aa r X i v : . [ m a t h . A T ] J un A METHOD FOR INTEGRAL COHOMOLOGY OF POSETS
ANTONIO D´IAZ RAMOS Introduction and summary
Homotopy type of partially ordered sets (poset for short) play a crucial role in alge-braic topology. In fact, every space is weakly equivalent to a simplicial complex which,of course, can be considered as a poset. Posets also arise in more specific contextsas homological decompositions [10, 6, 16, 20] and subgroups complexes associated tofinite groups [7, 26, 4]. The easy structure of a poset has led to the development ofseveral tools to study their homotopy type, including the remarkable Quillen’s theo-rems [25] and their equivariant versions by Th´evenaz and Webb [28]. In spite of theirapparent simplicity posets are the heart of many celebrated problems: Webb’s con-jecture (proven in [27] and generalized in [21]), the unresolved Quillen’s conjecture onthe p -subgroup complex (see [1]), or the fundamental Alperin’s conjecture (see [19]).In this paper we propose a method to compute integral cohomology of posets. Thistoolbox will be applicable as soon as the poset has certain local properties. Moreprecisely, we will require certain structure on the category under each object of theposet. By means of homological algebra of functors we prove that, in the presence ofthese local structures, the cohomology of the poset is that of a co-chain complex(1) 0 → M d → M d → M d → . . . , where M n is free with one generator for each object of “degree n ” of the poset. If thelocal structure is shared at a global level by the whole poset, further developmentsshow that the cohomology of the poset is that of a co-chain complex(2) 0 → B → B → B → . . . , where B n is free with one generator for each “critical” object of “degree n ” of theposet. We also obtain the inequalities(3) b n ≤ rk B n and(4) b n − b n − + . . . + ( − n b ≤ rk B n − rk B n − + . . . + ( − n rk B . where the b · are the Betti numbers of the geometrical realization of the poset.The complex (1) applies to simplex-like posets, i.e. posets such that the categoryunder any object is isomorphic to (the inclusion poset of) a simplex. The notion ofsimplex-like poset is half-way between semi-simplicial complexes (as defined originallyby Eilenberg and Zilber in [11]) and simplicial complexes in the classical sense. In Date : October 28, 2018. this latter case the co-chain complex (1) reduces to the usual simplicial co-chaincomplex for integral cohomology. Notice that all the examples in the first paragraphare simplex-like posets, but not all of them are simplicial complexes.Equations (3) and (4) closely resemble weak and strong Morse inequalities and, infact, whenever the poset is a simplicial complex equipped with a Morse function (see[24, 13]) our complex (2) is similar to the associated Morse complex on the criticalsimplices. It is interesting to point that while the Morse complex is obtained afterreconstructing the poset by homotopical gluing through critical points of the Morsefunction, our complex here is directly achieved through homological algebra.As first application we give an alternative proof of (the cohomological part of)Webb’s conjecture for saturated fusion systems (already proven in [21]). Furtherapplications are related to Coxeter groups: we prove that the Coxeter complex hasthe cohomology of a sphere if the group is finite and that of a point if the group isinfinite.The layout of the paper is as follows: Section 2 contains preliminaries about gradedposets and homological algebra on the category of functors. In Section 3 we definethe local structure (“local covering family”) we require on the under categories of aposet and study its properties. Further application of these features leads, throughSection 4, to a sequence of functors to compute the integral cohomology of a gradedposet. This culminates in the co-chain complex (1). The global structure on the poset(“global covering family”) is defined and used in Section 5 to obtain co-chain complex(2) and the inequalities (3) and (4). In Section 6 we show how simplex-like posets fitin this context. As an example we give a poset model of the real projective plane.Next, Section 7 is devoted to show the interplay between Morse theory and globalcovering families. Finally, Webb’s conjecture is proven in Section 8 while Coxetergroups are treated in Section 9.
Notation:
By the symbol P we denote a category which is a poset (see Section2 below), and their objects will be denoted by p, q, . . . ∈ Ob( P ). All posets will begraded, which means roughly that objects have an assigned degree (see below forprecise definitions). Then P n = Ob n ( P ) denotes the set of objects of degree n . ByAb and Ab P we denote the category of abelian groups and of functors from P to Abrespectively. If S is a set then | S | denotes the number of elements in S . For a category C , we denote its opposite category by C op and its realization by |C| (the geometricalrealization of the (simplicial set) nerve of the category C ). The functor c Z : C →
Abis the functor which sends any object to Z and any morphism to the identity on Z . Acknowledgements:
I would to thank my Ph.D. supervisor A. Viruel for hissupport during the development of this work, and for reading and listening to so manydifferent versions of it. Also, thanks to S. Shamir and F. Xu for making commentson it. 2.
Preliminaries
In this section we introduce some homological algebra on the abelian category offunctors Ab P from a given category P to the category Ab of abelian groups. We shall METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 3 work over a special kind of categories P called graded partial ordered sets (definedbelow). As well we define the cohomology of P with coefficients F ∈ Ab P and showa “shifting argument” to compute this cohomology. This method will be developedsystematically in the next section in case F takes free abelian groups as values. Definition 2.1. A poset is a category P in which, given objects p and p ′ , • there is at most one arrow p → p ′ , and • if there are arrows p → p ′ and p ′ → p then p = p ′ .Clearly a poset defined as above is exactly the same as a set endowed with a partialorder. If P is a poset we will write sometimes p ≤ p ′ to denote that there is an arrow p → p ′ , and p < p ′ if this is the case and p = p ′ . To define graded posets we first needto introduce preceding relations: Definition 2.2. If P is a poset and p < p ′ then p precedes p ′ if p ≤ p ′′ ≤ p ′ impliesthat p = p ′′ or p ′ = p ′′ . Definition 2.3.
Let P be a poset. P is called graded if there is a function deg : Ob( P ) → Z , called the degree function of P , which is order reversing andsuch that if p precedes p ′ then deg ( p ) = deg ( p ′ ) + 1. If p ∈ Ob( P ) then deg ( p ) iscalled the degree of p .Notice that for a given poset P and an object p ∈ Ob( P ) the under category( p ↓ P ) (see [22]) is exactly the full subcategory with objects { p | p ≤ p } . We alsodefine Definition 2.4.
Let P be a graded poset, let p ∈ Ob( P ), let n ∈ Z and let S ⊆ Z .Then we define ( p ↓ P ), ( p ↓ P ) ∗ , ( p ↓ P ) n and ( p ↓ P ) S as the full subcategoriesof P with objects { p | p ≤ p } , { p | p < p } , { p | p ≤ p, deg ( p ) = n } and { p | p ≤ p, deg ( p ) ∈ S } respectively.From the homotopy viewpoint, restricting to graded poset means no loss: anytopological space is weakly homotopy equivalent to a CW -complex which, in turn,is homotopy equivalent to a simplicial complex. This last can be seen as a gradedposet in which the degree function is the dimension of its simplices (more preciselyand according to our definition of order reversing degree function, the opposite of thesimplicial complex is the graded poset).The category Ab P , with objects the functors F : P →
Ab and functors the nat-ural transformations between them, is an abelian category in which the short exactsequences are the object-wise ones (see [23]). Because it contains enough injectivesobjects (see [29]) we can define the right derived functors of the inverse limit functorlim ←− : Ab P → Ab. For a given functor F ∈ Ab P we define the cohomology of P withcoefficients in F as H ∗ ( P ; F ) = lim ←− ∗ F. If F is the constant functor of value M ∈ Ab then H ∗ ( P ; F ) equals the cohomology ofthe topological realization |P| , of P , with trivial coefficients M . A functor F ∈ Ab P will be called acyclic if H ∗ ( P ; F ) = 0 for ∗ > ANTONIO D´IAZ RAMOS
Remark 2.5.
In the rest of the paper we assume the following on any graded poset P : • the set { p | p ≤ p } is finite for any p ∈ Ob( P ), and • the degree function deg of P takes values { . . . , , , , } .The second condition above is equivalent, by definition, to the poset P being boundedabove , as we can always consider translations of a degree function ( deg ′ = deg + c ).These conditions will be clearly fulfilled in the applications.Next we introduce some acyclic objects in Ab P . Definition 2.6.
Let P be a graded poset and F ∈ Ab P . Then F ′ is the functordefined by F ′ ( p ) = Y p ∈ ( p ↓P ) F ( p )on objects p ∈ Ob( P ). For a morphism p → p the summand F ( p ) correspondingto p ≤ p is mapped by the identity map to itself at the summand corresponding to p ≤ p if p ≤ p ≤ p . Otherwise it is mapped to zero.Notice that F ′ is built in a similar way as enough injectives are shown to exist inAb P (see [8, 243ff.]). The next result summarizes some interesting properties of thefunctor F ′ : Theorem 2.7.
Let F : P → Ab be a functor over a graded poset. Then the followingholds:(a) for each G ∈ Ab P there is a bijection Hom Ab P ( G, F ′ ) ϕ ∼ = / / Q p ∈ Ob( P ) Hom Ab ( G ( p ) , F ( p )) , (b) lim ←− F ′ ∼ = Q p ∈ Ob( P ) F ( p ) , (c) F ′ is acyclic, and(d) F ′ is injective in Ab P if and only if F ( p ) is injective in Ab for each p ∈ Ob( P ) .Proof. For the first part, the bijection ϕ : Hom Ab P ( G, F ′ ) → Q p ∈ Ob( P ) Hom Ab ( G ( p ) , F ( p ))is given by ϕ ( µ ) p = π p η p , where π p : F ′ ( p ) → F ( p ) is the projection into the summand corresponding to p ≤ p .The second part is consequence of (a) and of the isomorphism of abelian groupslim ←− H ∼ = Hom Ab P ( Z , H )for any H ∈ Ab P , where Z is the functor P →
Ab of constant value the integers.Part (c) is proven in [9]. There it is also proven that F ′ is injective if and only if F takes as values injective abelian groups. (cid:3) METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 5
To finish this section we discuss shortly how to compute higher limits via a “shiftingargument”. Fix F ∈ Ab P and consider Ker F ∈ Ab P defined byKer F ( p ) = \ p ∈ ( p ↓P ) ∗ Ker F ( p → p )and such that sends non-identity morphisms to zero. By Theorem 2.7a) if we have afamily of maps { τ p : F ( p ) → Ker F ( p ) } p ∈ Ob( P ) then there is a natural transformation λ : F ⇒ Ker ′ F . If λ is object-wise injective then we obtain a short exact sequence inAb P ⇒ F ⇒ Ker ′ F ⇒ G ⇒ , where G is the object-wise co-image of λ . By Theorem 2.7c) Ker ′ F is acyclic, and thusthe long exact sequence of the derived functors lim ←− ∗ gives lim ←− i F = lim ←− i − G for i > ←− F = Coim { lim ←− Ker ′ F → lim ←− G } . The conditions in the next definition ensurethat we can build such a natural transformation λ which is object-wise injective: Definition 2.8.
Let P be a graded poset, let F : P →
Ab be a functor and let n ∈ Z .We say that F is n -condensed if(a) F ( i ) = 0 if deg ( i ) < n , and(b) Ker F ( i ) = 0 if deg ( i ) > n .If the functor F is n -condensed then we can consider the natural transformation λ : F ⇒ Ker ′ F given by Theorem 2.7a) for the maps τ p : F ( p ) → Ker F ( p ) τ p = (cid:26) F ( p ) if deg ( p ) = n F ( p ) = (cid:26) F ( p ) if deg ( p ) = n ′ F takes valueson objects(5) Ker ′ F ( p ) = Y p ∈ ( p ↓P ) n F ( p ) . The homomorphism λ p : F ( p ) → Ker ′ F ( p ) is given by λ i = Y p ∈ ( p ↓P ) n F ( p → p ) : F ( p ) → Ker ′ F ( p ) = Y p ∈ ( p ↓P ) n F ( p ) . So λ i is a kind of “diagonal”. An easy induction argument on deg ( p ) ∈ { n, n + 1 , . . . } shows that λ is a monic natural transformation and we obtain Lemma 2.9.
Let F : P → Ab be an n -condensed functor. Then there is a shortexact sequence + F λ + Ker ′ F + G + . ANTONIO D´IAZ RAMOS
On the object p , G takes the value(6) G ( p ) = Y p ∈ ( p ↓P ) n F ( p ) /λ p ( F ( p )) . It is clear that G satisfies condition (a) of Definition 2.8 for n + 1, but in generalcondition (b) does not hold for G and n + 1. More precisely, if deg ( p ) > n + 1 thenKer G ( p ) = 0 is equivalent to the natural map F ( p ) → lim ←− ( p ↓P ) ∗ F being an isomorphism. This natural map is a monomorphism by condition (b) of F being n -condensed. So, Ker G ( p ) = 0 if and only if F ( p ) → lim ←− ( p ↓P ) ∗ F is surjective.We summarize these results in the following: Lemma 2.10.
Let F : P → Ab be an n -condensed functor. Then there is a shortexact sequence + F λ + Ker ′ F + G + . Moreover, G is ( n + 1) -condensed if and only if for each object p of degree greaterthan n + 1 , we have F ( p ) ∼ = → lim ←− ( p ↓P ) ∗ F . Example 2.11.
Consider the graded poset P with shape · / / AAAAA · / / BBBBB · · / / > > }}}}} · / / > > ||||| · , where the subindexes denote the degree of the objects. The geometrical realization |P| has the homotopy type of a two dimensional sphere S . Consider now the functor F : P →
Ab with values Z × / / × (cid:25) (cid:25) Z × / / × (cid:26) (cid:26) Z Z × / / × E E (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) Z × / / × D D (cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) Z . The functors Ker F and Ker ′ F have values / / (cid:22) (cid:22) .......... / / (cid:24) (cid:24) Z Z ⊕ Z
12 1 / / @@@@@@@@@@@@@ Z ⊕ Z / / (cid:28) (cid:28) :::::::::::: Z and0 / / H H (cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16)(cid:16) / / F F (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) Z Z ⊕ Z
12 1 / / > > ~~~~~~~~~~~~~ Z ⊕ Z / / B B (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) Z METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 7 respectively. As Ker F is concentrated in degree 0 the functor F is 0-condensed. Thefunctor G from Lemma 2.9 is given by: Z ⊕ Z π ⊕ / / π ⊕ BBBBBBBBBBB Z ⊕ Z
12 0 / / (cid:28) (cid:28) Z ⊕ Z π ⊕ / / π ⊕ > > ||||||||||| Z ⊕ Z
12 0 / / B B (cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5) , where π : Z ։ Z is the quotient map. Notice that G is not 1-condensed as Ker G takes the value Z on the two objects of degree 2 of P . In fact, if deg ( p ) = 2, thenthe map F ( p ) = Z → lim ←− ( p ↓P ) ∗ F = Z clearly is not an isomorphism. A straightforward computation shows thatlim ←− F = Z andlim ←− F = Coim { lim ←− Ker ′ F → lim ←− G } = Coim { Z ⊕ Z → Z ⊕ Z ⊕ Z } = Z . The next section is devoted to finding “local” conditions on the shape of the poset P such that G and the subsequent functors obtained by applying Lemma 2.9 to afunctor F that takes free abelian groups as values turn out to be condensed.3. Local covering families
In this section we study a bit further the condition given in Lemma 2.10 whendealing with a functor F which satisfies the following condition Definition 3.1.
Let F : P →
Ab be a functor where P is a graded poset. Wesay that F is free if F ( p ) is a finitely generated free abelian group for each object p ∈ Ob( P ) (not to be confused with a free object in the abelian category Ab P ).We also shall need the following Definition 3.2.
Let A f → B be a map between free abelian groups. We say that f is pure if Coker( f ) is a free abelian group (see [15]).If A ∼ = Z n is a finitely generated free abelian group we call rk( A ) def = n . Thefollowing property of pure maps is straightforward, and will be used repeatedly inwhat follows, Lemma 3.3.
Let A f → B be a map in Ab between free abelian groups of the samerank. If f is pure and injective then it is an isomorphism. Now consider the condition in Lemma 2.10 again: fix p of degree greater than n + 1 and consider the map given by restrictionlim ←− ( p ↓P ) ∗ F = Hom ( p ↓P ) ∗ ( c Z , F ) → Y p ∈ J F ( p ) ANTONIO D´IAZ RAMOS over a subset J ⊆ ( p ↓ P ) n . If this restriction map turns out to be injective (noticethat it is injective for J = ( p ↓ P ) n because F is n -condensed) then the composition F ( p ) → lim ←− ( p ↓P ) ∗ F → Y p ∈ J F ( p )is also injective. If F is a free functor (Definition 3.1) then both groups F ( p ) and Q p ∈ J F ( p ) are free abelian groups (because we are assuming Remark 2.5). If the map F ( p ) → Y p ∈ J F ( p )is pure then, by Lemma 3.3, the condition rk F ( p ) = P p ∈ J rk F ( p ) implies that thiscomposition is an isomorphism and so F ( p ) ∼ = → lim ←− ( p ↓P ) ∗ F . Thus we study thesubsets J ⊆ Ob( P ) that make this restriction map a pure monomorphism: Definition 3.4.
Let P be a graded poset with degree function deg . A family ofsubsets J = { J p n } p ∈ Ob( P ), 0 ≤ n ≤ deg ( p ) with J p n ⊆ ( p ↓ P ) n is a local coveringfamily ifa) For each p and 0 ≤ n < deg ( p ) it holds that S p ∈ J p n +1 ( p ↓ P ) n = ( p ↓ P ) n b) For each p , 0 ≤ n < deg ( p ) and p ∈ J p n +1 it holds that J pn ⊆ J p n Notice that the definition above does not depend on a functor defined over thecategory P . Also, we have J p deg ( p ) = { p } by a). The next definition states therelation we expect between a local covering family and an n -condensed free functor Definition 3.5.
Let P be a graded poset, J be a local covering family and F : P →
Ab be an n -condensed free functor. We say that F is J -determined if for any object p of degree greater than n + 1 the restriction maplim ←− ( p ↓P ) ∗ F → Y p ∈ J p n F ( p )is a monomorphism and the map F ( p ) → Y p ∈ J p n F ( p )is pure. If deg ( p ) = n + 1 then we require the last map above to be a pure monomor-phism.The main feature of local covering families is that they allow freeness plus J -determinacy to pass from F to G . For an object p with deg ( p ) ≥ n + 1 notice thatthe map F ( p ) → Y p ∈ J p n F ( p )is a pure monomorphism as consequence of Definition 3.5. The condition of Definition3.5 for deg ( p ) = n + 1 is added in order to obtain that G is a free functor. Notice METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 9 that the following proposition restricts to functors which take free abelian groups asvalues.
Proposition 3.6.
Let P be a graded poset and J a local covering family. Assumethat F : P → Ab is n -condensed, free and J -determined and consider the functor G defined by + F λ + Ker ′ F + G + . If for each object p with deg ( p ) ≥ n + 1 it holds that rk F ( p ) = P p ∈ J p n rk F ( p ) ,then G is ( n + 1) -condensed, free and J -determined.Proof. Notice that the hypothesis implies that for any object p of degree deg ( p ) >n + 1 the two maps F ( p ) → lim ←− ( p ↓P ) ∗ F → Y p ∈ J p n F ( p )are isomorphisms. In particular, F ( p ) ∼ = → lim ←− ( p ↓P ) ∗ F and so G is ( n + 1)-condensed.If deg ( p ) = n + 1 then the map F ( p ) → Y p ∈ J p n F ( p )is an isomorphism by hypothesis. Next we prove that G is a free functor. Considerany p ∈ Ob( P ) with deg ( p ) ≥ n + 1 (if deg ( p ) < n + 1 then G ( p ) = 0) and the shortexact sequence of abelian groups0 → F ( p ) λ p → Ker ′ F ( p ) π p → G ( p ) → . Then it is straightforward that the map s p : Ker ′ F ( p ) = Y q ∈ ( p ↓P ) n F ( q ) ։ Y q ∈ J pn F ( q ) ∼ = → F ( p )is a section of λ p , i.e. s p ◦ λ p = 1 F ( p ) (since the restriction map F ( p ) → Q q ∈ J pn F ( q )is injective). This implies that the short exact sequence above splits and so G ( p ) is asubgroup of the free abelian group Ker ′ F ( p ), and thus it is free as well.Next we prove that G is J -determined. Take p of degree n = deg ( p ) greater than n + 2. We first check that the restriction maplim ←− ( p ↓P ) ∗ G → Y p ∈ J p n +1 G ( p )is injective. Consider any element ψ ∈ lim ←− ( p ↓P ) ∗ G = Hom Ab P ( Z , G ) which is in thekernel of the restriction map above. Notice that, as deg ( p ) > n + 2, we can considerthe subset J p n +2 ⊆ ( p ↓ P ) ∗ . If for any q ∈ J p n +2 it holds that ψ q (1) = 0 then ψ = 0because of Definition 3.4a) and because G is ( n + 1)-condensed. Thus take q ∈ J p n +2 . We want to see that x def = ψ q (1) = 0. Recall the short exactsequence of abelian groups0 → F ( q ) λ q → Ker ′ F ( q ) π q → G ( q ) → y ∈ Ker ′ F ( q ) such that π q ( y ) = x . Recall that Ker ′ F ( q ) = Q p ∈ ( q ↓P ) n F ( p ) anddenote by α p : q → p the unique arrow from q to p for p ∈ ( q ↓ P ) n .Now consider the restriction y | J qn ∈ Q p ∈ J qn F ( p ). Because deg ( q ) = n + 2 > n + 1the map F ( q ) → Q p ∈ J qn F ( p ) is an isomorphism by hypothesis. Then there exists aunique z ∈ F ( q ) with F ( α p )( z ) = y p for each p ∈ J qn ⊆ ( q ↓ P ) n . If we prove that F ( α p )( z ) = y p for each p ∈ ( q ↓ P ) n then λ q ( z ) = y . This implies that x = π q ( y ) = π q ( λ q ( z )) = 0 and completes the proof.Thus take p ∈ ( q ↓ P ) n . By Definition 3.4a) there is β p : p ′ → p with p ′ ∈ J qn +1 .Write β p ′ : q → p ′ for the unique arrow from q to p ′ . It holds that α p = β p ◦ β p ′ .By Definition 3.4b) we have that J qn +1 ⊆ J p n +1 . Thus G ( β p ′ )( x ) = G ( β p ′ )( ψ q (1)) = ψ p ′ (1) = 0 as ψ is in the kernel of the restriction map. The short exact sequence0 → F ( p ′ ) λ p ′ → Ker ′ F ( p ′ ) π p ′ → G ( p ′ ) → t p ′ ∈ F ( p ′ ) such that λ p ′ ( t p ′ ) = Ker ′ F ( β p ′ )( y ). Consider z p ′ = F ( β p ′ )( z ). We have that z p ′ and t p ′ have the same image by the restriction maplim ←− P p ′∗ F → Y p ∈ J p ′ n F ( p )because J p ′ n ⊆ J qn . Because F is J -determined then this restriction map is a monomor-phism and so z p ′ = t p ′ . This implies that F ( α p )( z ) = F ( β p ◦ β p ′ )( z ) = F ( β p )( z p ′ ) = F ( β p )( t p ′ ) = y p and the proof of the restriction map being injective is finished.Now we check that the map ω : G ( p ) → Y p ∈ J p n +1 G ( p )is pure. Take z ∈ Q p ∈ J p n +1 G ( p ) such that there exists x ∈ G ( p ) with m · z = ω ( x )for some m = 0. We have to check that there exists x ′ ∈ G ( p ) with z = ω ( x ′ ), orequivalently, that x = m · x ′ for some x ′ ∈ G ( p ). Recall once more the short exactsequence of abelian groups0 → F ( p ) λ p → Ker ′ F ( p ) π p → G ( p ) → y ∈ Ker ′ F ( p ) with π p ( y ) = x . We are going to build h ∈ F ( p ) such that y − λ p ( h ) = m · y ′ , i.e., such that for any p ∈ ( p ↓ P ) n the element ( y − λ p ( h )) p = y p − F ( p → p )( h ) ∈ F ( p ) is divisible by m . This implies that x = m · x ′ with x ′ = π p ( y ′ ).Notice that by hypothesis for each q ∈ J p n +1 , G ( p → q )( x ) = m · z q ∈ G ( q ).This implies that there exist h q ∈ F ( q ) and y q ∈ Ker ′ F ( q ) such that Ker ′ F ( p → METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 11 q )( y ) − λ q ( h q ) = m · y q , i.e., such that for each p ∈ ( q ↓ P ) n ⊆ ( p ↓ P ) n we have that y p − F ( q → p )( h q ) = m · ( y q ) p ∈ F ( p ) (it is enough to take y q with π q ( y q ) = z q ).To build h we use the map τ : Y p ∈ J p n F ( p ) ∼ = → F ( p )given by hypothesis, which is the inverse of the map F ( p ) → Y p ∈ J p n F ( p ) . For each p ∈ J p n ⊆ ( p ↓ P ) n choose, by Definition 3.4a), q ( p ) ∈ J p n +1 such that thereis an arrow q ( p ) → p . Then set η p = F ( q ( p ) → p )( h q ( p ) ) ∈ F ( p ), where h q ( p ) is builtas before. Define h def = τ ( η ). By construction F ( p → p )( h ) = F ( q ( p ) → p )( h q ( p ) ) foreach p ∈ J p n (but not for an arbitrary p ∈ ( p ↓ P ) n ).With this definition for h we check now that y p − F ( p → p )( h ) is divisible by m for each p ∈ ( p ↓ P ) n . This finishes the proof. Fix p ∈ ( p ↓ P ) n and q p ∈ J p n +1 suchthat there is an arrow q p → p (we are not assuming that q p = q ( p ) if p ∈ J p n ). Onthe one hand we have by hypothesis that y k − F ( q p → k )( h q p ) = m · ( y q p ) k for each k ∈ P q p n . In particular,(7) y k − F ( q p → k )( h q p ) = m · ( y q p ) k for each k ∈ J q p n . Set h ′ def = F ( p → q p )( h ). Because q p ∈ J p n +1 then, by Definition3.4b), J q p n ⊆ J p n and thus by construction for any k ∈ J q p n y k − F ( p → k )( h ) = y k − F ( q ( k ) → k )( h q ( k ) ) = m · ( y q ( k ) ) k . Notice that F ( p → k )( h ) = F ( q p → k )( F ( p → q p )( h )) = F ( q p → k )( h ′ ). On theother hand, we have obtained(8) y k − F ( q p → k )( h ′ ) = m · ( y q ( k ) ) k for each k ∈ J q p n .Now write η k = ( y q ( k ) ) k − ( y q p ) k for each k ∈ J q p n and write h ′′ = τ ( η ) ∈ F ( q p )where τ is the inverse of the map F ( q p ) → Y p ∈ J qpn F ( p ) . By Equations (7) and (8) it is straightforward that the elements h q p − m · h ′′ and h ′ have the same image by this map. Then, as this map is injective by hypothesis, h ′ = h q p − m · h ′′ . As p ∈ J q p n we have y p − F ( p → p )( h ) = y p − F ( q p → p )( h ′ ) = y p − F ( q p → p )( h q p − m · h ′′ ) , and this equals m · ( y q p ) p + m · F ( q p → p )( h ′′ ) . Thus y p − F ( p → p )( h ) is divisible by m . If deg ( p ) = n + 2 we have to see that the map ω : G ( p ) → Y p ∈ J p n +1 G ( p )is a pure monomorphism. To prove that ω is a monomorphism use the proof abovestarting where ψ q is considered for an arbitrary object q of degree n + 2. The proofof ω being pure is exactly the same as above. (cid:3) Remark 3.7.
Notice that in the conditions of the proposition we have the followingformula for the rank of the free abelian group G ( p ) for deg ( p ) ≥ n + 1rk( G ( p )) = X p ∈ ( p ↓P ) n rk F ( p ) − rk F ( p ) . This is so because of the short exact sequence of free abelian groups0 → F ( p ) λ p → Ker ′ F ( p ) π p → G ( p ) → . Remark 3.8.
In the conditions of the proposition there are isomorphisms F ( p ) ∼ = → Y p ∈ J p n F ( p )for each p with deg ( p ) ≥ n + 1. Moreover, we have built a map s p : Ker ′ F ( p ) → F ( p ) with s p ◦ λ p = 1 F ( p ) . To the homomorphism s p corresponds the monomor-phism G ( p ) δ p / / Ker ′ F ( p )given by π p ( x ) (cid:31) / / x − ( λ p ◦ s p )( x ) , which satisfies π p ◦ δ p = 1 G ( p ) . It is straightforward that, by construction, Im δ p = Q p ∈ ( p ↓P ) n \ J p n F ( p ), and thus G ( p ) δ p ∼ = Y p ∈ ( p ↓P ) n \ J p n F ( p ) . Moreover, x = δ p ( y ) is the only preimage of y by π p which verifies x p = 0 for p ∈ J p n .The main consequence of the previous proposition is that it reduces the problem ofwhether G is ( n + 1)-condensed to some integral equations. Moreover, this procedurecan be applied recursively because the resulting functor G turns out to be ( n + 1)-condensed, free and J -determined, and so the proposition applies to G too. Noticeagain that the ranks of G are given by Remark 3.7. METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 13
Example 3.9.
Consider the graded poset P with shape a / / AAAAA (cid:23) (cid:23) c / / ! ! BBBBBB (cid:24) (cid:24) f d / / ! ! BBBBB = = ||||| g b / / > > }}}}} G G (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) e / / = = |||||| F F (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h , where the subindexes denote the degree of the objects. Notice that P has the homo-topy type of a wedge of 4 2-dimensional spheres W i =1 S i . The sets J f = { f } , J g = { g } , J h = { h } ,J c = { c } , J c = { f } , J d = { d } , J d = { g } , J e = { e } , J e = { h } ,J a = { a } , J a = { c } , J a = { f } , J b = { b } , J b = { e } , J b = { h } . define a local covering family J for P . Consider now the the functor F : P →
Abwith values Z / / (cid:31) (cid:31) ????? (cid:23) (cid:23) ////////// Z / / @@@@@ (cid:23) (cid:23) ////////// ZZ / / (cid:31) (cid:31) ????? > > ~~~~~ ZZ / / @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) Z / / ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) Z , such that all the arrows arriving and departing from c and e are the identity andall the arrows arriving and departing from d are minus the identity. Then F is 0-condensed, free and J -determined. Moreover, equations in Proposition 3.6 hold forany object of degree greater or equal to 1. Thus we obtain a functor G which is1-condensed, free and J -determined. By Remark 3.7 we know the ranks of the freeabelian groups that G takes as values: Z / / BBBBB (cid:24) (cid:24) Z / / @@@@@ (cid:23) (cid:23) Z / / @@@@@ > > ~~~~~ Z / / > > ||||| F F (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z / / > > ~~~~~ G G (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) . Moreover, equations in Proposition 3.6 applied to G hold for any object of degreegreater or equal to 2, and thus we obtain a functor H which is 2-condensed, free and J -determined: Z / / @@@@@ (cid:23) (cid:23) / / (cid:31) (cid:31) >>>>> (cid:23) (cid:23) ////////// / / (cid:31) (cid:31) >>>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Z / / > > ~~~~~ G G (cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14) / / ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) G G (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) , By Remark 3.8 we can identify G ( a ) ∼ = F ( g ) ⊕ F ( h ) = Z g ⊕ Z h . Also G ( c ) ∼ = Z g ⊕ Z h , G ( d ) ∼ = Z f ⊕ Z h and G ( e ) ∼ = Z f ⊕ Z g . By the definition of G as a co-image it is easy to see that G ( a → c ) : Z g ⊕ Z h → Z g ⊕ Z h ( g, h ) ( g, h ) . Also G ( a → d )( g, h ) = ( − g, h − g ) and G ( a → e )( g, h ) = ( − h, g − h ) with respect tothe ordered basis mentioned above. Additional computations lead tolim ←− F = Z , lim ←− F = Coim { lim ←− Ker ′ F → lim ←− G } = Coim { Z → Z } = 0, andlim ←− F = Coim { lim ←− Ker ′ G → lim ←− H } = Coim { Z → Z } = Z . Integral cohomology
In this section we apply the work developed in the preceding sections to compute thecohomology with integer coefficients of the realization of a graded poset P equippedwith a local covering family J .To compute the abelian group H n ( |P| ; Z ) for n ≥ ←− n c Z where c Z : P →
Ab is the functor of constant value Z which sends everymorphism to the identity 1 Z . We begin studying the extra conditions that the localcovering family J must satisfy to apply iteratively the Proposition 3.6 beginning on c Z .First, notice that c Z is 0-condensed (we are assuming 2.5) and free (Definition 3.1).By Definition 3.5, c Z is J -determined as 0-condensed functor if and only if for each p ∈ Ob( P ) with deg ( p ) ≥ J p intersects each connected component of ( p ↓P ) ∗ . The dimensional equation in Proposition 3.6 for p ∈ Ob( P ) with deg ( p ) ≥ c Z ( p ) = 1 = | J p | = P p ∈ J p rk c Z ( p ). Thus, c Z is J -determined asa 0-condensed functor if and only if ( p ↓ P ) ∗ is connected for deg ( p ) ≥ | J p | = 1 for deg ( p ) ≥
1. The successive applications of Proposition 3.6 give, by thedimensional equation in the statement of the Proposition 3.6, the following:
Definition 4.1.
Let P be a graded poset. Define, inductively on n , the number R p n for each object p with deg ( p ) ≥ n by R p = 1 and by R p n = X p ∈ ( p ↓P ) n − R pn − − R p n − for n ≥ Definition 4.2.
Let P be a graded poset and J be a local covering family for P .We say that J is adequate if ( p ↓ P ) ∗ is connected for deg ( p ) ≥
2, and if we havethe equality R p n = X p ∈ J p n R pn for n ≥ deg ( p ) ≥ n + 1. METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 15
Proposition 4.3.
Let P be a graded poset and let J be an adequate local coveringfamily. Then there is a sequence of functors F , F , F , . . . defined by F def = c Z : P → Ab and by the short exact sequence + F n − λ n − + Ker ′ F n − π n + F n + for n = 1 , , , . . . . Moreover, F n is n -condensed, free and J -determined for n ≥ .For deg ( p ) ≥ n we have rk F n ( p ) = R p n . The local properties of a graded poset P equipped with an adequate local coveringfamily J give rise to a sequence F = c Z , F , F ,. . . of functors. Now, we study someproperties of these functors which are independent of the local covering family J .The first point to notice is that the sequence of functors from Proposition 4.3 doesnot depend on the adequate local covering family J . Thus, two or more adequatelocal covering families can be considered for the the same graded poset and they stillgive rise to the same sequence of functors. Next we focus on the short exact sequence0 + F n − λ n − + Ker ′ F n − π n + F n + n ≥ / / lim ←− F n − ι n − / / lim ←− Ker ′ F n − ω n / / lim ←− F n / / H n ( P ; Z ) / / ι n − = g λ n − and ω n = f π n are the induced maps. Notice thatthe three inverse limits appearing above are free abelian groups as the correspondingfunctors take free abelian groups as values. In fact, for the middle term we have theexact description(10) lim ←− Ker ′ F n − ∼ = Y p ∈ Ob n − ( P ) F n − ( p )given by Theorem 2.7. It turns out that there is also a simpler description for lim ←− F n ,which can be interpreted as the analogue in the context of CW -complexes to thefact that the cohomology on degree n depends upon the n + 1 skeleton (recall that F n ( p ) = 0 if deg ( p ) < n ): Lemma 4.4.
Let P be a graded poset and let J be an adequate local covering family.Let c Z , F , F , . . . be the sequence of functors given by Proposition 4.3. Then lim ←− F n ∼ = lim ←− F n | P { n +1 ,n } for each n ≥ .Proof. Consider the restriction maplim ←− F n → lim ←− F n | P { n +1 ,n } . This map is clearly a monomorphism because F n is an n -condensed functor. To seethat it is surjective take ψ ∈ lim ←− F n | P { n +1 ,n } . We want to extend ψ to each p ∈ Ob( P )with deg ( p ) > n + 1. We do it inductively on deg ( p ). Notice that (see the beginning of the proof of Proposition 3.6) F n ( p ) → lim ←− ( p ↓P ) ∗ F n is an isomorphism for deg ( p ) > n + 1. For deg ( p ) = n + 2 we have that j ∈ ( p ↓ P ) ∗ implies that deg ( j ) ≤ n + 1. Then there is a unique way of extending ψ to ψ ( p ).Once we have extended ψ to P { n +2 ,n +1 ,n } we proceed with an induction argument.That the extension that we are building is actually a functor is again due to that F n is n -condensed. (cid:3) Also, from Equations (9) and (10), we have the following formula, analogue to thatof the Euler characteristic for CW -complexes: Lemma 4.5.
Let P be a poset for which exists an adequate local covering family.Then X i ( − i rk H i ( P ; Z ) = X i ( − i X p ∈ Ob i ( P ) R pi . Take up again Equations (9) and (10). We can form the sequence of abelian groups0 → Y p ∈ Ob ( P ) F ( p ) d → Y p ∈ Ob ( P ) F ( p ) d → Y p ∈ Ob ( P ) F ( p ) d → . . . where d n = ι n +1 ◦ w n +1 for n ≥
0. Then it is straightforward that this sequence is aco-chain complex and its cohomology is exactly the cohomology of P with integerscoefficients: Theorem 4.6.
Let P be a graded poset for which exists an adequate local coveringfamily. Then there exists a co-chain complex → Y p ∈ Ob ( P ) F ( p ) d → Y p ∈ Ob ( P ) F ( p ) d → Y p ∈ Ob ( P ) F ( p ) d → . . . of which cohomology is H ∗ ( P ; Z ) . Global covering families
Recall that local covering families were defined as subsets of the local categories( p ↓ P ) for p ∈ Ob( P ), where P is a graded poset. In this section we define globalcovering families by subsets of the whole category P , imitating some of the localfeatures of the local covering families. Definition 5.1.
Let P be a graded poset for which there exists an adequate localcovering family, and consider the sequence of functors F = c Z , F , F ,. . . given byProposition 4.3. A global covering family is a family of subsets K = { K n } n ≥ with K n ⊆ Ob n ( P ) such that(1) the morphism lim ←− F n → Y p ∈ K n F n ( p )is injective for each n ≥
0, and
METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 17 (2) the morphism Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) → Y p ∈ K n F n ( p )is pure for each n ≥ Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) → Y p ∈ K n F ( p )used in the definition is the composition Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) ֒ → Y p ∈ Ob n − ( P ) F n − ( p ) d n − → Y p ∈ Ob n ( P ) F n ( p ) → Y p ∈ K n F n ( p ) . We also have maps Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) ֒ → Y p ∈ Ob n − ( P ) F n − ( p ) d n − → Y p ∈ Ob n ( P ) F n ( p ) → Y p ∈ Ob n ( P ) \ K n F n ( p )and Y p ∈ K n − F n − ( p ) ֒ → Y p ∈ Ob n − ( P ) F n − ( p ) d n − → Y p ∈ Ob n ( P ) F n ( p ) → Y p ∈ K n F n ( p ) , obtained by pre and post composing the differential d n − with appropiate inclusionsand projections. We denote all of them by d n − . Also we fix the following Notation.
We will write elements x ∈ Q p ∈ Ob n ( P ) F n ( p ) as x = y ⊕ z , where y ∈ Q p ∈ Ob n ( P ) \ K n F n ( p ) and z ∈ Q p ∈ K n F n ( p ). Also, d n ( y ⊕ z ) = ( y ⊕ z ) ⊕ ( y + z )where d n ( y ⊕
0) = y ⊕ y and d n (0 ⊕ z ) = z ⊕ z .Next we introduce the Betti numbers associated to a global covering family Definition 5.2.
Let P be a graded poset for which there exists an adequate localcovering family, and let K be a global covering family for P . Then we define, for n ≥
0, the n -th Betti number of the family K as b K = X p ∈ K R p and b K n = X p ∈ K n − R pn − − X p ∈ Ob n − ( P ) R pn − + X p ∈ K n R pn for n ≥ Remark 5.3.
It turns out that a local covering family gives, for each subcategory( p ↓ P ), a global covering family K for ( p ↓ P ) with b K n = 0 for n ≥ b K = 1.More precisely, let P be a graded poset and let J be an adequate local coveringfamily for P . Fix the graded poset P ′ = ( p ↓ P ) for some object p of P . Define K n = J p n ⊆ Ob n ( P ′ ) for 0 ≤ n ≤ deg ( p ) and empty otherwise. Then, using that p is an initial object in P ′ and Remark 3.8, it is straightforward that K = { K · } n ≥ is a global covering family for P ′ . In fact, all the the maps in Definition 5.1 becomeisomorphisms. The integral equalities in Definition 4.2 (adequateness) correspondexactly to the statements b K = 1 and b Kn = 0 for n ≥ b n , the n -thBetti number of P , equals b n = rk lim ←− F n − − X p ∈ Ob n − ( P ) R pn − + rk lim ←− F n . Because the map lim ←− F n → Q p ∈ K n F n ( p ) is injective then we haverk lim ←− F n ≤ rk Y p ∈ K n F n ( p ) = X p ∈ K n R pn Thus, by Definition 5.2, b n ≤ b K n for any n ≥
0. An easy induction argument proves
Proposition 5.4.
Let P be a poset for which exists an adequate local covering familyand a global covering family K . Then b n ≤ b K n and b n − b n − + . . . + ( − n b ≤ b K n − b K n − + . . . + ( − n b K for any n ≥ . Now we focus on the main theorem of this section, which states that we can computethe cohomology of P using a cochain complex which in degree n has a free abeliangroup of rank b K n : Theorem 5.5.
Let P be a graded poset for which exists an adequate local coveringfamily and a global covering family K . Then there exists a cochain complex / / B K / / B K / / B K / / . . . with B K n ∼ = Z b K n for n ≥ , of which cohomology is H ∗ ( P ; Z ) . Before proving the propositon we prove the following definition-lemma
METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 19
Lemma 5.6.
Let P be a graded poset for which there exists an adequate local coveringfamily and a global covering family K . Fix n ≥ , then there is a split short exactsequence → Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) → Y p ∈ K n F n ( p ) → B K n → , where B K n ∼ = Z b K n .Proof. The map λ : Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) → Q p ∈ K n F n ( p ) is pure by definition ofglobal covering family, and thus its cokernel B K n is free and we have a section. So,if we show that this map is a monomorphism we obtain that the sequence is exactand the appropriate rank of the cokernel by the definition of the number b K n . Take x ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) with λ ( x ) = 0. Recall the sequencelim ←− F n − ι n − / / Q p ∈ Ob n − ( P ) F n − ( p ) ω n / / lim ←− F n ι n / / Q p ∈ Ob n ( P ) F n ( p ) , which is exact at Q p ∈ Ob n − ( P ) F n − ( p ). Then ι n ◦ ω n ( x ⊕ p = 0 for each p ∈ K n and, by Definition 5.1 (1), we obtain that ω n ( x ⊕
0) = 0. By exactness there exist y ∈ lim ←− F n − with ι n − ( y ) = x ⊕
0. But, by definition, ( x ⊕ p = 0 for each p ∈ K n − .Then, by Definition 5.1 (1) again, we obtain that y = 0 and x = 0. (cid:3) The leftmost square in the following diagam commutes, and thus, we can find anarrow which closes the rightmost square. We denote this arrow by Ω n , and it is thedifferential on B · induced by the differential d n :0 / / Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) d n − / / − d n − (cid:15) (cid:15) Q p ∈ K n F n ( p ) / / d n (cid:15) (cid:15) B K n / / Ω n (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) / / Q p ∈ Ob n ( P ) \ K n F n ( p ) d n / / Q p ∈ K n +1 F n +1 ( p ) / / B K n +1 / / . The maps Ω · verify Ω n +1 ◦ Ω n = 0 for each n ≥
0, as can be seen by using thepreceding diagram. We have then a chain complex0 / / B K / / B K / / B K / / . . . , and we claim that the homology of this chain complex is the same as that of the chaincomplex in Theorem 4.6.Thus to prove the proposition we shall build for each n ≥ ψ : Ker Ω n → Ker d n / Im d n − of which kernel is Im Ω n − . We denote the class of x ∈ Q p ∈ K n F n ( p ) in B K n as x , andthe class of x ∈ Ker d n in Ker d n / Im d n − by x .Thus, take x ∈ B K n such that Ω n ( x ) = 0. Then there is y ∈ Q p ∈ Ob n ( P ) \ K n F n ( p )such that d n ( x ) = d n ( y ). If d n ( y ⊕
0) = y ⊕ y and d n (0 ⊕ x ) = x ⊕ x then y = x .Define ψ ( x ) = − y ⊕ x. Then d n ( − y ⊕ x ) = − ( y ⊕ y )+( x ⊕ x ) = x − y ⊕
0. Therefore ι n +1 ( ω n +1 ( − y ⊕ x )) = x − y ⊕ ←− F n +1 ι n +1 → Y p ∈ Ob n +1 ( P ) F n +1 ( p ) → Y p ∈K n +1 ( P ) F n +1 ( p )is injective by definition of global covering family, we obtain that x = y and d n ( − y ⊕ x ) = 0, i.e., − y ⊕ x ∈ Ker d n . Notice that the element y chosen above is unique: if y ′ ∈ Q p ∈ Ob n ( P ) \ K n F n ( p ) and d n ( x ) = d n ( y ) then d n ( y − y ′ ) = 0 and by Lemma 5.6 y = y ′ .Now we prove that ψ ( x ) does not depend on the chosen representative x . Thustake x ′ = x + z , where z ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) and d n − (0 ⊕ z ) = z ⊕ z . Then0 = d n ( d n − (0 ⊕ z )) = z , + z , ⊕ z , + z , . Moreover, d n ( y − z ⊕
0) = y − z , ⊕ y − z , and thus d n ( y − z ) = y − z , = y + z , = x + z , = d n ( x ′ ). Then, ψ ( x ′ ) = − y + z ⊕ x ′ . But notice that d n − ( z ⊕
0) = z ⊕ z = ( − y + z ⊕ x ′ ) − ( − y ⊕ x ) and thus − y + z ⊕ x ′ = − y ⊕ x in Ker d n / Im d n − .It is clear that ψ is a homomorphism. Next we prove that it is an epimorphism.Take a ⊕ b ∈ Ker d n / Im d n − . Then 0 = d n ( a ⊕ b ) = a + b ⊕ a + b . Consider b ∈ Q p ∈ K n F n ( p ). Notice that d n ( a ⊕
0) = a ⊕ a and d n (0 ⊕ b ) = b ⊕ b . Hence d n ( b ) = b = − a = d n ( − a ), and thus Ω n ( b ) = 0 and ψ ( b ) = a ⊕ b .To finish the proof of the proposition we show that Ker ψ = Im Ω n − . First take x ∈ B K n with x = Ω n − ( y ) and y ∈ B K n − . There is z ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p )with d n − ( z ) = x − d n − ( y ). If d n − ( z ⊕
0) = z ⊕ z and d n − (0 ⊕ y ) = y ⊕ y then z = x − y and d n ( x ) = x = z , + y , . Take t = − z − y ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p ).Therefore d n ( t,
0) = − z , − y , ⊕ − z , − y , , d n ( t ) = − z , − y , = z , + y , = x and ψ ( x ) = − t ⊕ x with d n ( z ⊕ y ) = z + y ⊕ z + y = − t ⊕ x . This shows thatIm Ω n − ⊆ Ker ψ .Finally we prove that Ker ψ ⊆ Im Ω n − . Take x ∈ B K n such that ψ ( x ) = 0 inKer d n / Im d n − . This means that if y ∈ Q p ∈ Ob n ( P ) \ K n F n ( p ) is such that d n ( x ) = d n ( y ) then there is a ⊕ b ∈ Q p ∈ Ob n − ( P ) F n − ( p ) with d n − ( a ⊕ b ) = − y ⊕ x . Hence a ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) and b ∈ Q p ∈K n − ( P ) F n − ( p ) are such that d n − ( b ) = x − d n − ( a ) and thus Ω n − ( b ) = x .6. Simplex-like posets
In this section we will show that for a big family of posets, which includes simplicialcomplexes, there exists local covering families. We start defining posets which locallyare like simplicial complexes:
METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 21
Definition 6.1.
The category △ n has as objects the faces of the standard n -dimensionalsimplex and arrows the inclusions among faces. For example △ is the category: − o o ^ ^ >>>>>>>> ·△ o o (cid:127) (cid:127) ~~~~~~~ _ _ @@@@@@@ − (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ^ ^ >>>>>>>> ·− o o (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) · . Definition 6.2.
Let P be a poset. Then P is simplex-like if for all p ∈ Ob( P ) thecategory ( P ↓ p ) is isomorphic to △ n for some n .Of course simplicial complexes are simplex-like posets. In fact ([14, 3.1]), simplicialcomplexes are exactly the simplex-like posets such that any two elements which havea lower bound have an infimum, i.e., a greatest lower bound. Another examples ofsimplex-like posets are barycentric subdivisions and, in general, subdivision categoriesin the sense of [21]. We have: Lemma 6.3.
Let P be a simplex-like poset. Then P op is a graded poset.Proof. To see that P is graded recall that for any p ∈ Ob( P ) the subcategory ( P ↓ p )is isomorphic to △ n for some n ≥
0. Define deg ( p ) = n . Then deg : Ob( P op ) → Z isa degree function which assigns 0 to maximal elements and P is graded. (cid:3) Next, we will define local covering families for △ opn . Due to the local shape ofsimplex-like posets this local covering family for △ opn will allow us to define a localcovering family for the opposite category of a simplex-like poset. Lemma 6.4.
There exists an adequate local covering family for △ opn for each n ≥ .Proof. Fix a total order v < v < . . . < v n for the vertices of △ n . For each face σ of △ n and each 0 ≤ m ≤ deg ( σ ) we must define a subset J σm of the m -dimensionalfaces of σ . Take the greatest vertex contained in σ and then define J σm as those m -dimensional faces of σ which contain this vertex. Then it is straightforward that J is an adequate local covering family of △ opn , and that R σm = m X l =0 ( − ( m − l ) (cid:0) deg ( σ )+1 l (cid:1) for σ ∈ △ n and 0 ≤ m ≤ deg ( σ ). (cid:3) Now we reach the main result of this section:
Lemma 6.5. If P is a simplex-like poset then there is an adequate local coveringfamily for the graded poset P op . Proof.
By definition there are isomorphisms of categories ( P ↓ p ) op ≃ ( p ↓ P op ) ≃ △ opn for each p ∈ Ob( P ), and we know that △ opn can be equipped with an adequate localcovering family. To build an adequate local covering family K = { K i p } i ∈ Ob( P op ), 0 ≤ p ≤ deg ( i ) we just have to choose appropriately the isomorphisms ( P ↓ p ) ≃ △ n .Consider the degree function deg : Ob( P op ) → Z defined in Lemma 6.3 and the set T of the maximal elements of P op , i.e., T = { p ∈ Ob( P op ) | deg ( p ) = 0 } . Choose atotal order < for T (suppose T is finite or use the Axiom of Choice [18]). Then, given p ∈ Ob( P op ), consider the subset ( p ↓ P op ) ⊆ T and the restriction (( p ↓ P op ) , < )of the total order from T . There is a unique isomorphism ϕ p : ( p ↓ P op ) ≃ △ opdeg ( p ) which induces an order preserving map(( p ↓ P op ) , < ) ≃ ( △ deg ( p ) ) = { v , v , v , . . . , v deg ( p ) } . Denote by J the local covering family for △ opdeg ( p ) of Lemma 6.4 and define, for0 ≤ m ≤ deg ( p ), K pm = ϕ − p ( J ϕ ( p ) m ) . Then K fulfills condition a) of Definition 3.4 because for 0 ≤ m < deg ( p )( p ↓ P op ) m = ϕ − p (( △ opdeg ( p ) ) m ) = ϕ − p ( [ i ∈ J ϕp ( p ) m +1 ( i ↓ △ opdeg ( p ) ) m ) == [ ϕ − p ( i ) ∈ ϕ − p ( K pm +1 ) ϕ − p (( i ↓ △ opdeg ( p ) ) m ) = [ i ∈ K pm +1 ( i ↓ P op ) m . To check condition b) of Definition 3.4 take q ∈ K pm +1 for some 0 ≤ m < deg ( p )and call J ′ to the local covering family for △ opm +1 of Lemma 6.4. We want to see that K qm ⊆ K pm . Recalling the natural inclusion ( q ↓ P op ) ⊆ ( p ↓ P op ) this is equivalent to ϕ − q ( J ′ ϕ q ( q ) m ) ⊆ ϕ − p ( J ϕ p ( p ) m )and to ψ ( J ′ ϕ q ( q ) m ) ⊆ J ϕ p ( p ) m where ψ = ϕ p ◦ ϕ − q . By construction ψ is order preserving and thus this inclusionholds. (cid:3) Notice that comparing the general expression for R pn in the proof of Lemma 6.4with the binomial expansion of (1 − deg ( p )+1 = 0 we obtain that R pdeg ( p ) = 1 for anyobject p in a simplex-like poset. The following are re-statements of results about localcovering families applied to simplex-like posets: Lemma 6.6.
Let P be a simplex-like poset and consider the graded poset P op (forwhich exists an adequate local covering family by Lemma ). Let K be a globalcovering family for P op . Then the Betti numbers of the family K are given by b K = | K | , METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 23 and for n ≥ , by b K n = | K n − | − | Ob n − ( P ) | + | K n | . Lemma 6.7.
Let P a simplex-like poset. Then X i ( − i rk H i ( P ; Z ) = X i ( − i | Ob i ( P ) | . Again because R pdeg ( p ) = 1, Proposition 4.3 gives rk F deg ( p ) ( p ) = 1 and threfore F deg ( p ) ( p ) = Z . We may identify the object p ∈ P n with its image v v . . . v n underthe isomorphism ( P ↓ p ) ∼ = △ n built in Lemma 6.6. Then the expression for thedifferential in Theorem 4.6 becomes the familiar expression for simplicial complexes: Proposition 6.8.
Let P be a simplex-like poset. Then there exists a co-chain complex → Z Ob ( P ) d → Z Ob ( P ) d → Z Ob ( P ) d → . . . of which cohomology is H ∗ ( P ; Z ) . Under the identifications described above we havefor p = v v . . . v n ∈ Ob n ( P ) , n ≥ and x ∈ Z Ob n − ( P ) d n − ( x ) p = n X j =0 ( − n − j x v v ... ˆ v j ...v n Proof.
Take p and x as in the statement. The short exact sequence considered inRemark 3.7 0 → F n − ( p ) λ p → Ker ′ F n − ( p ) π p → F n ( p ) → → F n − ( v v . . . v n ) λ p → n Y j =0 F n − ( v . . . ˆ v j . . . v n ) π p → F n ( v v . . . v n ) → . By the definition of the differential d n − we have that d n − ( x ) p = π p ( y ), where y = x | { v ... ˆ v j ...v n , j = 0 , . . . , n } ∈ Q nj =0 F n − ( v . . . ˆ v j . . . v n ). As in Remark 3.8 we identify π p ( y )with its unique pre-image y ′ ∈ Q nj =0 F n − ( v . . . ˆ v j . . . v n ) such that y ′ v ... ˆ v j ...v n = 0 for j = 0 , . . . , n −
1. Thus, if we obtain z ∈ F n − ( v v . . . v n ) such that λ p ( z ) v ... ˆ v j ...v n = y v ... ˆ v j ...v n for j = 0 , . . . , n −
1, we will have π p ( y ) ≡ y ′ = y − λ p ( z ).The rest of the proof is by induction on n . For n = 1 the short exact sequence is0 → Z v v λ p → Z v × Z v π p → Z v v → , where λ p ( n ) = ( n, n ). Then, if y = ( y v , y v ), we have z = y v and y ′ = y − λ p ( z ) = ( y v , y v ) − ( y v , y v ) = ( y v − y v , . Now we make the inductive step for n ≥
2. Recall that we want z such that λ p ( z ) v ... ˆ v j ...v n = y v ... ˆ v j ...v n for j = 0 , . . . , n −
1. Again by Remark 3.8 we identify(11) F n − ( v v . . . v n ) ∼ = n − Y j =0 F n − ( v v . . . ˆ v j . . . v n − ) , and, for j = 0 , . . . , n − F n − ( v v . . . ˆ v j . . . v n ) ∼ = F n − ( v v . . . ˆ v j . . . v n − ) , and(13) F n − ( v v . . . v n − ) ∼ = F n − ( v v . . . v n − ) . Recall that the “diagonal” λ p is given by Q nj =0 F n − ( v . . . v n → v . . . ˆ v j . . . v n ). For j ∈ { , . . . , n − } , and with the identifications (11) and (12), the map F n − ( v . . . v n → v . . . ˆ v j . . . v n ) becomes exactly the projection n − Y j =0 F n − ( v v . . . ˆ v j . . . v n − ) → F n − ( v v . . . ˆ v j . . . v n − ) . Thus, z ∈ F n − ( v v . . . v n ) is such that z v v ... ˆ v j ...v n − = y v v ... ˆ v j ...v n (again under theidentifications (11) and (12)). The only thing left to compute is F n − ( v . . . v n → v v . . . v n − )( z ) under the identifications (11) and (13). Define q = v v . . . v n − andconsider the short exact sequence given by Remark 3.70 → F n − ( v v . . . v n − ) λ q → n − Y j =0 F n − ( v v . . . ˆ v j . . . v n − ) π q → F n − ( v v . . . v n − ) → . Now, by the induction hypothesis, π q ( z ) = P n − j =0 ( − n − − j z v v ... ˆ v j ...v n − . Thus, fi-nally, y ′ = y − z = ( y v ...v n − − n − X j =0 ( − n − − j z v v ... ˆ v j ...v n − , , . . . , y ′ = ( n X j =0 ( − n − j y v v ... ˆ v j ...v n , , . . . , . (cid:3) Before finishing this section we show how the symmetry of the category △ n isuseful when checking if a given family of subsets of objects K = { K n } n ≥ fulfillsthe properties of a global covering family. Consider a simplex like poset P and call J to the local covering family for ( p ↓ P op ) ≃ △ opn built in Lemma 6.5. Denote p = v v . . . v n and assume deg ( p ) = n ≥
1, then there are isomorphisms (Remark3.8) F n − ( v v . . . v n ) ∼ = → n − Y j =0 F n − ( v . . . ˆ v j . . . v n )and F n ( v v . . . v n ) ∼ = → F n − ( v v . . . v n − ) . By letting the symmetric group Σ n +1 act on △ opn we map J to other local coveringfamilies for ( p ↓ P op ). In particular we can permute v n with any of the objects METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 25 { v , . . . , v n − } . Recall that the functors F · do not depend on the local covering familychosen. We may summarize this as Remark 6.9.
Let P be a simplex-like poset and let p = v v . . . v n be an object with deg ( p ) = n ≥
1. The maps F n − ( v v . . . v n ) → n Y j =0 ,j = j F n − ( v . . . ˆ v j . . . v n )and F n ( v v . . . v n ) → F n − ( v . . . ˆ v j . . . v n − ) . are both isomorphisms for j = 0 , . . . , n . Example 6.10.
In this example we consider the real projective plane R P and aposet model P of it. It has four 2-cells, six edges and three vertices: w d ??????????????? B o o b (cid:15) (cid:15) e (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) vAa x aCv c (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) O O b / / D f ??????????????? w A o o d d IIIIIIIIIIIII ^ ^ >>>>>>>>>>>>>>> a o o i i TTTTTTTTTTT vB o o d d IIIIIIIIIIIII ^ ^ >>>>>>>>>>>>>>>> b u u kkkkkkkkkkkk o o wc z z uuuuuuuuuuuuu ^ ^ ================ d z z uuuuuuuuuuuuu c c HHHHHHHHHHHHH C (cid:4) (cid:4) (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8) o o i i SSSSSSSSSSS e (cid:4) (cid:4) (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8) i i SSSSSSSSSSS D (cid:4) (cid:4) (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) o o f (cid:4) (cid:4) (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8) o o x This poset is a simplex-like poset and thus there is a local covering family for itsopposite category P op . Notice that it is not a simplicial complex as, for example,there are two edges ( c and e ) with common vertices v and x . Applying Proposition6.8 we obtain that its cohomology is computed by a co-chain complex:0 → Z d → Z d → Z → . Proposition 6.8 gives a description of the differentials d and d . Now consider thefollowing family K = { K , K , K } of subsets of objects of P : K = { x } ,K = { d, e, f } ,K = { A, B, C, D } . Using Remark 6.9 it is straightforward that K is a global covering family. Its Bettinumbers are b K = 1, b K = 1 and b K = 1 and thus, by Theorem 5.5, the cohomologyof R P is that of a cochain complex0 → Z Ω → Z Ω → Z → . The induced differentials are easily computed to be Ω ≡ ≡ × Morse theory
In this section we show how any Morse function on a simplicial complex gives riseto a global covering family. The setup for this section is the discrete Morse theory for CW -complexes that Forman introduces in [12]. We will restrict ourselves to simplicialcomplexes, as the same author does in the user’s guide [13].We start introducing the concept of Morse function in this context. Suppose P is agiven simplicial complex (in this section all simplicial complexes are assumed to havea finite number of vertices). Also, for a simplex p we write p n to denote that n is thedimension of p . A function f : P → R is called a Morse function if for each p n ∈ P|{ q n +1 > p | f ( q ) ≤ f ( p ) }| ≤ |{ q n − < p | f ( p ) ≤ f ( q ) }| ≤ . This means roughly that f increases as the dimension of the simplices increase, withat most one exception, locally, at each simplex. We reproduce here the basic result[12, Lemma 2.5] Lemma 7.1.
Let P be a simplicial complex and f : P → R a Morse function. Thenfor any simplex p n either |{ q n +1 > p | f ( q ) ≤ f ( p ) }| = 0 or |{ q n − < p | f ( p ) ≤ f ( q ) }| = 0 . If both conditions in the lemma hold for a simplex then we call it critical, i.e., for p n in P we say that p is critical if |{ q n +1 > p | f ( q ) ≤ f ( p ) }| = 0and |{ q n − < p | f ( p ) ≤ f ( q ) }| = 0 . Now, we come back to the setup of posets. Because P is a simplicial complex thenit is a simplex-like poset, and by Lemma 6.5 there is an adequate local covering familyfor P op . In the next proposition we see how any Morse function f : P → R gives riseto a global covering family on P op . Proposition 7.2.
Let P be a simplicial complex and f : P → R a Morse function.Then there is a global covering family K for P op . Moreover, for each n ≥ , b K n = |{ critical simplices of dimension n }| . Proof.
According to Lemma 7.1 we can divide the simplices of dimension n ≥ P n ,in the following disjoint sets C n = { p n ||{ q n +1 > p | f ( q ) ≤ f ( p ) }| = 0 and |{ q n − < p | f ( p ) ≤ f ( q ) }| = 0 } ,D n = { p n ||{ q n − < p | f ( p ) ≤ f ( q ) }| = 1 } , and E n = { p n ||{ q n +1 > p | f ( q ) ≤ f ( p ) }| = 1 } . METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 27
The set C n consists of the n -dimensional critical simplices. For n ≥ E n − → D n which maps p n − ∈ E n − to the unique n -simplex q n with f ( q ) ≤ f ( p ). Now define K n = C n ∪ D n (disjoint union) for each n ≥
0. Notice that if K = { K n } n ≥ were aglobal covering family then b K = | K | = | C | + | D | = | C | as D = ∅ and, for n ≥ b K n = | K n − | − |P n − | + | K n | . Because of the bijection E n − ∼ = D n this equals b K n = | C n − | + | D n − | − ( | C n − | + | D n − | + | E n − | ) + | C n | + | D n | = | C n | . Fix n ≥
0. We show that the restriction map w : lim ←− F n → Y p ∈ K n F n ( p )is a monomorphism, where F n : P op → Ab are the functors obtained from Lemma4.3 applied to P op .Take ψ ∈ lim ←− F n = hom Ab P ( c Z , F n ) which goes to zero by the restriction map w .If we prove that ψ ( q ) = 0 for each simplex of dimension n + 1 then ψ ( p ) = 0 forevery simplex of dimension n . To see this consider any simplex p n . If p is not theface of any ( n + 1)-simplex then p ∈ K n and ψ ( p ) = 0 (as ψ is in the kernel of w ).If there exists q n +1 with q > p then ψ ( q ) = 0 by hypothesis and then ψ ( p ) = ψ ( q → p )( ψ ( q )) = ψ ( q → p )(0) = 0. Finally, as F n is n -condensed, if ψ ( p ) = 0 for each n -simplex p then ψ = 0.Now we prove that ψ ( q ) = 0 for any ( n + 1)-simplex q . Recall that we are assumingthat the set of vertices of P , and thus P itself, is finite. We consider the total orderedfinite set f ( P n +1 ) and make induction on it. The base case is a ( n + 1)-simplex q suchthat f ( q ) = min { f ( P n +1 ) } . There are n + 2 n -dimensional faces of q . If at least n + 1of these faces are in K n then ψ ( p ) = 0 for each of these faces and, by 6.9, ψ ( q ) = 0.Now suppose that there exists at least two n -dimensional faces of q which are notin K n . By the definition of Morse function one of the values of f in these faces isstrictly smaller than f ( q ). Call this face p so f ( q ) > f ( p ). As p belongs to E n thereexists an ( n + 1)-dimensional simplex q ′ such that q ′ > p and f ( q ′ ) ≤ f ( p ). Butthen we obtain f ( q ) > f ( p ) ≥ f ( q ′ ). This contradicts the fact that f ( q ) is minimunamong ( n + 1)-simplices and thus it has to be the case that there are at least n + 1 n -dimensional faces of q which are in K n .Next we do the induction step: take an ( n +1)-dimensional simplex q . By definitionof Morse function at least n + 1 of the n + 2 n -dimensional faces p of q satisfy f ( q ) > f ( p ). For each one of these n + 1 faces p either p ∈ K n and ψ ( p ) = 0 or p ∈ E n and there exists a ( n + 1)-dimensional simplex q ′ with f ( q ′ ) ≤ f ( p ). In this last casewe obtain f ( q ) > f ( p ) ≥ f ( q ′ ) and then by the induction hypothesis ψ ( q ′ ) = 0 and ψ ( p ) = ψ ( q ′ → p )( ψ ( q ′ )) = 0. Thus, ψ is zero in at least n + 1 n -dimensional faces of q and by 6.9 ψ ( q ) = 0.Now we prove that for n ≥ w : Y p ∈ Ob n − ( P ) \ K n − F n − ( p ) → Y p ∈ K n F n ( p )is pure. Take y ∈ Q p ∈ K n F n ( p ) , m ≥ z ∈ Q p ∈ Ob n − ( P ) \ K n − F n − ( p ) with m · y = w ( z ). We want to prove that m | z p for each p ∈ Ob n − ( P ) \ K n − . We dothis by induction on f (Ob n − ( P ) \ K n − ). The base case is a ( n − p not in K n − such that f ( p ) is a minimum value. As p is in E n − there is q n ∈ D n ⊂ K n with f ( q ) ≤ f ( p ). For any of the n faces p ′ of q different from p we have f ( q ) > f ( p ′ ) andthus f ( p ) ≥ f ( q ) > f ( p ′ ). As f ( p ) is minimum then p ′ ∈ K n − and, by Remark 6.9, m | z p as m | w ( z ) q . The induction step runs in a similar way to the earlier case. (cid:3) Webb’s conjecture
In [7], Brown introduces the so called Brown’s complex of a finite group. Given afinite group G and a prime p , its associated Brown’s complex S p ( G ) is the geometricalrealization of the poset of non trivial p -subgroups of G . Webb conjectured that theorbit space S p ( G ) /G (as topological space) is contractible. This conjecture was firstproven by Symonds in [27], generalized for blocks by Barker [2, 3] and extended toarbitrary (saturated) fusion systems by Linckelmann [21].The works of Symonds and Linckelmann prove the contractibility of the orbit spaceby showing that it is simply connected and acyclic, and invoking Whitehead’s The-orem. Both proofs of acyclicity work on the subposet of normal chains (introducedby Knorr and Robinson [19] for groups). Symonds uses that the subposet of normalchains is G -equivalent to Brown’s complex, as was proved by Th´evenaz and Webb in[28]. Linckelmann proves on his own that, also for fusion systems, the orbit space andthe orbit space on the normal chains has the same cohomology [21, Theorem 4.7]. Inthis chapter we shall apply the results of Section 5 to prove, in an alternative way,that the orbit space on the normal chains is acyclic.Let ( S, F ) be a saturated fusion system where S is a p -group [6]. Consider itssubdivision category S ( F ) (see [21, Section 2]) and the poset [ S ( F )]. An object in[ S ( F )] is an F -isomorphism class of chains[ Q < Q < . . . < Q n ]where the Q i ’s are subgroups of S . The subcategory ([ S ( F )] ↓ [ Q < . . . < Q n ]) hasobjects [ Q i < . . . < Q i m ] with 0 ≤ m ≤ n and 0 ≤ i < i < . . . < i m ≤ n (see [21, METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 29
Section 2] again). For example, ([ S ( F )] ↓ [ Q < Q < Q ]) is[ Q ] / / % % JJJJJJJJJJ [ Q < Q ] ( ( QQQQQQQQQQQQ [ Q ] tttttttttt % % JJJJJJJJJJ [ Q < Q ] / / [ Q < Q < Q ] . [ Q ] / / tttttttttt [ Q < Q ] mmmmmmmmmmmm Then it is clear that [ S ( F )] is a simplex-like poset. Following Linckelmann’s no-tation, denote by S ⊳ ( F ) the full subcategory of S ( F ) which objects Q < . . . < Q n with Q i ⊳ Q n for i = 0 , . . . , n . Also, denote by [ S ⊳ ( F )] the subdivision categoryof S ⊳ ( F ), which is a sub-poset of [ S ( F )]. By [21, Theorem 4.7], H ∗ ([ S ( F )]; Z ) = H ∗ ([ S ⊳ ( F )]; Z ). Our goal in this section is to prove that H n ([ S ⊳ ( F )]; Z ) = 0 for n ≥ H ([ S ⊳ ( F )]; Z ) = Z . It is straightforward that [ S ⊳ ( F )] is a simplex-likeposet and thus, by Lemma 6.5, there exists an adequate covering family for the gradedposet [ S ⊳ ( F )] op . We shall build a global covering family K for [ S ⊳ ( F )] op . This family K will verify b K n = 0 for n ≥ b K = 1, and so Theorem 5.5 will give the desiredresult.The definition of the global covering family is as follows, and it is related with thepairing defined by Linckelmann in [21, Definition 5.5]. The notion of paired chainswas used by Kn¨orr and Robinson in several forms throughout [19]. Definition 8.1.
For the graded poset [ S ⊳ ( F )] op define the subsets K = { K n } n ≥ by K n = (cid:8) [ Q < . . . < Q n ] | [ Q < . . . < Q n ] = [ Q ′ < . . . < Q ′ n ] ⇒ ∩ ni =0 N S ( Q ′ i ) = Q ′ n (cid:9) . The fact that K is defined through a pairing provides (see below) a bijection ψ :Ob n ([ S ⊳ ( F )] op ) \ K n → K n +1 for each n ≥
0. This proves that b K n = 0 for each n ≥ K = { [ S ] } and thus b K = 1. Next we prove that the family K = { K n } n ≥ defined in 8.1 is a global covering family for [ S ⊳ ( F )] op . We use terminology andresults from [6, Appendix].For any chain Q < . . . < Q n in S ⊳ ( F ) define the following subgroup of automor-phisms of Q n A Q <... Definition 8.2. For the graded poset [ S ⊳ ( F )] op define the subsets K = { K n } n ≥ by K n = { [ Q ′ < . . . < Q ′ n ] | Q ′ n fully A Q ′ <... For any n ≥ there is a bijection ψ : Ob n ([ S ⊳ ( F )] op ) \ K n → K n +1 . Proof. Take [ Q < . . . < Q n ] ∈ Ob n ([ S ⊳ ( F )] op ) \ K n and a representantive Q ′ <. . . < Q ′ n which is fully A Q ′ <... The family K = { K n } n ≥ defined in 8.1 is a global covering family for [ S ⊳ ( F )] op .Proof. We start proving that for any n ≥ ←− F n → Y i ∈ K n F n ( i )is a monomorphism. Take ψ ∈ lim ←− F n such that ψ ( i ) = 0 for each i ∈ K n . Ifthere is no object of degree greater than n then K n = Ob n ([ S ⊳ ( F )] op ) and we aredone. If not, we prove that ψ ( j ) = 0 for each j of degree n + 1 by induction on | Q n +1 | . This is enough to see that ψ is zero as F n is n -condensed. The base case is j = [ Q < . . . < Q n +1 ] with | Q n +1 | maximal. This implies that J jn ⊆ K n . Then ψ ( j )goes to zero by the monomorphism F n ( j ) → Y i ∈ J jn F n ( i ) , and thus ψ ( j ) = 0. For the induction step consider j = [ Q < . . . < Q n +1 ] and j ′ = [ Q < . . . < c Q l < . . . < Q n +1 ] ∈ J jn with 0 ≤ l < n . Then, either j ′ ∈ K n and ψ ( j ′ ) = 0, or j ′ / ∈ K n and there is an arrow in [ S ⊳ ( F )] op j ′′ = [ Q ′ < . . . < c Q ′ l < . . . < Q ′ n +1 < ∩ mi =0 ,i = l N S ( Q ′ i )] → j ′ = [ Q < . . . < c Q l < . . . < Q n +1 ] . In the latter case ψ ( j ′′ ) = 0 by the induction hypothesis, and thus ψ ( j ′ ) = 0 too. Asbefore, since the map F n ( j ) → Q i ∈ J jn F n ( i ) is a monomorphism, ψ ( j ) = 0.Now we prove that for any n ≥ ω : Y i ∈ Ob n − ( P ) \ K n − F n − ( i ) → Y i ∈ K n F n ( i )is pure. Take y ∈ Q i ∈ K n F n ( i ), m ≥ x ∈ Q i ∈ Ob n − ( P ) \ K n − F n − ( i ) with m · y = ω ( x ) . We want to find x ′ with m · x ′ = x . We prove that x i is divisible by m for each i = [ Q < . . . < Q n − ] ∈ Ob n − ( P ) \ K n − by induction on | Q n − | . The base case is i = [ Q < . . . < Q n − ] with | Q n − | maximal. Consider the arrow in [ S ⊳ ( F )] op j = [ Q ′ < . . . < Q ′ n − < ∩ n − i =0 N S ( Q ′ i )] → [ Q < . . . < Q n − ] . As Q n − is maximal then J jn − ⊆ K n − . Then m · y j = ω ( x ) j is the image of( x i , , . . . , 0) by the mapKer ′ F n − ( j ) = Y l ∈ ( j ↓ [ S ⊳ ( F )]) n − F n − ( i ) π j → F n ( j ) . As F n ( j ) ∼ = Q l ∈ ( j ↓ [ S ⊳ ( F )]) n − \ J jn − F n − ( l ) = F n − ( i ) by Remark 3.8 then m divides x i .For the induction step consider i = [ Q < . . . < Q n − ] ∈ Ob n − ( P ) \ K n − and j = [ Q ′ < . . . < Q ′ n − < ∩ i = n − i =0 N S ( Q ′ i )]. As before, m · y j = ω ( x ) j is the image of˜ x = x | ( j ↓ [ S ⊳ ( F )]) n − by the mapKer ′ F n − ( j ) = Y l ∈ ( j ↓ [ S ⊳ ( F )]) n − F n − ( i ) π j → F n ( j ) . By Remark 3.8 again, m · y j = π j (˜ x − ( λ j ◦ s j )(˜ x )) = π j (( x i − ( λ j ◦ s j )( x | J jn ) , , . . . , . Now, by the induction hypothesis, for each l ∈ J jn either l ∈ K n − and x l = 0, either l / ∈ K n − and thus, by the induction hypothesis, m divides x l . Then m divides x | J jn and so, by the equation above and the isomorphism F n ( j ) ∼ = F n − ( i ), m divides x i too. (cid:3) Coxeter groups It is well known that the Coxeter complex associated to a finite Coxeter group( W, S ) is a simplicial decomposition of a sphere of dimension | S | − 1, and thus hasthe homotopy type of a sphere. In case W is infinite then this complex becomescontractible. In this section we prove that the cohomology of the Coxeter complexis that of a sphere if W is finite or trivial if W is infinite. We do it by using thetechniques of earlier sections. More precisely, we construct a global covering family K for the Coxeter complex with b K = b K | S |− = 1 and b Kn = 0, n = 0 , | S | − W finite, and with b K = 1 and b Kn = 0, n = 0 for W infinite. We will use general factsabout Coxeter groups which can be found in [5], [17] or [14], as well as we will recallsome basic definitions and statements.Let ( W, S ) be a Coxeter system where W is a Coxeter group and S = { s , ..., s N } is a set of generators (which we always assume is finite). For any word w ∈ W itslength l ( w ) is the minimum number of generators from S that are needed to write it.If W is finite there is a unique element of maximal length which we denote by w .For every subset I ⊂ S we have the parabolic subgroup W I ≤ W generated by thegenerators belonging to I . We also define the following subset W I = { w ∈ W | l ( ws ) > l ( w ) for all s ∈ I } . In [17, 5.12] and [5, excersise 3, p. 37] is proven the following: Lemma 9.1. Fix I ⊆ S . Given w ∈ W , there is a unique u ∈ W I and a unique v ∈ W I such that w = uv . Then l ( w ) = l ( u ) + l ( v ) . Moreover, u is the unique elementof smallest length in the coset wW I . From this lemma it is straightforward that there is a bijection(14) W/W I → W I which sends the coset wW I ∈ W/W I to the unique element u ∈ W I of smallest lengthin the coset wW I . Notice that uW I = wW I for w and u as in the lemma. Next we METHOD FOR INTEGRAL COHOMOLOGY OF POSETS 33 describe the Coxeter complex associated to ( W, S ): it is a simplicial complex whichsimplices are the cosets of proper parabolic subgroups with inclusion reversed: wW I ← w ′ W I ′ ⇔ I ⊂ I ′ and w − w ′ ∈ W I ′ . The dimension of the simplex wW I is n = | S | − | I | − 1. Thus, vertices correspondto maximal subsets I of S and facets, i.e., maximal dimensional faces, correspond to I = ∅ . Notice that it is a pure simplicial complex. By the bijection (14) we can writethe coset wW I as uW I with u ∈ W I , where w and u are as in the previous lemma.We will do this in the rest of the section. For any u ∈ W define (cf. [17, 1.11]) S u = { s ∈ S | l ( us ) > l ( u ) } and let s u = max { S u } be the maximum element in S u with respect to the order in S : s < ... < s N . Notice that S u = S if and only if u = 1. Moreover, if W is finite then S u = ∅ if and only u = w , the unique word of maximal length. It is also clear that(15) u ∈ W I ⇔ I ⊆ S u .Before defining the global covering family notice that, by Lemma 6.5, there is anadequate local covering family for P op . Now, for each n ≥ K n = { uW I || I | = | S | − − n and s u / ∈ I } . The condition | I | = | S | − − n in the definition above just states that uW I correspondto a simplex of dimension n . Lemma 9.2. The family K = { K n } n ≥ is a global covering family for P op . Moreover, b K = 1 . If W is finite then b K| S |− = 1 and b K n = 0 for n = 0 , | S | − . If W is infinitethen b K n = 0 for n = 0 . By Theorem 5.5 this lemma imply that the integral cohomology of P is that of asphere of dimension | S | − W is finite, and that of a point if W is infinite. Proof. First we compute the numbers b Kn and afterwards we will prove K satisfiesDefinition 5.1. Fix u ∈ W and consider the contribution which it makes to P op . ByEquation (15) u ∈ W I if and only if I ⊆ S u . Thus, denoting by P u the sub-poset withobjects { uW I , I ⊆ S u } we have that P is contained in the disjoint union S u ∈ W P u . Itis a combinatorial exercise (cf. Lemma 6.4) that there is a bijection between K n +1 ∩P u and {P n \ K n } ∩ P u for n ≥ u = w . This gives, in case W is infinite, b Kn = 0for n ≥ 1. If W is finite then P w = { w W ∅ } and w W ∅ is a facet which belongs to K | S |− . Then we obtain b K | S |− = 1 and b Kn = 0 for | S | − > n ≥ 1. Finally, it is aneasy consequence of the definition that K = { · W S \{ s N } } and thus b K = 1.Now fix n ≥ 0. We show that the restriction maplim ←− F n → Y uW I ∈ K n F n ( uW I )is a monomorphism, where F n : P op → Ab are the functors obtained from Lemma4.3 applied to P op .Take ψ ∈ lim ←− F n = hom Ab P ( c Z , F n ) which is mapped to zero by the restriction map.To prove that ψ = 0 it is enough to prove that ψ ( uW I ) = 0 for each simplex uW I of dimension n + 1 (as in the proof of Theorem 7.2). We do this by induction on thelength l ( u ).We start with uW I with l ( u ) = 0, i.e., u = 1. Then S u = S . If s u = s N / ∈ I , i.e., u ∈ K n +1 , then there n +1 n -faces of u which are in K n . These are the cosets W J where J = I ∪ { s } with s ∈ S \ { I ∪ { s N }} . The remaining face, i.e., W J with J = I ∪ { s N } ,is not in K n . By Remark 6.9, ψ ( W I ) = 0. Now assume that s u = s N ∈ I , i.e., u / ∈ K n +1 . Consider any of the n + 2 n -faces W J of W I , where J = I ∪ { s } with s ∈ S \ I . Then W J is also a face of then n + 1 simplex W I \{ s N }∪{ s } . By the precedingargument for the case W I \{ s N }∪{ s } ∈ K n +1 we obtain ψ ( W I \{ s N }∪{ s } ) = 0, and thus ψ ( W J ) = 0. Then ψ is zero in all the faces of W I . By Remark 6.9 again we obtain ψ ( W I ) = 0.Next we do the induction step: take an n +1 dimensional simplex uW I with l ( u ) > u ∈ K n +1 , i.e., s u / ∈ I . The faces of uW I are the n + 2 n -simplices uW J where J = I ∪ { s } and s ∈ S \ I . Notice that, as I ⊆ S u , S \ I = S u \ I ∪ S \ S u ,where the union is disjoint. Take first s ∈ S \ S u . Then l ( us ) < l ( u ) and, as us ∈ uW J ,the unique element u ′ of minimal length in uW J is different from u and has smallerlength. Then uW J = u ′ W J is also a face of u ′ W I . By the induction hypothesis andbecause l ( u ′ ) < l ( u ) we have ψ ( u ′ W I ) = 0 and thus ψ ( uW J ) = 0. Now take s ∈ S u \ I .Then uW J ∈ K n unless s = s u . Then ψ is zero in all but one face of uW I and, byRemark 6.9, ψ ( uW I ) = 0.Now assume that u / ∈ K n +1 , i.e., s u ∈ I , and take a face uW J as before. If J = I ∪ { s } with s ∈ S \ S u then arguing as before we obtain that ψ ( uW J ) =0. If J = I ∪ { s } with s ∈ S u \ I then uW J is also a face of the n + 1 simplex uW I \{ s N }∪{ s } . By the preceding argument for the case uW I \{ s N }∪{ s } ∈ K n +1 we have ψ ( uW I \{ s N }∪{ s } ) = 0, and thus ψ ( W J ) = 0. 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Q ′ n . Define ψ ([ Q < . . . < Q n ]) = [ Q ′ < . . . < Q ′ n < ∩ ni =0 N S ( Q ′ i )]. The proof is divided in foursteps: a) ψ is well defined. Take another representantive Q ′′ < . . . < Q ′′ n which is fully A Q ′′ <...