –simplex is a sequence v = ( n , . . . , n k ) < · · · < v p = ( n p , . . . , n pk ) in ( N k , (cid:54) ) such that v i − v ∈ { , } k for all 1 (cid:54) i (cid:54) p , i.e. n ij − n j ∈ { , } for all j .Note that the condition v i − v ∈ { , } k for all 1 (cid:54) i (cid:54) p is equivalent to the condition v i − v j ∈ { , } k forall 0 (cid:54) j (cid:54) i (cid:54) p given that v < · · · < v p in the poset ( N k , (cid:54) ) .For any a tuple of positive numbers ( r , . . . , r k ) , we can identify M ( k ) with a semi-simplicial subset of L [ ∑ r i ] :to a vertex v = ( n , . . . , n k ) of M ( k ) , we can associate the set [ r ] × A n (cid:116) · · · (cid:116) [ r k ] × A n k . which we think of as an expansion of the set [ r ] = [ r ] ⊕ · · · ⊕ [ r k ] . Under this identification, the condition fora sequence ( n , . . . , n k ) < · · · < ( n p , . . . , n pk ) to form a p –simplex in M ( k ) is precisely that the correspondingexpansions of [ r ] form a p –simplex in L [ r ] . But note that here we are only considering level expansions of aspecific form, allowing only expansions along the subsets [ r i ] × A n i , or unions of such. Lemma 4.19.
For any k > , the semi-simplicial set M ( k ) is contractible.Proof. We consider the restriction of the rank filtration of the poset ( N k , (cid:54) ) to M ( k ) : the rank of a ver-tex ( n , . . . , n k ) is ∑ i n i . Let F i M ( k ) be the semi-simplicial subset on the vertices of rank at most i . Wehave { ( , . . . , ) } = F M ( k ) ⊂ F M ( k ) ⊂ · · · ⊂ F i M ( k ) ⊂ · · · ⊂ M ( k ) . By compactness, any map from a sphere into M ( k ) will land in a finite filtration, so it is enough to showthat F i M ( k ) is contractible for each i . Clearly F M ( k ) is contractible, as it is just a point. Assuming thatwe have proved that F i M ( k ) is contractible, we will show that F i + M ( k ) is also contractible. As vertices ofany simplex in M ( k ) necessarily have distinct rank, we see that F i + M ( k ) is obtained from F i M ( k ) by addinga cone on each vertex v of rank equal to i +
1, attached to F i M ( k ) along Link ( v ) ∩ F i M ( k ) . By definition,a simplex w < · · · < w q of M ( k ) lies in Link ( v ) precisely if w < · · · < w p < v < w p + < · · · < w q is asimplex of M ( k ) . As v has rank i +
1, we see that only w , . . . , w p have rank at most i , and hence thatLink ( v ) ∩ F i M ( k ) identifies with the semi-simplical subset M ( k ) < v ⊂ M ( k ) of simplices w < · · · < w p suchthat w < · · · < w p < v is also a simplex of M ( k ) .So we are left to show that each M ( k ) < v is contractible. Note first that the semi-simplicial set M ( k ) < v is the (small) nerve of a poset: for vertices w , w (cid:48) of M ( k ) < v , set w ≺ w (cid:48) if w < w (cid:48) is a 1–simplexin M ( k ) < v . This defines a poset structure on M ( k ) < v as w ≺ w (cid:48) ≺ w (cid:48)(cid:48) implies that w < w (cid:48)(cid:48) is also a 1–simplex in M ( k ) < v , because w < v and w (cid:48)(cid:48) < v are simplices in M ( k ) , which means that v − w ∈ { , } k and 0 (cid:54) w (cid:48)(cid:48) i − w i (cid:54) v i − w i (cid:54) { , } for each i . This allows us to finish the proof,because that poset has a least element: indeed, if v = ( n , . . . , n k ) , then w = ( n , . . . , n k ) − ( ε , . . . , ε k ) with ε i = min { , n i } is the least element.As in the case of L [ r ] , we will need to pass to the poset of simplices of M ( k ) . We start by giving a descriptionof M ( k ) that is more suitable for our needs. The idea is, just as in the case of L ( X ) , that we can encode theinformation of a simplex ( n , . . . , n k ) < · · · < ( n p , . . . , n pk )
37y remembering the first tuple of elements ( n , . . . , n k ) , and then remembering for each 1 (cid:54) i (cid:54) p which setof indices J i ⊂ { , . . . , k } was raised by 1 when going from ( n i − , . . . , n i − k ) to ( n i , . . . , n ik ) , that is for whichindex (cid:96) the difference n i (cid:96) − n i − (cid:96) is equal to 1. This idea formalizes to the following: Lemma 4.20.
The semi-simplicial set M ( k ) is isomorphic to the semi-simplicial set M (cid:48) ( k ) whose p–simplicesare the tuples (( n , . . . , n k ) , J , . . . , J p ) for ( n , . . . , n k ) ∈ N k and J , . . . , J p a collection of p (cid:62) disjoint non-empty subsets of [ k ] = { , . . . , k } , andwith the face maps defined byd i (( n , . . . , n k ) , J , . . . , J p ) = (( n + ε , . . . , n k + ε k ) , J , . . . , J p ) i = (( n , . . . , n k ) , J , . . . , J i ∪ J i + , . . . , J p ) < i < p (( n , . . . , n k ) , J , . . . , J p − ) i = p , where ε j = if j ∈ J and otherwise.Proof. Define a map F : M ( k ) → M (cid:48) ( k ) on p –simplices by setting F (cid:0) ( n , . . . , n k ) < · · · < ( n p , . . . , n pk ) (cid:1) = (cid:0) ( n , . . . , n k ) , J , . . . , J p (cid:1) for J i ⊂ { , . . . , k } the set of indices of ( n i , . . . , n ik ) − ( n i − , . . . , n i − k ) that are equal to 1. By definition of M ( k ) ,this difference lies in { , } k \ { } k so J i is non-empty. Also the subsets J i must be pairwise disjoint since theexistence of (cid:96) ∈ J i ∩ J j for i < j would contradict that n j (cid:96) − n (cid:96) ∈ { , } . Hence F has image inside M (cid:48) ( k ) . Theface maps on the latter semi-simplicial set are defined so that F is semi-simplicial. An inverse is obtained bytaking a tuple (cid:0) v = ( n , . . . , n k ) , J , . . . , J p (cid:1) to the simplex v < v + (cid:104) J (cid:105) < · · · < v + (cid:104) J p (cid:105) with (cid:104) J i (cid:105) = ( ε , . . . , ε k ) for ε j = ε j ∈ J i and 0 otherwise. Part 2 of the proof of Theorem 4.17: The functor H : Lev A → Tho A is an equivalence. We will again con-sider the homotopy fibers of the functor H . We will show that they are homotopy equivalent to the poset M ( k ) of simplices of M (cid:48) ( k ) , from which the result will follow as a consequence of Lemma 4.19 and Lemma 4.20.For any object ([ r ] , . . . , [ r k ]) in Tho A , we will define functors Λ : M ( k ) op ←→ ([ r ] , . . . , [ r k ]) \ H : Π and show that they define an equivalence. We start by defining Λ . Definition of Λ . To a p –simplex (( n , . . . , n k ) , J , . . . , J p ) of the semi-simplicial set M (cid:48) ( k ) we can associate theset E = [ r ] × A n (cid:116) · · · (cid:116) [ r k ] × A n k , considered as an expansion of [ ∑ i r i ] via the sum [ r ] (cid:116) · · · (cid:116) [ r k ] ⊕ −→ [ ∑ i r i ] ,and canonically identified with [ ∑ i r i n n i ] via λ E . For 1 (cid:54) j (cid:54) p , let P j = λ E ( (cid:71) i ∈ J j [ r i ] × A n i ) ⊂ [ ∑ (cid:54) i (cid:54) k r i n n i ] be the subset of [ ∑ i r i n n i ] corresponding to J j . We define Λ on objects by setting Λ (( n , . . . , n k ) , J , . . . , J p ) = (([ ∑ (cid:54) i (cid:54) k r i n n i ] , P , . . . , P p ) , ( ψ , µ , id )) with ( ψ , µ , id ) : ([ r ] , . . . , [ r k ]) −→ H ([ ∑ (cid:54) i (cid:54) k r i n n i ] , ( P , . . . , P p )) = ([ ∑ i ∈ J r i n n i ] , . . . , [ ∑ i ∈ J p r i n n i ] , ([ ∑ i / ∈∪ j J j r i n n i ])) defined by setting ψ ( i ) = j if i ∈ J j , and ψ ( i ) = p + i / ∈ ∪ j J j , and µ ( i ) = n i , and with id = ( id , . . . , id ) denoting a p (cid:48) –tuple of identity maps; here, and in the rest of the proof, the last component of the rightmostterm above is dropped if it is an empty set. And as before, we use p (cid:48) to denote p or p + Λ on morphisms, note first that X = (( m , . . . , m k ) , I , . . . , I p ) (cid:62) (( n , . . . , n k ) , J , . . . , J q ) = Y in M ( k ) if there are 0 (cid:54) p , p (cid:54) p and κ , . . . , κ q (cid:62) p + ∑ κ j + p = p such that n i = m i + ε i for ε i = i ∈ I ∪ · · · ∪ I p and 0 otherwise, and J j = I p + κ + ··· + κ j − + ∪ · · · ∪ I p + κ + ··· + κ j . To such an inequality, weassociate the morphism α : Λ ( X ) = ([ ∑ (cid:54) i (cid:54) k r i n m i ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ E −→ ([ ∑ (cid:54) i (cid:54) k r i n n i ] , ( Q , . . . , Q q )) = Λ ( Y ) in Lev A with E the level expansion of [ ∑ i r i n m i ] along the subset P ∪ · · · ∪ P p , and Q (cid:48) j = ∪ i ∈ J j P i . For thisto make sense, we need that λ E ( Q (cid:48) j ) = Q j , which holds by the compatibility of the lexicographic order-ings (Lemma 4.8). Analyzing diagram (4.7) in the definition of H , we see that H ( α ) = ( ϕ , ν , σ , . . . , σ q (cid:48) ) with σ j : ∑ i ∈ ψ − ( j ) ∑ (cid:96) ∈ I i r (cid:96) n m (cid:96) −→ ∑ i ∈ J j r (cid:96) n m (cid:96) if j (cid:54) q reordering the summands (corresponding to the natural map (cid:116) i ∈ ψ − ( j ) P i → Q (cid:48) j as subsets of [ ∑ i r i n m i ] ),and similarly for j = q + ([ r ] , . . . , [ r k ]) ( ψ , µ , id ) (cid:47) (cid:47) ( ψ (cid:48) , µ (cid:48) , id ) (cid:43) (cid:43) ([ ∑ i ∈ I r i n m i ] , . . . , [ ∑ i ∈ I p r i n m i ] , ([ ∑ i / ∈∪ pj = I j r i n m i ])) H ( α )=( ϕ , ν , σ ) (cid:15) (cid:15) ([ ∑ i ∈ J r i n n i ] , . . . , [ ∑ i ∈ J q r i n n i ] , ([ ∑ i / ∈∪ qj = I j r i n n i ])) commutes, as ϕ ◦ ψ = ψ (cid:48) , both surjections keeping track of which component each expanded [ r i ] goes to,and µ (cid:48) = µ + ( ν ◦ ψ ) as ν records which additional expansions occurred from µ to µ (cid:48) , and because the maps σ j precisely take care of the necessary reordering of the summands. Hence we have defined a morphism inthe fiber from Λ ( X ) to Λ ( Y ) . Left is to check that this assignment is compatible with composition, which isa direct computation. Definition of Π . We define Π on objects of the fiber as follows. Let X = (cid:0) ([ s ] , ( P , . . . , P p )) , ( ψ , µ , f ) (cid:1) be anobject of the fiber, with ([ r ] , . . . , [ r k ]) ( ψ , µ , f ) −→ H ([ s ] , ( P , . . . , P p )) = ([ s ] , . . . , [ s p ] , [ s − ∑ j s j ]) = ([ s ] , . . . , [ s p (cid:48) ]) for f = ( f , . . . , f p (cid:48) ) with f j : (cid:116) i ∈ ψ − ( j ) [ r i n µ ( i ) ] ∼ = −→ [ s j ] , where we set [ s p + ] : = [ s − ∑ j s j ] if the latter set isnon-empty. We define Π ( X ) = (( µ ( ) , . . . , µ ( k )) , ψ − ( ) , . . . , ψ − ( p )) . This is well-defined as the sets [ s ] , . . . , [ s p ] are non-empty. Given a morphism X → Y in the fiber, we needto check that we have the relation Π ( X ) (cid:62) Π ( Y ) in M ( k ) . If X = (cid:0) ( ψ , µ , f ) , ([ s ] , ( P , . . . , P p )) (cid:1) and Y = (cid:0) ( ψ (cid:48) , µ (cid:48) , f (cid:48) ) , ([ s (cid:48) ] , ( Q , . . . , Q q )) (cid:1) , such a morphism is given by a morphism in Lev A α : ([ s ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ −→ ([ s (cid:48) ] , ( Q , . . . , Q q )) with E the level expansion of [ s ] along P ∪ · · · ∪ P p and each Q (cid:48) j a union of P (cid:48) i , and λ : E → [ s (cid:48) ] a bijection,such that the diagram ([ r ] , . . . , [ r k ]) ( ψ , µ , f ) (cid:47) (cid:47) ( ψ (cid:48) , µ (cid:48) , f (cid:48) ) (cid:41) (cid:41) ([ s ] , . . . , [ s p (cid:48) ]) H ( α ) (cid:15) (cid:15) ([ s (cid:48) ] , . . . , [ s (cid:48) q (cid:48) ]) (4.8)39ommutes. Because α is a level expansion along the first p (cid:62) P i , we get that µ (cid:48) = µ + ε for ε i = i ∈ ψ − { , . . . , p } and ε i = ψ (cid:48)− ( j ) is the union of the ψ − ( i ) for those i forwhich Q (cid:48) j is the union of the P i ’s. This precisely gives that Π ( X ) = (( µ ( ) , . . . , µ ( k )) , ψ − ( ) , . . . , ψ − ( p )) (cid:62) (( µ (cid:48) ( ) , . . . , µ (cid:48) ( k )) , ψ (cid:48)− ( ) , . . . , ψ (cid:48)− ( q )) = Π ( Y ) in M ( k ) , as required. The functors define an equivalence.
The composition ΠΛ is the identity: As M ( k ) op is a poset, it is enoughto check this on objects, where ΠΛ (( n , . . . , n k ) , J , . . . , J q ) = (( µ ( ) , . . . , µ ( k )) , ψ − ( ) , . . . , ψ − ( p )) for the function µ defined by setting µ ( i ) = n i and ψ defined by setting ψ ( i ) = j if i ∈ J j . Onthe other hand, the composition ΛΠ is not the identity, but we will see that, just like in the previ-ous cases, the information forgotten by that composition can be used to define a natural transformationbetween ΛΠ and the identity functor. Indeed, given an object X = (cid:0) ([ s ] , ( P , . . . , P p )) , ( ψ , µ , f ) (cid:1) in the fiber,we have ΛΠ ( X ) = (cid:0) ([ s ] , ( ¯ P , . . . , ¯ P p )) , ( ψ , µ , id ) (cid:1) with s = ∑ i r i n µ ( i ) and ¯ P j the subset of [ s ] corresponding tothe inclusion [ ∑ i ∈ ψ − ( j ) r i n µ ( i ) ] ⊂ [ ∑ i r i n µ ( i )] = [ s ] coming from the inclusion of indexing sets ψ − ( j ) ⊂ [ k ] .Consider the following bijection f : [ s ] −→ P (cid:116) · · · (cid:116) P p (cid:116) [ s \ ∪ i P i ] (cid:116) j λ Pj −−−→ [ s ] (cid:116) · · · (cid:116) [ s p (cid:48) ] (cid:116) j f j −−→ [ s ] (cid:116) · · · (cid:116) [ s p (cid:48) ] (cid:116) j λ − Pj −−−−→ ¯ P (cid:116) · · · (cid:116) ¯ P p (cid:116) [ s \ ∪ i ¯ P i ] −→ [ ∑ i r i n µ ( i ) ] = [ s ] where the first map splits [ s ] into components according to the P j ’s and the last map reassembles the com-ponents using the inclusion of the sets ¯ P i instead. As f ( P j ) = ¯ P j , it defines a morphism ([ s ] , ( P , . . . , P p ) , f ) in Lev A from ([ s ] , ( P , . . . , P p )) to ([ s ] , ( ¯ P , . . . , ¯ P p )) . Moreover, applying H to this morphism gives a commu-tative diagram in Tho A ([ r ] , . . . , [ r k ]) ( ψ , µ , id ) (cid:47) (cid:47) ( ψ , µ , f ,..., f p (cid:48) ) (cid:43) (cid:43) ([ ∑ i ∈ ψ − ( ) r i n n i ] , . . . , [ ∑ i ∈ ψ − ( p (cid:48) ) r i n n i ]) H ([ s ] , ( P ,..., P p ) , f )=( id , , f ,..., f p (cid:48) ) (cid:15) (cid:15) ([ s ] , . . . , [ s p ] , [ s − ∑ j s j ]) . Hence we have defined a morphism η in the fiber from ΛΠ ( X ) back to X . We are left to checkthat these morphisms together form a natural transformation. Given X = (cid:0) ([ s ] , ( P , . . . , P p )) , ( ψ , µ , f ) (cid:1) and Y = (cid:0) ([ s (cid:48) ] , ( Q , . . . , Q q )) , ( ψ (cid:48) , µ (cid:48) , f (cid:48) ) (cid:1) , a morphism X → Y in the fiber is defined by a map α : ([ s ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ −→ ([ s (cid:48) ] , ( Q , . . . , Q q )) in Lev A making the diagram (4.8) commute. We need to check that ([ s ] , ( P , . . . , P p )) (cid:62) ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) λ (cid:47) (cid:47) f (cid:15) (cid:15) ([ s (cid:48) ] , ( Q , . . . , Q q )) f (cid:48) (cid:15) (cid:15) ([ s ] , ( P , . . . , P p )) (cid:62) ( ¯ E , ( ¯ Q (cid:48) , . . . , ¯ Q (cid:48) q )) λ ¯ E (cid:47) (cid:47) ([ s (cid:48) ] , ( Q , . . . , Q q )) (4.9)commutes in Lev A , where we have suppressed the trivial inequalities in the vertical morphisms andwhere the bottom row is ΛΠ ( α ) . Now going along the top of the diagram, the morphisms com-pose to give ( E , ( Q (cid:48) , . . . , Q (cid:48) q ) , f (cid:48) ◦ λ ) whereas going along the bottom, they compose to the morphism40 ˆ f − ¯ E , ( ˆ f − ¯ Q (cid:48) , . . . , ˆ f − ¯ Q (cid:48) q ) , λ E ◦ f ) . So we are left to check that f (cid:48) ◦ λ = λ E ◦ f and that ˆ f takes ( E , ( Q (cid:48) , . . . , Q (cid:48) q )) to ( ¯ E , ( ¯ Q (cid:48) , . . . , ¯ Q (cid:48) q )) . The latter fact follows from the fact that f takes ([ s ] , ( P , . . . , P p )) to ([ s ] , ( ¯ P , . . . , ¯ P p )) and that the fact that E and the Q (cid:48) j ’s are build from [ s ] and the P i ’s in exactly the sameway as ¯ E from [ s ] and the ¯ P i ’s. Finally, the fact that the morphisms in (4.9) are morphisms in the fiber, givesa commutative diagram in Tho A : ([ s ] , . . . , [ s p ] , [ s − ∑ i s i ]]) H ( η X ) (cid:15) (cid:15) H ( α ) (cid:47) (cid:47) ([ s (cid:48) ] , . . . , [ s (cid:48) q ] , [ s (cid:48) − ∑ j s (cid:48) j ]) H ( η Y ) (cid:15) (cid:15) ([ r ] , . . . , [ r k ]) (cid:107) (cid:107) (cid:51) (cid:51) (cid:115) (cid:115) (cid:43) (cid:43) ([ s ] , . . . , [ s p ] , [ s − ∑ i s i ]]) H ( ΛΠ α ) (cid:47) (cid:47) ([ s (cid:48) ] , . . . , [ s (cid:48) q ] , [ s (cid:48) − ∑ j s (cid:48) j ]) Now the maps in this diagram assemble to give a commutative diagram of bijections on the set [ s (cid:48) ] where s (cid:48) = | E | = | ¯ E | = ∑ i r i n µ (cid:48) ( i ) , where the left vertical map is induced by f , the right vertical map is induced by f (cid:48) , the top one by λ and the bottom one by λ E . Because this is now a commutative diagram of invertible maps,we can conclude that the outer square commutes, which precisely gives the equality f (cid:48) ◦ λ = λ E ◦ f . There is a canonical functor
Tho A to Cantor × A coming from the universal property of the homotopy col-imit Tho A and the defining structure of Cantor algebras. We give it here explicitly and check that it issymmetric monoidal using the universal property. Finally, we prove that it is compatible with the zig-zag offunctors we have so far defined between the two categories.Let F : Tho A −→ Cantor × A be the functor defined on objects by F ([ r ] , . . . , [ r m ]) = C A [ r + · · · + r m ] and on morphisms by taking ( ψ , µ , f ) : ([ r ] , . . . , [ r m ]) −→ ([ s ] , . . . , [ s n ]) with f j : (cid:71) i ∈ ψ − ( j ) [ r i n µ ( i ) ] ∼ = −→ [ s j ] to the isomorphism of Cantor algebras represented by ( E , [ s + · · · + s n ] , f ) : C A [ r + · · · + r m ] −→ C A [ s + · · · + s n ] for E = (cid:70) i r i × A µ ( i ) considered as an expansion of [ r ] ⊕ · · · ⊕ [ r k ] = [ r + · · · + r m ] and with f : E = (cid:71) (cid:54) i (cid:54) m r i × A µ ( i ) ∼ = −→ (cid:71) (cid:54) j (cid:54) n (cid:71) i ∈ ψ − ( j ) r i × A µ ( i ) (cid:116) j f j −−→ (cid:71) (cid:54) j (cid:54) n [ s j ] ⊕ −→ [ s + · · · + s n ] where the first map reorders the factors. Proposition 4.21.
The functor F : Tho A −→ Cantor × A is symmetric monoidal.Proof. The functor F is the functor induced, via the universal property of the homotopy colimit Tho A [Tho82, Prop 3.21], from the free Cantor algebra functor C A : Set × → Cantor × A and the natural transfor-mation η : C A → C A ◦ Σ A defined on objects as the morphism η r : C A [ r ] → C A [ rn ] represented by the triple41 E = [ r ] × A , [ rn ] , λ E ) . Indeed, for the naturality, we need to check that for any bijection λ : [ r ] → [ r ] , thediagram C A [ r ] C A ( λ ) (cid:47) (cid:47) η r (cid:15) (cid:15) C A [ r ] η r (cid:15) (cid:15) C A [ rn ] C A Σ A ( λ ) (cid:47) (cid:47) C A [ rn ] commutes, which precisely follows from the commutativity of diagram (4.6) defining Σ A ( λ ) . (One can alsocheck directly that F is a symmetric monoidal functor. In fact it is strict symmetric monoidal.)Note that F has image in the subcategory I : Exp A (cid:44) → Cantor × A . We will denote by F : Tho A → Exp A thefunctor considered with Exp A as target category.We have constructed a diagram of functors Tho
A F (cid:47) (cid:47) F (cid:40) (cid:40) Cantor × A Lev AH ∼ (cid:79) (cid:79) J ∼ (cid:47) (cid:47) Exp AI ∼ (cid:79) (cid:79) and we are left to check that the diagram commutes up to homotopy, which will follow once we have shownthat the bottom triangle commutes up to homotopy. Proposition 4.22.
Consider the diagram
Lev
A J (cid:35) (cid:35) H (cid:47) (cid:47) Tho AF (cid:15) (cid:15) Exp A of categories. There is a natural transformation η : J → F ◦ H. In particular, the corresponding diagram ofclassifying spaces commutes up to homotopy.Proof.
Given an object ([ r ] , ( P , . . . , P p )) of Lev A , we have that J ([ r ] , ( P , . . . , P p )) = C A [ r ] = F ◦ H ([ r ] , ( P , . . . , P p )) . Indeed, J forgets the subsets ( P , . . . , P p ) , while H takes ([ r ] , ( P , . . . , P p )) to ([ r ] , . . . , [ r p ] , [ r − ∑ i r i ]) in Tho A for r i = | P i | (dropping the last component if it is zero), which is then taken by F to the Cantor alge-bra C A [ ∑ i r i + r − ∑ i r i ] = C A [ r ] . However we see that the subsets P i have been “permuted” in the com-position F ◦ H , which will affect the value of the composed functor F ◦ H on morphisms. The naturaltransformation η is given by that permutation: Let η ([ r ] , ( P , . . . , P p )) : C A [ r ] −→ C A [ r ] be induced by thebijection [ r ] −→ P (cid:116) · · · (cid:116) P p (cid:116) ([ r ] \ (cid:116) i P i ) ( lexi ) −→ [ r ] (cid:116) · · · (cid:116) [ r p ] (cid:116) [ r − ∑ i r i ] ⊕ −→ [ ∑ r i + ( r − ∑ r i )] = [ r ] . We need to check that this is natural with respect to the morphisms of
Lev A . It is enough to check naturalityfor each of the generating morphisms d , d i , d p and λ . Recall that d expands along P , that d i takes theunion of P i and P i + , and that d P forgets P p . We see that η is precisely the map that ensures that we applythese operations to the “same” subsets of [ r ] : indeed, J considers each P i as the original P i ⊂ [ r ] , while F ◦ H considers P i as the subset of P (cid:116) · · · (cid:116) P p (cid:116) ([ r ] \ (cid:116) i P i ) ∼ = [ r ] , and η precisely takes the one version of P i to theother. Likewise, applying a bijection λ will be compatible by the definition of the functors involved: Given42 bijection λ : [ r ] → [ r ] inducing a morphism λ : ([ r ] , ( P , . . . , P p )) → ([ r ] , ( λ P , . . . , λ P p )) in Lev A , we get acommutative diagram C [ r ] J ( λ )= λ (cid:47) (cid:47) η ∼ = (cid:15) (cid:15) C [ r ] η ∼ = (cid:15) (cid:15) (cid:0) C A [ r ] ⊕ · · · ⊕ C A [ r p ] (cid:1) ⊕ C A [ r − ∑ i r i ] ⊕ i λ i (cid:47) (cid:47) (cid:0) C A [ r ] ⊕ · · · ⊕ C A [ r p ] (cid:1) ⊕ C A [ r − ∑ i r i ] C A [ r ] F ◦ H ( λ ) (cid:47) (cid:47) C A [ r ] where λ i is the map induced by λ on each P i (resp. on [ r ] \(cid:116) i P i ), through their canonical identification with [ r i ] via the lexicographic ordering map λ P i . In this section, we identify the spectrum K ( Tho A ) from the preceding section with the Moore spectrumfor Z / ( n − ) , where n is the cardinality of the set A as above. We will use this and the relationship betweenthe category Tho A and the Higman–Thompson groups to give, in Section 6, concrete computations of thehomology of the Higman–Thompson groups. For any integer n (cid:62) M n denote the homotopy cofiber of the multiplication by n map on the spherespectrum S , so that there is a homotopy cofiber sequence as follows. S n −→ S −→ M n (5.1)The spectrum M n is the Moore spectrum for Z / n , also known as the mod n Moore spectrum . Theorem 5.1.
Let A be a finite set of cardinality n (cid:62) . There is an equivalence of spectra K ( Tho A ) (cid:39) M n − . Proof.
Recall from Section 4.3 that the category
Tho A is defined from the diagram of categories on N whichtakes the unique object to the category Set × and the arrow k to the functor Σ kA that corresponds to taking theproduct with A k . By the Barratt–Priddy–Quillen theorem [BP72], we have that K ( Set × ) : = K ( Set × ) (cid:39) S andthe functor Σ A induces multiplication with the cardinality n of A on S . We apply Thomason’s formula (4.1)to our definition (4.2) and get K ( Tho A ) = K ( hocolim N ( Set × , Σ A )) (cid:39) hocolim N ( K ( Set × ) , K ( Σ A )) (cid:39) hocolim N ( S , n ) . There is a spectral sequence E p , q = H p ( N ; ( π q S , n )) = ⇒ π ∗ hocolim N ( S , n ) that computes the homotopy groups of the homotopy colimit, see [Tho82, Sec. 3]. Here ( π q S , n ) is thediagram of abelian groups on N that takes the object to π q S and the morphism 1 to multiplication by n . Themonoid ring Z [ N ] ∼ = Z [ T ] is polynomial on one generator T, so that we can use the standard Koszul resolution Z [ T ] T − −→ Z [ T ] −→ Z
43o compute the E page. As the resolution has length one, the spectral sequence degenerates at E , and yieldsa long exact sequence · · · −−−→ π ∗ S n − −−−→ π ∗ S −−−→ π ∗ hocolim N ( S , n ) −−−→ · · · . (5.2)Let ε ∈ π hocolim N ( S , n ) = [ S , hocolim N ( S , n ) ] be the image of 1 ∈ π S = [ S , S ] in the long exact sequence.From the long exact sequence we get that ( n − ) ε =
0. Therefore, this map factors through the Moorespectrum to give a map ε : M n − −→ hocolim N ( S , n ) . Comparison of the long exact sequence obtained from the homotopy cofibration sequence (5.1) with thelong exact sequence (5.2) shows that ε induces an isomorphism on homotopy groups, and thus that it is anequivalence.Together with Corollary 4.2 the proposition gives the following result. Corollary 5.2.
For any finite set A of cardinality n (cid:62) there is an equivalence of spectra K ( Cantor × A ) (cid:39) M n − . Remark 5.3.
It is instructive to work out the implications of Corollary 5.2 on the level of compo-nents. The cofiber sequence (5.1) shows that the group π M n − is the cokernel of the multiplicationby n − π S = Z , so that it is cyclic of order n −
1. On the other hand the abeliangroup π Ω ∞ K ( Cantor × A ) = π K ( Cantor × A ) is the group completion of the abelian monoid π | Cantor × A | .The latter can be identified with { , , . . . , n − } as a set, where an integer r corresponds to the free Can-tor algebra C A [ r ] of type A on r generators. The monoid structure is dictated by C A [ r ] ⊕ C A [ s ] = C A [ r + s ] and C A [ r + ( n − )] ∼ = C A [ r ] if r (cid:62)
1. Note that the neutral element 0 is the only invertible element in thismonoid. The group completion of this monoid is Z / ( n − ) by the theorem, but this can of course also beworked out by hand: Once 1 is inverted, the element n − Z / ( n − ) . If X is a spectrum, its associated infinite loop space Ω ∞ X is a group-like E ∞ –space: the monoid of componentsis a group. It follows that all its components are homotopy equivalent, and in the following, we denoteby Ω ∞ X the component corresponding to the zero element 0 ∈ π X . We will now relate the homology ofthe zeroth component Ω ∞ K ( Cantor A ) to that of the Higman–Thompson groups. Together with Corollary 5.2this will yield the main identification we are after, namely the isomorphism of the homology of the Higman–Thompson groups with that of Ω ∞ M n − .Recall from Section 3.1 the stabilization homomorphism s r : V n , r → V n , r + and letV n , ∞ = colim ( V n , −→ V n , −→ · · · ) denote the associated stable group . The homology of V n , ∞ is usually called the stable homology of thegroups V n , r for r (cid:62)
1, but a direct consequence of our stability theorem, Theorem 3.6, is thatH ∗ ( V n , r ) ∼ = H ∗ ( V n , ∞ ) . So in the case at hand, all the homology is stable and hence it is enough to identify the homology of V n , ∞ .Given a monoid M , one can form its bar construction B M and the loop space Ω B M is a group-like space,known as the group completion of M . The group completion theorem of McDuff and Segal [McDS76] iden-tifies in good cases the homology of Ω B M with that of its “stable part.” We will use this theorem to computethe homology of Ω ∞ K ( Cantor × A ) (cid:39) Ω B | Cantor × A | , the group completion of the E ∞ monoid | Cantor × A | .44 heorem 5.4. Let A = { a , . . . , a n } with n (cid:62) . There is a map BV n , ∞ −→ Ω ∞ K ( Cantor × A ) which induces an isomorphism in homology with all systems of local coefficients on Ω ∞ K ( Cantor × A ) .Proof. We apply the group completion theorem to the monoid M = | Cantor × A | . More precisely, we willuse Theorem 1.1 in [R-W13], which makes explicit the relevant result in [McDS76]. The monoid M ishomotopy commutative, as it is the classifying space of a permutative category. It has components indexedby 0 , . . . , n −
1, forming a monoid in the way describe in Remark 5.3. To apply Theorem 1.1 of [R-W13], weuse the constant sequence of elements of M given by C A [ ] , C A [ ] , . . . . We need to check that for every m ∈ M ,the component of m in M is a right factor of the component of some finite sum C A [ ] ⊕ · · · ⊕ C A [ ] , which isobvious as every component is reached this way except for the zero component which is a right factor of anysuch sum.Form the colimit M ∞ = colim (cid:0) | Cantor × A | ⊕ C A [ ] −→ | Cantor × A | ⊕ C A [ ] −→ · · · ) . Theorem 1.1 in [R-W13] says that there is a homology isomorphism M ∞ −→ Ω B | Cantor × A | (cid:39) Ω ∞ K ( Cantor A ) with respect to all local coefficient systems on the target. Now the zeroth component of M ∞ can be identifiedwith the colimit on classifying space of the mapsV n , −→ V n , s −→ · · · −→ V n , n − s n − −→ V n , n s n −→ V n , n + −→ · · · of groups , and this colimit of groups is the group V n , ∞ in the stability statement. The result follows.We are now ready to prove the main result of this text. Proof of Theorem A.
As a consequence of our stability result, Theorem 3.6, for all r (cid:62)
1, the stabiliza-tion map s r : V n , r → V n , r + induces an isomorphism in homology with coefficients in any H ( V n , ∞ ) –module. Note that, in particular, we have an isomorphism H ( V n , r ) ∼ = H ( V n , ∞ ) , so that H ( V n , ∞ ) –modules are the same as H ( V n , r ) –modules. It follows that the map BV n , r → BV n , ∞ induces an iso-morphism in homology with abelian coefficients for all r (cid:62)
1. Theorem 5.4 gives that there is amap BV n , ∞ → Ω ∞ ( Cantor × A ) which induces an isomorphism in homology with all local coefficients on thetarget. (Note that π Ω ∞ ( Cantor × A ) ∼ = H ( Ω ∞ ( Cantor × A )) ∼ = H ( V n , ∞ ) , so these are again H ( V n , r ) –modulesas above.) By Corollary 5.2, we have a homotopy equivalence Ω ∞ K ( Cantor × A ) (cid:39) Ω ∞ M n − , proving theresult. We will in this section explain how Theorem A can be used to give explicit consequences for the homologyof the Higman–Thompson groups. Concretely, we will compute the abelianizations and Schur multipliers ofthe Higman–Thompson groups directly from Theorem A, and we will also completely decide which of thegroups are integrally or rationally acyclic. We recover old results with new methods, as well as prove newresults. In particular, we prove the acyclicity of the Thompson group V.The results in this section are based on computations of the homology groups of the infinite loop space of theMoore spectrum M n with classical methods from homotopy theory. Given a spectrum X , the stable homotopygroups π ∗ X agree with the (unstable) homotopy groups π ∗ Ω ∞ X of the underlying infinite loop space Ω ∞ X .45he situation is different for homology, however. The homology of the Moore spectrum M n is, up to a shift,the homology of the mod n Moore space: H M n ∼ = Z / n and H d M n = d (cid:54) =
0. In contrast, the homologyof the underlying infinite loop space Ω ∞ M n is more difficult to compute. We will here give some partialcomputations of these homology groups. Further computations can be obtained by working harder. In this section, we compute H and H as well as the first non-trivial homology group of V n , r by computingthese groups for Ω ∞ M n − . We confirm and extend the known results from the literature. Proposition 6.1.
For all n (cid:62) and r (cid:62) there are isomorphisms H ( V n , r ) ∼ = (cid:40) n even Z / n odd. Proposition 6.2.
For all n (cid:62) , we have that H d ( V n , r ) ∼ = (cid:40) < d < p − Z / p d = p − for p the smallest prime dividing n − , and H q − ( V n , r ) (cid:54) = for q any prime dividing n − . Proposition 6.1 is essentially the case p = Proof of Propositions 6.1 and 6.2.
By Theorem A, it is equivalent to compute these homology groupsfor Ω ∞ M n − . The space Ω ∞ M n − is a connected infinite loop space, and soH Ω ∞ M n − ∼ = π Ω ∞ M n − ∼ = π M n − . As π S ∼ = Z /
2, the latter can easily be computed from the cofiber sequence (5.1) of spectra to be the cokernelof multiplication by n − π S = Z /
2, which proves the first proposition.For the second proposition, we assume that n (cid:62) n − (cid:62)
2. Let p be a prime. The p -parts of the homotopy groups of the sphere spectrum are zero between dimension 0 and 2 p − π p − ( S ) ⊗ Z ( p ) ∼ = Z / p . Now multiplication by n − Z / p if and only if p doesnot divide n −
1, and if it does, the map is zero. The result follows using the same long exact sequence nowfor homotopy groups with coefficients and Hurewicz’s theorem.The following result recovers and extends the computation of Kapoudjian [Kap02], who worked out thecase r = Proposition 6.3.
For all n (cid:62) and r (cid:62) there are group isomorphisms H ( V n , r ) ∼ = n even Z / n ≡ mod Z / ⊕ Z / n ≡ mod . roof. Again, by Theorem A, it is equivalent to compute these homology groups for Ω ∞ M n − . Let us firsttick off the case when n is even, so that n − n (cid:62) n − n , r is in degree 2 · − = n − vanishes. If n =
2, the group vanishes because multiplication by n − M the trivialspectrum. (See also the more general Theorem 6.4.)Let us now assume that n is odd, and write X = Ω ∞ M n − . We have that π X ∼ = Z /
2. Consider the Postnikovtruncation X , with the same first and second homotopy groups as X . We have H X ∼ = H X . As X is aninfinite loop space, its first k -invariant vanishes (see [Arl90]), so that X (cid:39) K ( π X , ) × K ( π X , ) . Now the K¨unneth theorem givesH X ∼ = H K ( π X , ) ⊕ H K ( π X , ) ∼ = H K ( π X , ) , given that π X ∼ = Z / . As H K ( π X , ) ∼ = π X , we obtain thatH Ω ∞ M n − ∼ = π Ω ∞ M n − ∼ = π M n − , which reduces the question to stable homotopy theory as above. When n is odd, the homotopy cofibresequence (5.1) only shows that this group is of order 4. We need a further computation to identify the group.Let us assume that n ≡ n − M ( n − ) / −→ M n − −→ M from the octahedral axiom to see that π M n − ∼ = π M , and the latter group is known to be cyclic of order 4,see [Muk66, Thm. 3.2].Lastly, if n ≡ n − k = k · j : M → M k that has even degree on the bottom cell and odd degree on the top cell. Analyzing the diagram · · · KO ( Σ S ) ∼ = Z / (cid:111) (cid:111) KO ( Σ M ) ∼ = Z / (cid:111) (cid:111) KO ( Σ S ) ∼ = Z / (cid:111) (cid:111) · · · (cid:111) (cid:111) · · · KO ( Σ S ) ∼ = Z / (cid:111) (cid:111) (cid:79) (cid:79) KO ( Σ M k ) ∼ = ? (cid:111) (cid:111) j ∗ (cid:79) (cid:79) KO ( Σ S ) ∼ = Z / (cid:111) (cid:111) ∼ = (cid:79) (cid:79) · · · (cid:111) (cid:111) shows that j ∗ cannot be zero or epi, so that KO Σ M k must split into Z / We now deduce global results about the homology of the groups V n , r .The following result has been suggested by Brown [Bro92, Sec. 6]. Theorem 6.4.
For all r (cid:62) , the Thompson group V ∼ = V , r is integrally acyclic: H d ( V ) = H d ( V , r ) = for all d (cid:54) = .Proof. For n = n − = M , is contractible, and the homologyof the infinite loop space vanishes. 47 heorem 6.5. For all n (cid:62) and r (cid:62) , the group V n , r is rationally but not integrally acyclic: H d ( V n , r ) ⊗ Q = for all d (cid:54) = , but H p − ( V n , r ) (cid:54) = for any prime p such that p divides n − .Proof. For n (cid:62) n − M n − isrationally contractible, and the rational homology groups vanish. This proves the first part of the statement.The second part of the statement is given by Proposition 6.2. Remark 6.6.
In the case n =
2, rationally acyclicity of the Thompson group has earlier been shown byBrown [Bro92, Thm. 4], where the author also indicated that his proof can be adapted to prove the case n (cid:62) n =
2, still only rationally, was later reproved by Farley in [Far05].We end by mentioning a consequence of our work for the commutator subgroups. When n is odd, Proposi-tion 6.1 implies that the commutator subgroup V + n , r of V n , r is an index-two subgroup. Let ˜ Ω ∞ K ( Cantor × A ) denote the universal cover of the space Ω ∞ K ( Cantor × A ) . Shapiro’s Lemma, Theorem A and Theorem 3.6applied to the twisted coefficients M = Z H ( V n , r ) ∼ = Z H ( Ω ∞ K ( Cantor × A )) give the following: Corollary 6.7.
There are homology isomorphisms H ∗ ( V + n , r ) ∼ = H ∗ ( V + n , ∞ ) ∼ = H ∗ ( ˜ Ω ∞ K ( Cantor × A )) . In particular, the groups V + n , r and V + n , ∞ are not acyclic when n is odd.Proof. See Sections 3.1 and 3.2 of [R-WW17].This answers a question of Sergiescu.
Acknowledgment
This research has been supported by the Danish National Research Foundation through the Centre for Sym-metry and Deformation (DNRF92) in Copenhagen. Parts of this paper were conceived while the authorswere visiting the Hausdorff Research Institute for Mathematics (HIM) in Bonn, and the paper was revisedduring a visit at the Newton Institute in Cambridge. We thank both institutes for their support. The authorswould like to thank Dustin Clausen and Oscar Randal-Williams for pointing out gaps in early versions ofthis paper, and the referee for a report that helped us improving the paper. The first author would also like tothank Ricardo Andrade, Ken Brown, Bjørn Dundas, Magdalena Musat, Martin Palmer, and Vlad Sergiescufor conversations related to the subject of this paper.
References [Arl90] D. Arlettaz. The first k -invariant of a double loop space is trivial. Arch. Math. 54 (1990) 84–92.[BP72] M. Barratt, S. Priddy. On the homology of non-connected monoids and their associated groups.Comment. Math. Helv. 47 (1972) 1–14. 48Bro87] K.S. Brown. Finiteness properties of groups. J. Pure Appl. Algebra 44 (1987) 45–75.[Bro92] K.S. Brown. The geometry of finitely presented infinite simple groups. Algorithms and classifi-cation in combinatorial group theory (Berkeley, CA, 1989) 121–136. Math. Sci. Res. Inst. Publ.23. Springer, New York, 1992.[BG84] K.S. Brown, R. Geoghegan. An infinite-dimensional torsion-free FP ∞∞