SMALL RATIONAL MODEL OF SUBSPACE COMPLEMENT

SERGEY YUZVINSKY

1. Introduction

This paper concerns the cohomology ring of the complement of a complex subspace arrangement. Let A be a finite set of proper subspaces of a finite dimensional complex linear space V. Put C(A) = V \ ⋃_{A∈A} A. We can assume that there are no inclusions among elements of A. A rough characterization of A is given by the lattice X = X(A) consisting of all the intersections of the elements from A ordered opposite to inclusion and labeled by their complex codimensions.

If each subspace A ∈ A is a hyperplane then A can be given by a set {α_{A}}_{A∈A} ⊂ V^{∗} where ker α_{A} = A. In this case, the lattice X is geometric and the codimension of its each element can be recovered from the ordering as the rank of this element. There are two main results describing the structure of the ring R = H^{∗}(C(A), ZZ) in this case. The first one is the Arnold-Brieskorn theorem [1, 4] stating that the algebra of differential forms generated by the closed forms \frac{1}{2πi} \frac{dα_{A}}{α_{A}}, A ∈ A, is isomorphic to R under the de Rham homomorphism. The second one is the Orlik-Solomon theorem stating that R depends only on X, that is, on the set of intersections of hyperplanes and its partial order by (opposite) inclusion. Moreover the theorem gives a presentation of R by generators and relations defined by X.

In the general case, X is not necessarily geometric and the codimensions cdA of A ∈ X can not be anymore recovered from the order on X. Again, there are two main results about H^{∗}(C(A)). The first one is the Goresky-MacPherson theorem [9] stating that the groups H^{p}(C(A)) are determined by the labeled lattice X. More precisely these groups are expressed in terms of the local homology groups of X(A) as follows. For every p

H̃^{p}(C(A), ZZ) = ⊕_{A∈X} H̃_{2cdA−p−2}((V, A), ZZ)

where (V, A) = {B ∈ X|V < B < A} and H_{∗}((V, A)) the homology of the complex of flags in (V, A). This is an example of application of Stratified Morse theory developed in [9]. Later, this result was obtained by a different method in [16]. A completely different approach was used recently by De Concini and Procesi [5]. They constructed a rational model for C(A) using only the labeled lattice X (see also section 2) and proved that the rational cohomology algebra and rational homotopy type of C(A) are defined by this lattice. The natural problem that was left open is how to recover the ring structure explicitly from the combinatorics, in particular how to relate the multiplication to the local homology of X that occurs in the Goresky-MacPherson formulas.

The present paper solves this problem (Theorem 6.5). To be more specific here we need to introduce more notation. For every pair (A, B) (A, B ∈ X) denote by A ∨ B the join (i.e., the least upper bound) of the pair. The joins of sets of atoms from (V, A) ∪ (V, B) form a subposet X_{A,B} of (V, A ∨ B). The following condition on A, B is an important breaking point

cd(A ∨ B) = cdA + cdB.  (∗)

If this condition holds then the flag complex of X_{A,B} has the homotopy type of the suspension of the join of the flag complexes of (V, A) and (V, B). Then there is a natural isomorphism

α_{∗} = α(A, B)_{∗} : H̃_{r}((V, A), Ql ) ⊗ H̃_{s}((V, B), Ql ) → H̃_{r+s+2}(X_{A,B}, Ql ).

Theorem. The multiplication in H^{∗}(C(A), Ql ) is given by bilinear pairings

θ_{A,B} : H̃_{r}((V, A), Ql ) ⊗ H̃_{s}((V, B), Ql ) → H̃_{r+s+2}((V, A ∨ B), Ql )

for every A, B ∈ X. Here ψ_{A,B} is the composition of ϕ_{A,B} with the embedding X_{A,B} ⊂ (V, A ∨ B) If (*) holds and ψ_{A,B} = 0 otherwise.

Notice that the set of flag complexes of X(A) for all the subspace arrangements A includes the homotopy types of all finite simplicial complexes. In this generality, it is impossible to give any explicit description of local homology classes of X(A) and the multiplication on them. Instead we describe a multiplication on chain groups of the local flag complexes that induces the right multiplication on the homology.

Let A, B ∈ X(A) with condition (*). Let F_{A} = (A_{1} < A_{2} < ⋯ < A_{i}) and F_{B} = (B_{1} < B_{2} < ⋯ < B_{j}) be flags in the open intervals (V, A) and (V, B) respectively. Augment the flags by A and B respectively and take the shuffle product of them. The result is a linear combination (with coefficients equal to signs of permutations) of certain sequences of A_{r} and B_{s}. Create a flag from each such sequence taking the intersections of all initial subsequences. Equate flags with repetitions to 0 and delete A ∨ B at the ends of all others. The resulting linear combination of flags is the product F_{A} ·F_{B}. This product generates the product on linear combinations of flags by bilinearity that commutes with the differentials. Hence it induces a bilinear pairing

H̃_{i−1}((V, A)) ⊗ H̃_{j−1}((V, B)) → H̃_{i+j}((V, A ∨ B)).

This is the multiplication on local homology of X. The multiplication can be defined even easier on so called Whitney (or relative atomic) complexes but we will present it elsewhere.

For the classes of lattices whose local homology can be described explicitly our theorem gives presentations of H^{∗}(C(A)). We consider two such classes; first one where X is a geometric lattice and the other one where C(A) is a so called k-equal manifold. In the former case, the local homology groups of X were computed in [7, 13]; in the latter case, they were computed in [3]. We give presentations of the algebra H^{∗}(C(A), Ql ) in these cases that in particular solves a problem stated in [3]. The representation for the former case was obtained independently and by completely different method by E.M. Feichtner in [6].

The method we use in this paper is as follows. We start with the De Concini - Procesi differential graded algebra M that is a specialization of the Morgan rational model of the complement of a divisor with normal crossings (see [10]). The algebra M is a rational model of C(A). We find inside M a significantly smaller subalgebra CM quasi-isomorphic to M whence also a rational model of C(A) (that may be of interest by itself). The algebra CM gives a multiplicative structure on the flag complexes of X that induces the ring structure on H^{∗}(C(A)), Ql ) described above. Moreover CM has a natural integral version CM(ZZ) that gives the right groups H^{p}(C(A)) and is a good candidate for an integral model of C(A) (see Remark 5.4 and Conjecture 6.7). ^{1}

The paper is organized as follows. In section 2, we recall the De Concini-Procesi model M and a monomial basis of it found in [15]. In section 3, we switch to generators more convenient for the purpose of the paper and exhibit a basis of monomials in the new generators. In section 4, we define the smaller subcomplex CM that is a deformation retract of M. In section 5, we prove that CM is a subalgebra of M and study the multiplicative structure on it. In section 6, we describe the multiplication induced on its cohomology. In section 7 and 8, we give a presentation of the algebra H^{∗}(C(A)) for the cases where X is a geometric lattice and where C(A) is a k-equal manifold.

This work was started when I was visiting Technische Universität in Berlin. I want to express my gratitude to G. Ziegler for his hospitality. I am also grateful to C. De Concini for useful communications and to Eva Feichtner for finding an error in the first version of Proposition 7.2..

2. De Concini - Procesi model

In this section, we recall the construction of rational models from [5]. To follow closer the notation in [5] we dualize our initial definitions from

Introduction. Let V be a finite dimensional complex linear space and A a finite set of nonzero linear subspaces of V. Without any loss of generality we can assume that their sum is V. The topological space we are interested in is C(A) = V^{∗} \ ⋃_{A∈A} A^{∘} where A^{∘} is the annihilator of A in V^{∗}. The space C(A) does not change if we include all nonempty sums of elements of A in A. Therefore we will be dealing with the lattice X = X(A) of all sums of elements of A ordered by inclusion and labeled by the dimensions of its elements. Notice that C(A) = C(A) = V^{∗} \⋃_{A∈X,A≠0} A^{∘}. Let us emphasize that the only structure on X used in this paper is that of a labeled lattice. We also denote by P(A) the projectivization of a linear space A and put PC(A) = PV^{∗} \ ⋃_{A∈X,A≠0} P(A^{∘}).

1November, 2000. This paper was witten at the end of 1996 and has since been circulating as a preprint. The Conjecture 6.7 has been taken up repeatedly and was recently proven independently by de Longueville and Schultz [17] by direct topological methods and by Deligne, Goresky, and MacPherson [18] via a sheaf-theoretic approach. Also author’s note [19] has appeared where even smaller and more natural rational model for subspace complements is constructed, using results of this paper.

Now we describe two differential graded algebras ( DGA) M^{∗}(X) and M(X). There are different (though quasi-isomorphic to each other) versions of these algebras depending on the choice of a building set of X. Recall that a subset G of X is a building set if every A ∈ X is the direct sum of those elements in G that are maximal among elements of G contained in A. For the most part of the paper we take the whole X as the building set. The only time a different building set is used is for X′ in this Section.

Let G be a building set containing V. Start with the polynomial algebra P over Ql with indeterminates c_{A} corresponding to A ∈ G and the exterior algebra Λ over Ql generated by e_{A} corresponding to A ∈ G \ {V }. Consider the DGA P ⊗ Λ with the grading induced by deg c_{A} = 2 and deg e_{A} = 1 and the differential generated by d(e_{A}) = c_{A}. Now consider the elements

r(X_{1}, X_{2}, B) = ∏_{A∈X_{1}} e_{A} ∏_{A∈X_{2}} c_{A}( ∑_{C⊇B} c_{C})^{d(Y,B)}  (2.1)

of P ⊗ Λ where X_{1}, X_{2} ⊆ G \ {V }, Y = X_{1} ∪ X_{2}, B ∈ G, B ⊋ A for all A ∈ Y and d(Y, B) = dim B − dim(∑_{A∈Y} A). Notice that in order to define the first product and thus r(X_{1}, X_{2}, B) precisely one needs to introduce a linear order on X_{1}. Otherwise these elements are defined only up to ±1; this suffices for the following definition. Let J be the ideal generated by all the elements r(X_{1}, X_{2}, B). Clearly it is homogeneous and invariant with respect to d. Thus M^{∗}(X) = (P ⊗ Λ)/J is a DGA. DGA M(X) is defined similarly with the only difference that both e_{A} and c_{A} are indexed by all the elements of G (including V). We will denote by M^{∗}(X)_{r} and M(X)_{r} the homogeneous components of degree r of the algebras.

As it is customary, we will keep the notation c_{A} and e_{A} for the images of these elements under factorization and the term “monomials” for the images of monomials in c_{A} and e_{A}. Theorem 2.1. ([5], Theorem 5.3.) There is a Ql -algebra isomorphism H^{∗}(M^{∗}(X)) ≈ H^{∗}(PC(A), Ql ).

In [5], the problem of the naturality of this isomorphism is left open. It follows from [10] that after tensoring with C this isomorphism becomes canonical. Over Ql it is not functorial in the category of smooth algebraic varieties ([10]). It is probably functorial in the category of arrangement complements but a proof is not known to me.

Now we want a model for C(A) ⊂ V^{∗}. We can view A as a subspace arrangement in V′ = V ⊕ W with dim W = 1 (this is a generalization of the coning construction from [13], p.14). Then we can consider the new arrangement A′ = A ∪ {W } ∪ {A ⊕ W |A ∈ A} in V′. Notice that PC(A′) = C(A). Thus M^{∗}(X′) can serve as a model of C(A).

To simplify definition of M^{∗}(X′) we choose X ∪ {W, V′} as the building set for X′. Then to obtain M^{∗}(X′) one needs to adjoin e_{W} , e_{V} , c_{V′} and c_{W} to the generators of M^{∗}(X). Taking X_{1} = X_{2} = ∅ and B = W, we obtain the relation

c_{W} + c_{V′} = 0.

Thus c_{W} = −c_{V′} and we can forget about c_{W}.

The next observation is that we can substitute simplier DGA M(X) for M^{∗}(X′). Proposition 2.2. The assignment e_{A} ↦ e_{A} (A ∈ X \ {V }), c_{B} ↦ c_{B} (B ∈ X), e_{W} ↦ 0, and c_{V′} ↦ 0 generates a surjective DGA morphism ϕ : M^{∗}(X′) → M(X) that induces an isomorphism in cohomology.

Proof. A direct check shows that the assignment from the statement sends every relation for M^{∗}(X′) to a relation for M(X). Thus it generates an algebra homomrphism ϕ. Clearly ϕ preserves the gradings, commutes with the differentials, and is surjective.

The kernel of ϕ is the ideal I of M^{∗}(X′) generated by e = e_{W} and c = c_{V′}. Using the relations for this algebra it is easy to exhibit a linear basis of I: {ec^{k}, c^{k+1}, e_{A}c^{m}, c_{A}c^{m}} where 0 ≤ k < dim V′−1, A ∈ X, and for each A we have 1 ≤ m < dim V′−dim A (cf. Proposition 2.3). This basis is invariant for d up to ±1; more exactly d(ec^{k}) = −c^{k+1} and d(e_{A}c^{m}) = c_{A}c^{m}. This shows that I is an acyclic cochain complex and the last claim of the proposition follows from the exact cohomology sequence of the exact sequence

0 → I → M^{∗}(X′) → M(X) → 0.
□

Proposition 2.2 implies that M(X) can be taken as a model of C(A). Keeping this in mind we will focus our attention on M = M(X) from now on. Let us emphasize again that the building set for M is the whole X.

Our goal for the rest of the section is to exhibit a linear basis of M consisting of monomials in the generators c_{A} and e_{A}. For each T ⊆ X denote by μ(T) the monomial of Λ corresponding to T (defined up to ±). Then consider the grading Λ = ⊕_{T ⊆X}Λ^{T} of Λ where Λ^{T} = Ql μ(T). It generates a grading on Λ ⊗ P and, since every r(X_{1}, X_{2}, B) is homogeneous in this grading, a grading M = ⊕_{T ⊆X}M^{T} on M. Then notice that for every A_{1}, A_{2} ∈ X neither of which is a subset of the other

a_{A_{1}}a_{A_{2}} = 0

in M where a_{A} denotes either c_{A} or e_{A}. To see this it is enough to take in (2.1) Y = {A_{1}, A_{2}} and B = A_{1} + A_{2} that gives d(Y, B) = 0. Hence all the monomials in e_{A} identified up to the ordering (or ±1) are in oneto-one correspondence with flags (by inclusion) in X. In particular, in the representation M = ⊕_{T ⊆X}M^{T}, a summand M^{T} is nonzero only if T is a flag in X. In any case M^{T} has the following structure of a graded algebra (although not a subalgebra of M). Denote by X^{T} the set of all elements of X each one of which forms a flag with T. Then M^{T} is the factor of the polynomial ring Ql [c_{A}|A ∈ X^{T}] by the ideal generated by

r(Y, B) = ∏_{A∈Y} c_{A}( ∑_{C⊇B} c_{C})^{d^{T} (Y,B)}

for every Y ⊂ X^{T} \ T, B ∈ X^{T} such that B ⊋ A for every A ∈ Y, and d^{T} (Y, B) = dim B − dim(∑_{A∈Y ∪T,A⊂B} A). (In fact in [5] the algebras M^{T} appeared first and M^{∗}(X) was constructed from them following [10].)

A monomial basis of algebra M^{T} was first constructed in [15] (see [15], Remark 3.11) and then generalized in [8]. Combining the monomial bases of M^{T} for all flags T we obtain the following.

Proposition 2.3. Suppose S, T ⊆ X (not necessarily disjoint), T = {A_{1}, …, A_{k}} is ordered by inclusion and S ∪ T is a flag. Suppose m : S → Z_{+} satisfies m(A) < dim A − dim A′ for every A ∈ S where A′ = 0 if A is the smallest element of T ∪ S and A′ is the predecessor of A in T ∪ S otherwise. Then the monomials λ(S, T, m) = e_{A_{1}} ⋯ e_{A_{k}} ∏_{B∈S} c_{B}^{m(B)} form a basis of M.

We will need later one more idea from [15]. Consider an arbitrary monomial λ = e_{A_{1}} ⋯ e_{A_{k}}c_{B_{1}}^{m_{1}} ⋯ c_{B_{ℓ}}^{m_{ℓ}}. Then define the weight of λ as

wt(λ) = ∑_{i=1}^{k} dim A_{i} + 2 ∑_{i=1}^{ℓ} m_{i} dim B_{i}.

It is easy to see that for any monomial λ(S, T, m) that appears in the linear decomposition of λ with a nonzero coefficient we have

wt(λ(S, T, m)) ≥ wt(λ)

(cf. [15], proof of Proposition 2.2).

3. New generators

In this section, we switch to new generators of algebra M. For every A ∈ X put

σ_{A} = ∑_{B⊇A} c_{B} and τ_{A} = ∑_{B⊇A} e_{B}.

Clearly {σ_{A}, τ_{A}} is a generating set of M.

The new generators satisfy many relations that follow from the relations for c_{A} and e_{A}. We notice some of them for future use. We use symbol ρ_{A} meaning either σ_{A} or τ_{A}.

(3.1) All σ_{A} have degree 2 and thus belong to the center of M. The elements τ_{A} are degree 1 generators of an exterior subalgebra of M. In particular τ_{A}^{2} = 0 and τ_{A}τ_{B} = −τ_{B}τ_{A}. Thus all monomials in τ_{A} and σ_{A} can be parametrized by the triples (S, T, m) where S, T ⊆ X, a linear ordering on T is fixed and m : S → Z_{+}. We put

μ(S, T, m) = ∏_{B∈T} τ_{B} ∏_{A∈S} σ_{A}^{m(A)}

where the first product is taken according to the order on T. Notice that deg μ(S, T, m) = |T | + 2 ∑_{A∈S} m(A).

(3.2) For every A, B ∈ X we have

τ_{A}τ_{B} = τ_{A}τ_{A+B} − τ_{B}τ_{A+B}

Proof. Represent τ_{A} = Δ_{A,B} + τ_{A+B} where

Δ_{A,B} = ∑_{C⊇A,C⊉B} e_{C}.

Similarly τ_{B} = Δ_{B,A} + τ_{A+B}. Since for every C_{1} ⊇ A, C_{1} ⊉ B and C_{2} ⊇ B, C_{2} ⊉ A the set {C_{1}, C_{2}} is not a flag we have e_{C_{1}}e_{C_{2}} = 0 whence Δ_{A,B}Δ_{B,A} = 0. Using besides that τ_{A+B}^{2} = 0 we obtain the result.

(3.3) For every A ∈ X we have

σ_{A}^{dim A} = 0

and

ρ_{A}σ_{B}^{k} = ρ_{B}σ_{B}^{k}

for every A ⊆ B and k ≥ dim B − dim A.

Proof. The first relation follows immediately from (2.1) if one takes Y = ∅ and B = A. In order to prove the second one it suffices to take k = dim B − dim A. Fix A, B ∈ X such that A ⊂ B and put k = dim B − dim A. We have

ρ_{A}σ_{B}^{k} = ∑_{C⊇A} a_{C}σ_{B}^{k} = ∑_{C⊇A} a_{C}( ∑_{D⊇B} c_{D})^{k}  (3.4)

where a_{A} = c_{A} or e_{A} respectively for ρ_{A} = σ_{A} or τ_{A}. Consider the sum b_{C} = a_{C}(∑_{D⊇B} c_{D})^{k} for some C such that C ⊇ A but C ⊉ B. If there is no inclusion between C and D then a_{C}c_{D} = 0. So it suffices to sum over D such that D ⊇ C, i.e. D ⊇ B + C. In other words

b_{C} = a_{C}( ∑_{D⊇B+C} c_{D})^{k}.

Now we have

dim(B + C) − dim C = dim B − dim B ∩ C ≤ dim B − dim A = k.

Due to (2.1) b_{C} = 0 and the summation in (3.4) can be taken over C such that C ⊇ B. □

Now we are ready to exhibit a basis of M consisting of monomials in σ_{A} and τ_{A}.

Proposition 3.1. Suppose that S, T ⊆ X, S ∪T is a flag (by inclusion), the ordering on T coincides with the inclusion order, m(A) < dim A−dim A′ for every A ∈ S and A′ as in Proposition 2.3, and finally |T |+2 ∑_{A∈S} m(A) = r for a fixed nonnegative r. Then the set of monomials μ(S, T, m) is a linear basis of M_{r} (we call the monomials in this set basic monomials).

Proof. Comparing the set of basic monomials of degree r with the basis of M_{r} from Section 2 we see that they have the same number of elements. Thus it suffices to proof that every monomial λ = λ(S, T, m) in c_{A} and e_{A} with m(A) < dim A − dim A′ and 2 ∑_{A∈S} m(A) + |T | = r is a linear combination of basic monomials of degree r. Apply downward induction on weight of λ. Clearly the weight of any monomial of degree r is limited from above by nr where n = dim V and the monomial of weight nr is either c_{V}^{r/2} = σ_{V}^{r/2} if r is even or e_{V} c_{V}^{(r−1)/2} = τ_{V} σ_{V}^{(r−1)/2} otherwise. Since this monomial is 0 if ⌊r/2⌋ ≥ n (cf. (3.3)) and basic otherwise we have the base of the induction.

Let λ = λ(S, T, m) be an arbitrary monomial in c_{A} and e_{A} of degree r with m(A) < dim A − dim A′ and wt(λ) < nr. Consider the polynomial p = λ − μ(S, T, m) in c_{A} and e_{A}. Clearly every monomial of p has a weight larger than the weight of λ whence the same is true for each monomial of the decomposition of p into the basis of Section 2 (compare the remarks on weights after Proposition 2.3). Thus the induction works. □

4. Critical monomials

Whenever we consider a monomial μ(S, T, m) in the rest of the paper, if S ⊂ T we extend m to T by 0.

Definition 4.1. A basic monomial μ(S, T, m) is critical if T ⊇ S and m(A) = dim A − dim A′ − 1 for every A ∈ T (so S = {A ∈ T | dim A − dim A′ > 1}).

Notice that the set of all critical monomials is linearly independent in M. Notice also that a critical monomial is defined by a flag T provided with the dimensions of its elements. Denote by cμ(T) the critical monomial defined by T. We have

deg cμ(T) = 2 dim A(T) − |T |

where A(T) is the maximal element of T. Denote by CM the linear subspace of M spanned by all the critical monomials.

Proposition 4.2. Let T = {A_{1}, A_{2}, …, A_{k}} where A_{i} ⊂ A_{i+1} and let T_{i} = {A_{1}, …, Â_{i}, …, A_{k}} for every i = 1, 2, …, k. Then

d(cμ(T)) = ∑_{i=1}^{k−1}(−1)^{i}cμ(T_{i}).

Proof. To simplify notation put ρ_{i} = ρ_{A_{i}} and m(i) = dim A_{i} − dim A_{i−1} − 1. Using the Leibniz formula for d and the relations (3.3) repeatedly we have

d(cμ(T)) = d(τ_{1} ⋯ τ_{k}σ_{1}^{m(1)} ⋯ σ_{k}^{m(k)})

= ∑_{i=1}^{k}(−1)^{i−1}τ_{1} ⋯ τ̂_{i} ⋯ τ_{k}σ_{1}^{m(1)} ⋯ σ_{i}^{m(i)+1} ⋯ σ_{k}^{m(k)}

= ∑_{i=2}^{k}(−1)^{i−1}τ_{1} ⋯ τ̂_{i−1} ⋯ τ_{k}σ_{1}^{m(1)} ⋯ { \atop σ_{i−1}^{m(i−1)}}σ_{i}^{m(i)+m(i−1)+1} ⋯ σ_{k}^{m(k)}

= ∑_{i=1}^{k−1}(−1)^{i}cμ(T_{i})

where the third equality follows from

τ_{i−1}σ_{i−1}^{m(i−1)}σ_{i}^{m(i)+1} = τ_{i}σ_{i}^{m(i−1)+m(i)+1}

for i ≥ 2 by the second relation in (3.3) and

σ_{1}^{m(1)+1} = 0

by the first relation in (3.3).

□

Corollary 4.3. The linear space CM is a subcomplex of M.

Remark 4.4. Proposition 4.2 implies that the complex CM affords the grading CM = ⊕_{A∈X}CM_{A} where CM_{A} is generated by monomials cμ(T) with A(T) = A. The deletion of A from all the flags gives an isomorphism of degree -1 from CM_{A} to the usual flag complex F (A) of the poset (0, A). This relates the De Concini - Procesi model for C(A) and the Goresky- MacPherson formula via

H^{p}(CM) = ⊕_{A∈X}H_{2 dim A−p−2}(F (A))

(see Section 6).

Our next goal is to prove that CM is quasi-isomorphic to M. Denote by CM^{⊥} the linear space spanned by all the basic monomials that are not critical. Clearly M = CM ⊕CM^{⊥} as a linear space. The next lemma proves that it is true in the category of complexes also.

Lemma 4.5. The space CM^{⊥} is a subcomplex of M.

Proof. Let μ = μ(S, T, m) ∈ CM^{⊥}. This means that μ satisfies one of the following conditions.

1. S ⊈ T, i.e., there exists A ∈ X such that σ_{A} divides μ and τ_{A} does not. Then every monomial in d(μ) satisfies the same condition whence d(μ) ∈ CM^{⊥}.

2. S ⊆ T and there exists A ∈ T such that m(A) < dim A − dim A′ − 1. Suppose T = {A_{1}, …, A_{k}} and A = A_{i}. As in the proof of Proposition 4.2 we put ρ_{A_{j}} = ρ_{j}, m(A_{j}) = m(j) and have

d(μ) = ∑_{j=1}^{k}(−1)^{j−1}μ_{j}

where

μ_{j} = τ_{1} ⋯ τ̂_{j} ⋯ τ_{k}σ_{1}^{m(1)} ⋯ σ_{j}^{m(j)+1} ⋯ σ_{k}^{m(k)}.

Clearly μ_{j} = μ(T_{j}, S^{j}, m^{(j)}) for some S^{j} and a function m^{(j)}.

Consider several cases. If m(j) < dim A_{j} − dim A_{j−1} − 1 then μ_{j} is basic and S^{j} ⊈ T_{j} whence μ_{j} ∈ CM^{⊥}. Suppose m(j) = dim A_{j} − dim A_{j−1} − 1, in particular j ≠ i. For j = 1, the exponent of σ_{1} in μ_{1} is dim A_{1} − 1, whence μ_{1} = 0 by the first relation in (3.3). For j > 1, using the same computation as in the proof of Proposition 4.2 we have

μ_{j} = τ_{1} ⋯ τ̂_{j−1} ⋯ τ_{k}σ_{1}^{m(1)} ⋯ { \atop σ_{j−1}^{m(j−1)}}σ_{j}^{m(j)+m(j−1)+1} ⋯ σ_{k}^{m(k)}.

The monomial μ_{j} is again basic. If i ≠ j − 1 then the exponent of σ_{i} in μ_{j} is m(i) whence μ_{j} ∈ CM^{⊥}. Finally if i = j − 1 then the exponent of σ_{j} in μ_{j} is

m(j) + m(i) + 1 < dim A_{j} − dim A_{i} − 1 + dim A_{i} − dim A_{i−1} − 1 + 1

= dim A_{j} − dim A_{j−2} − 1

whence again μ_{j} ∈ CM^{⊥}. That completes the proof. □

Now we are to construct a homotopy on CM^{⊥} between the identity map and 0.

For every basic monomial μ = μ(S, T, m) and A ∈ S \ T put h_{A}μ = μ(S, T ∪ {A}, m)/σ_{A}. Then put hμ = ∑_{A∈S\T} (−1)^{(A,T)}h_{A}μ where (A, T) = |{B ∈ T |B ⊂ A}|. Also for every B ∈ T put d^{B}(μ) = μ(S, T \ {B}, m)σ_{B}. Notice that d(μ) = ∑_{B∈T} (−1)^{(B,T)}d^{B}(μ) where (B, T) = |{C ∈ T |C ⊂ B}|. Call B ∈ T critical for μ if m(B) = dim B − dim B′ − 1 (as usual put m(B) = 0 if B ∉ S). Denote by CT the set of all critical elements of T. Now put |μ(S, T, m)| = |S ∪ T \ CT |. Notice that |μ| ≠ 0 if and only if μ ∈ CM^{⊥}.

Proposition 4.6. The linear map CM^{⊥} → CM^{⊥} defined by μ ↦ (1/|μ|)hμ is a homotopy between the identity map and 0.

Proof. It suffices to check that for every basic monomial μ we have

hd(μ) + d(hμ) = |μ|μ.  (4.1)

To make the check easier we state several simple observations. In them and in the rest of the proof we always mean that μ = μ(S, T, m). Also all the equalities mean in particular that the expressions in them are defined.

(a) For every A ∈ S \ T and B ∈ T \ CT we have h_{A}d^{B}(μ) = d^{B}(h_{A}μ).

(b) For every A and B as in (a) we have d^{A}(h_{A}μ) = μ = h_{B}d^{B}(μ).

(c) For every A ∈ S \ T the set of all critical for h_{A}μ elements coincides with CT.

(d) For B ∈ CT, let μ(S′, T′, m′) be the basic monomial equal to d^{B}(μ). (d.1) If B′ ∈ T then S′ \ T′ = S \ T. (d.2) If B′ ∉ T then S′ \ T′ = (S \ T ∪ {B}) \ {B′}.

(e) Let again B ∈ CT and A ∈ S \ T.

(e.1) If B′ ∈ T or A ≠ B′ then h_{A}d^{B}(μ) = d^{B}(h_{A}μ).

(e.2) If B′ ∉ T then h_{B}d^{B}(μ) = −d^{B}h_{B′}(μ) and d^{B′}h_{B′}(μ) = μ.

The statements (a)-(c) are straightforward. To check (d) and (e) apply relations of type (3.3) several times.

Combining (a)-(e) and using the rule for signs in the Leibniz property of d one gets (4.1). □

Corollary 4.7. The embedding CM ⊂ M is a quasi-isomorphism of the complexes.

This Corollary together with Remark 4.4 imply the Goresky-MacPherson formula for the cohomology groups of C(A).

5. Multiplicative structure of CM

In this section, we prove that CM is a subalgebra of M and compute the product of two critical monomials explicitly in terms of the poset X labeled by dimension.

In fact, we do first a more general computation. Consider the C-linear space W of all formal linear combinations of the pairs (T, m) where T ⊆ X is a flag and m : T → N. We want to convert W into an associative algebra. For that we recall the shuffle product on the free Abelian group generated by flags (cf. [13], 3.4). Let T_{1} = (A_{1} ⊂ A_{2} ⊂ ⋯ ⊂ A_{p}) and T_{2} = (B_{1} ⊂ B_{2} ⊂ ⋯ ⊂ B_{q}) and m_{i} : T_{i} → N (i = 1, 2). Denote by λ the natural operator that turns an arbitrarily subset of X into a flag, where the subset is equipped with a linear order independent of the order on the poset. More precisely, for {C_{1}, …, C_{k}} ⊆ X, linearly ordered along increasing indexes

λ(C_{1}, C_{2}, …, C_{k}) = (C_{1}, C_{1} + C_{2}, …, C_{1} + C_{2} + ⋯ + C_{k})

if all the elements in the right-hand side are distinct and 0 (of the group) otherwise. Then define

T_{1} ∘ T_{2} = ∑_{π}(signπ)λπ(T_{1} ∪ T_{2})

where π runs through all (p, q)-shuffles of T_{1}∪T_{2} (these are the permutations preserving the order among A_{i}’s and the order among B_{j}’s). It is easy to check that the defined multiplication is associative (see [13], p.88). For the future use put for a (p, q)-shuffle π of T_{1} ∪ T_{2}

(T_{1} ∪ T_{2})^{π} = λπ(T_{1} ∪ T_{2}).

We want to extend this multiplication to W. For that fix a (p, q)-shuffle π and notice that the flag (T_{1} ∪ T_{2})^{π} consists of some of the spaces A_{i} + B_{j} where 0 ≤ i ≤ p and 0 ≤ j ≤ q (we always put A_{0} = B_{0} = 0). Notice also that the predecessor of A_{i} + B_{j} is either A_{i−1} + B_{j} or A_{i} + B_{j−1} (where the predecessor of the smallest element A_{1} = A_{1} + B_{0} (or B_{1} = A_{0} + B_{1}) is taken to be A_{0} + B_{0} = 0). Define (m_{1} ∘ m_{2})^{π} : (T_{1} ∪ T_{2})^{π} → N via

(m_{1} ∘ m_{2})^{π}(A_{i} + B_{j}) = m_{1}(A_{i}) or m_{2}(B_{j})

if the predecessor of A_{i} + B_{j} is A_{i−1} + B_{j} or A_{i} + B_{j−1} respectively. Now put

(T_{1}, m_{1}) ∘ (T_{2}, m_{2}) = ∑_{π} sign(π)((T_{1} ∪ T_{2})^{π}, (m_{1} ∘ m_{2})^{π})

where π again runs through the group of (p, q)-shuffles of T_{1} ∪ T_{2}. The associativity of this product follows easily from the associativity of the shuffle product of flags. Hence this product defines a structure of an associative algebra on W (with the identity represented by the empty flag).

Define the linear map f : W → M via f(∅) = 1 and

f((T, m)) = τ_{A_{1}} ⋯ τ_{A_{p}} ∏_{i=1}^{p} σ_{A_{i}}^{m(A_{i})}

for T = (A_{1} ⊂ ⋯ ⊂ A_{p}).

The main theorem of this section is as follows.

Theorem 5.1. The map f is a homomorphism of the algebras.

Proof. Fix (T_{1}, m_{1}) and (T_{2}, m_{2}) from W where T_{1} = (A_{1} ⊂ ⋯ ⊂ A_{p}) and T_{2} = (B_{1} ⊂ ⋯ ⊂ B_{q}). We need to prove that

f(T_{1}, m_{1})f(T_{2}, m_{2}) = f((T_{1}, m_{1}) ∘ (T_{2}, m_{2})).  (5.1)

First we consider particular cases.

(a) Suppose m_{i} = 0 (i = 1, 2) and p = q = 1. Then (5.1) is a relation of type (3.2).

(b) Suppose m_{i} = 0 (i = 1, 2) and p = 1. Putting A = A_{1} we have (5.1) in the form

τ_{A}τ_{B_{1}} ⋯ τ_{B_{q}} = ∑_{r=0}^{q}(−1)^{r}τ_{B_{1}} ⋯ τ_{B_{r}}τ_{A+B_{r}} ⋯ τ_{A+B_{q}}.  (5.2)

We prove (5.2) by induction on q using (a) as the base for q = 1. For q > 1 by the inductive hypothesis we have

τ_{A}τ_{B_{1}} ⋯ τ_{B_{q}} = ∑_{r=0}^{q−1}(−1)^{r}τ_{B_{1}} ⋯ τ_{B_{r}}τ_{A+B_{r}} ⋯ τ_{A+B_{q−1}}τ_{B_{q}}.  (5.3)

Applying relation (3.2) to the last two factors of (5.3) we can substitute instead of them

τ_{A+B_{q−1}}τ_{A+B_{q}} − τ_{B_{q}}τ_{A+B_{q}}.  (5.4)

From the second summand of (5.4) we get q − 1 monomials ending with

τ_{A+B_{q−2}}τ_{B_{q}}τ_{A+B_{q}}  (5.5)

and the monomial

τ_{B_{1}} ⋯ τ_{B_{q−1}}τ_{B_{q}}τ_{A+B_{q}}  (5.6)

with coefficient (−1)^{q}. Applying again relation (3.2) to the first two factors from (5.5) and using that A + B_{q−2} + B_{q} = A + B_{q} and τ_{A+B_{q}}^{2} = 0 we obtain that all q − 1 monomials of the first kind are 0. Thus (5.2) follows.

(c) Suppose m_{i} = 0 (i = 1, 2). The proof is straightforward by induction on p using (b) as the base for p = 1 and the associativity of the shuffle product.

(d) To finish the proof we need the following claim:

τ_{A}τ_{A+B}σ_{B} = τ_{A}τ_{A+B}σ_{A+B}

for every A, B ∈ X. The proof is similar to the proof of relation (3.2) using also that τ_{A+B}^{2} = 0. The details are left to the reader.

(e) Consider the general case. By (c) the left hand side of (5.1) can be rewritten as

∑_{π}(signπ)f((T_{1} ∪ T_{2})^{π}, 0) ∏_{i=1}^{p} σ_{A_{i}}^{m_{1}(A_{i})} ∏_{j=1}^{q} σ_{B_{j}}^{m_{2}(B_{j})}.  (5.7)

Fix a π and denote the respective monomial of (5.7) by s(π). Suppose A_{i} + B_{j} ∈ (T_{1} ∪ T_{2})^{π}. If A_{i−1} + B_{j} ∈ (T_{1} ∪ T_{2})^{π} (and thus precedes A_{i} + B_{j}) then apply (d) to

τ_{A_{i−1}+B_{j}}τ_{A_{i}+B_{j}}σ_{A_{i}}^{m_{1}(A_{i})}  (5.8)

m_{1}(A_{i}) times. Since A_{i−1} + B_{j} + A_{i} = A_{i} + B_{j} we can substitute

τ_{A_{i−1}+B_{j}}τ_{A_{i}+B_{j}}σ_{A_{i}+B_{j}}^{m_{1}(A_{i})} = τ_{A_{i−1}+B_{j}}τ_{A_{i}+B_{j}}σ_{A_{i}+B_{j}}^{(m_{1}∘m_{2})^{π}(A_{i}+B_{j})}

instead of (5.8). The case where A_{i} + B_{j−1} ∈ (T_{1} ∪ T_{2})^{π} can be handled similarly. Since eventually every factor of the form σ_{C}^{k} is used we obtain

s(π) = f((T_{1} ∪ T_{2})^{π}, (m_{1} ∘ m_{2})^{π})

which completes the proof. □

Corollary 5.2. Recall that cμ(T) denotes the critical monomial defined by T (compare the beginning of Section 4). For every two flags T_{1} and T_{2} with the maximal elements A and B respectively we have

cμ(T_{1})cμ(T_{2}) = ∑_{π}(signπ)cμ(T_{1} ∪ T_{2})^{π}

if A ∩ B = 0 (equivalently dim(A + B) = dim A + dim B) and

cμ(T_{1})cμ(T_{2}) = 0

otherwise. Here again π runs through all the (|T_{1}|, |T_{2}|)-shuffles of T_{1} ∪ T_{2}.

Proof. Fix T_{1} and T_{2} as in the proof of the theorem above and fix a (p, q)- shuffle π. Define m_{k} : T_{k} → ZZ_{+} (k = 1, 2) via m_{1}(A_{i}) = dim A_{i}−dim A_{i−1}− 1 and m_{2}(B_{j}) = dim B_{j} − dim B_{j−1} − 1. By definition cμ(T_{k}) = f((T_{k}, m_{k})) (k = 1, 2).

Suppose first that A_{p}∩B_{q} = 0. We need to show that (m_{1}∘m_{2})^{π} coincides with the exponent function in cμ(T_{1} ∪ T_{2})^{π}. Let A_{i} + B_{j} ∈ (T_{1} ∪ T_{2})^{π}. If its predecessor is A_{i−1} + B_{j} then we have

(m_{1} ∘ m_{2})^{π}(A_{i} + B_{j}) = m_{1}(A_{i}) = dim A_{i} − dim A_{i−1} − 1
= dim(A_{i} + B_{j}) − dim(A_{i−1} + B_{j}) − 1  (5.9)

since A_{i} ∩ B_{j} = A_{i−1} ∩ B_{j} = 0. The case where the predecessor is A_{i} + B_{j−1} can be handled similarly. Thus

f((T_{1} ∪ T_{2})^{π}, (m_{1} ∘ m_{2})^{π}) = cμ((T_{1} ∪ T_{2})^{π})

which completes the proof of the first case.

Now suppose that A_{p}∩B_{q} ≠ 0. Then there exists A_{i}+B_{j} ∈ (T_{1}∪T_{2})^{π} that is the smallest (by inclusion) element such that dim(A_{i} + B_{j}) < dim A_{i} + dim B_{j}. To simplify notation put m = (m_{1} ∘ m_{2})^{π}, T = (T_{1} ∪ T_{2})^{π} and denote all the elements of T from the smallest to A_{i} + B_{j} by C_{1}, C_{2}, …, C_{k}. Then similar calculations as in (5.9) give m(C_{i}) = dim C_{i} − dim C_{i−1} − 1 for i < k and m(C_{k}) ≥ dim C_{k} − dim C_{k−1}. Applying several times relation (3.3) we see that f((T, m)) is equal to a monomial having σ_{C_{k}}^{ℓ} in it with ℓ ≥ dim C_{k}. Thus f((T, m)) = 0, which completes the proof. □

Now Corollaries 5.2 and 4.7 imply the following.

Corollary 5.3. CM is a (differential graded) subalgebra of M and a rational model of C(A).

Remark 5.4. DGA CM has a natural integer version (and a differential graded subring) CM(ZZ) generated as a free Abelian group by the monomials cμ(T).

6. Algebra H^{∗}(CM)

In this section, we use the combinatorial description of algebra CM from the previous section to give a combinatorial description of algebra H^{∗}(CM).

In the rest of the paper, we will work much with homology of lattices. Let us introduce the notation that we will use. A lattice L has two operations: the least upper bound ∨ (join) and the greatest lower bound ∧ (meet). For any subset σ ⊆ L put

⋁(σ) = ⋁_{A∈σ} A.

The smallest and largest elements of a lattice will be denoted by 0 and 1 respectively unless more specific symbols are appropriate. The set of all the atoms of L will be denoted by 𝔄 = 𝔄(L). An arbitrary linear ordering is fixed on this set and all subsets of atoms will be ordered by the induced order. For every T ∈ L the set of complements of T is C(T) = {S ∈ L|S∨T = 1, S ∧ T = 0}.

With an arbitrary poset P one can associate the abstract simplicial complex F (P) on P whose simplexes are all the flags of F (P). For a lattice L, one also considers another abstract simplicial complex, the atomic complex Δ(L) on 𝔄. Its simplexes are the subsets σ ⊆ 𝔄 with ⋁(σ) < 1. Let L_{0} = L \ {0, 1}. Then Δ(L) is homotopy equivalent to F (L_{0}) (see [2] and Lemma 6.1 below) whence their homology are isomorphic. This homology is called the homology of L and denoted by H_{∗}(L). For any simplicial complex Δ we denote by C(Δ) its chain complex over Ql augmented by Ql in degree -1 (although many results below hold over ZZ).

We need to fix a particular homotopy equivalence of the chain complexes C(Δ(L)) and C(F (L_{0})). For that denote by Δ′(L) the barycentric subdivision of Δ(L) and identify Δ′(L) with F (Q) where Q is the poset of all nonempty simplexes of Δ(L) ordered by inclusion. Denote by β the standard homotopy equivalence β : C(Δ(L)) → C(F (Q)) defined by

β({A_{1}, …, A_{p}}) = ∑_{δ} signδ({A_{δ(1)}}, {A_{δ(1)}, A_{δ(2)}}, ⋯, {A_{δ(1)}, …, A_{δ(p)}})

where {A_{1}, …, A_{p}} is an arbitrary simplex of Δ(L) (ordered according to the fixed order on 𝔄) and the summation is taken over all the permutations δ of rank p . Also define the order preserving map γ : Q → L_{0} via γ(σ) = ⋁(σ) for σ ∈ Δ(L) and keep the same symbol for the respective chain map C(F (Q)) → C(F (L_{0})). The map γ is a homotopy equivalence since for every C ∈ L_{0} the poset γ^{−1}({B ∈ L_{0}|B ≤ C}) has the unique maximal element {A ∈ 𝔄(L)|A ≤ C} whence is contractible (e.g., see [14]). Thus we can register the following lemma.

Lemma 6.1. For a lattice L the group homomorphism f_{L} = γβ : C(Δ(L)) → C(F (L_{0})) given on atomic simplexes by

f_{L}(A_{1}, …, A_{p}) = ∑_{δ} signδ(A_{δ(1)} ≤ A_{δ(1)} ∨ A_{δ(2)} ≤ ⋯ ≤ ⋁_{i=1}^{p} A_{δ(i)}),

where the summation is taken over all the permutations δ of rank p and every flag with repetitions considered to be 0, is a homotopy equivalence of chain complexes.

Let us introduce several more pieces of notation. For every A, B ∈ L such that B ≤ A there is the lattice (closed interval) [B, A] = {C ∈ L|B ≤ C ≤ A} whose homology is denoted simply by H_{∗}(B, A). The set of atoms of [0, A] we denote by 𝔄(A), the flag complex of [0, A]_{0} = (0, A) by F (A) and the atomic complex of [0, A] by Δ(A). For any simplicial complex Δ we denote by S(Δ) its suspension. For any simplicial complexes Δ_{1} and Δ_{2} we denote by Δ_{1} ∗ Δ_{2} their join. Also we use \overline{Δ} for the simplex on all the vertices of Δ. For every A, B ∈ L we denote by Δ(A, B) the full subcomplex of Δ(A ∨ B) on the vertex set 𝔄(A, B) = 𝔄(A) ∪ 𝔄(B). The joins of all subsets of 𝔄(A, B) form a sublattice of [0, A ∨ B] that we denote by L_{A,B} and we write i(A, B) for the inclusion L_{A,B} ⊆ [0, A∨B]. We denote by F (A, B) the flag complex of L_{A,B}.

Now we return to the poset X of subspaces. When we apply to X notation and results from lattice theory we always mean that 0 is adjoined to it.

Lemma 6.2. If A, B ∈ X are such that A ∩ B = 0 then

Δ(A, B) = (\overline{Δ(A)} ∗ Δ(B)) ∪ (Δ(A) ∗ \overline{Δ(B)}).

Proof. Condition A ∩ B = 0 implies the similar condition on any subspaces of A and B. In particular, for A_{1} ≤ A and B_{1} ≤ B we have A_{1} ∨B_{1} = A∨B only if A_{1} = A and B_{1} = B. The result follows immediately. □

Lemma 6.2 implies in particular that under the condition of the lemma, Δ(A, B) is homotopy equivalent to S(Δ(A) ∗ Δ(B)). To fix a homotopy equivalence it is convenient to pass to Δ(A, B)′. Denote the additional vertices of the suspended complex S(Δ(A) ∗ Δ(B)) by a and b and consider the chain map

ϕ = ϕ(A, B) : C(S(Δ(A) ∗ Δ(B))) → C(Δ(A, B)′)

defined as follows. First of all, it acts as β on simplexes in Δ(A) ∗ Δ(B). Recall that the images under β are flags of simplexes in Δ(A, B). Then ϕ maps a and b to the barycenters of \overline{Δ(A)} and \overline{Δ(B)} respectively, that are 𝔄(A) and 𝔄(B). Notice that the latter sets of atoms can, as simplexes in Δ(A, B) (cf. Lemma 6.2), be included into the flags that occur as images under β.

For this map to be well-defined we need to specify that the vertices of the suspension are taken last among vertices of simplexes containing them. For instance, if {A_{1}} ∈ Δ(A), {B_{1}} ∈ Δ(B) and A_{1} < B_{1} in the ordering on atoms then

ϕ({A_{1}, B_{1}, a})

= ({A_{1}} ⊂ {A_{1}, B_{1}} ⊂ 𝔄(A) ∪ {B_{1}}) − ({A_{1}} ⊂ 𝔄(A) ⊂ 𝔄(A) ∪ {B_{1}}) −({B_{1}} ⊂ {A_{1}, B_{1}} ⊂ 𝔄(A) ∪ {B_{1}}).

Lemma 6.3. The map ϕ is a homotopy equivalence of chain complexes.

This follows from Lemma 6.2 and from the fact that ϕ is really an algebraic version of the natural homotopy equivalence of realizations of the involved abstract complexes.

Finally we need a result from elementary algebraic topology. It is wellknown but hard to find in the literature. It follows immediately from an exact homology sequence for the join of two complexes ([11], p. 373), the isomorphism for the homology of the suspension ([11], p. 47), and the Künneth formula. Specialized for our situation, the result is as follows.

Lemma 6.4. For every A, B ∈ X the group homomorphism

ψ = ψ(A, B) : C(Δ(A)) ⊗ C(Δ(B)) → C(S(Δ(A) ∗ Δ(B))),

given by

σ ⊗ τ ↦ (−1)^{dim τ} signϵ(σ, τ)(σ ∪ τ ∪ {b} − σ ∪ τ ∪ {a})

for σ ∈ Δ(A) and τ ∈ Δ(B), is a quasi-isomorphism of degree +2 of chain complexes. Here ϵ(σ, τ) is the shuffle of σ ∪ τ putting all the elements of τ after elements of σ.

It is important to keep in mind that all our chain complexes are augmented and defined over a field. The sign for ψ(σ ⊗ τ) in Lemma 6.4 is chosen so that the diagram in the proof of Theorem 6.6 will be commutative.

If we put α(A, B) = α = ϕ(A, B)ψ(A, B) and keep in mind that the poset X_{A,B} is homotopy equivalent to the face poset of Δ(A, B) then Lemmas 6.3 and 6.4 and the Künneth formula imply the following.

Corollary 6.5. The map

α_{∗} : H̃_{r}((0, A), Ql ) ⊗ H̃_{s}((0, B), Ql ) → H̃_{r+s+2}(X_{A,B}, Ql )

is an isomorphism for every r and s.

Now we are ready to describe H^{∗}(C(A), Ql ) = H^{∗}(CM).

Theorem 6.6. (i) For every p we have isomorphisms

H̃^{p}(C(A), Ql ) ≈ H̃^{p}(CM) ≈ ⊕_{A∈X} H̃_{2 dim A−p−2}((0, A), Ql ).

(ii) Under the isomorphism from (i) the multiplication on H^{∗}(C(A), Ql ) is given by the compositions

H̃_{r}((0, A), Ql )⊗H̃_{s}((0, B), Ql )⟶^{α_{∗}}H̃_{r+s+2}(X_{A,B}, Ql )⟶^{i_{∗}}H̃_{r+s+2}((0, A∨B), Ql )

if A ∩ B = 0 (equivalently dim(A ∨ B) = dim A + dim B) and 0 otherwise.

Proof. (i) has been proved before in Section 4 and is included here just for completeness. Thus we need to prove (ii) only.

In the rest of the proof all homology groups have the rational coefficients and we suppress Ql . We know from Corollary 5.2 that the multiplication on H_{∗}(0, A) ⊗ H_{∗}(0, B) is 0 unless A ∩ B = 0. Hence let us fix A, B ∈ X such that A ∩ B = 0. Then consider the following diagram

C(Δ(A)) ⊗ C(Δ(B)) ⟶^{ψ(A,B)} C(S(Δ(A) ∗ Δ(B))) ⟶^{ϕ(A,B)} C(Δ(A, B)′)
↓^{f_{A}⊗f_{B}}  ↓^{γ(A,B)}
C(F (A)) ⊗ C(F (B))  ⟶^{ν(A,B)}  C(F (A, B))

where ν(A, B) is the shuffle multiplication of flags from the previous section and γ(A, B) is the natural map (introduced before Lemma 6.1) for the lattice X_{A,B}. Notice that before one applies the shuffle multiplication to two flags from (0, A) and (0, B) one should augment them by A and B respectively and then delete A ∨ B from all flags in the support of the product. A straightforward check shows that the diagram commutes. Since the vertical maps are homotopy equivalences, passing to the homology completes the proof. □

The statement (ii) of Theorem 6.6 makes sense over ZZ. I make the following conjecture.

Conjecture 6.7. Theorem 6.6 (ii) holds for the ring H^{∗}(C(A), ZZ) (using integer homology of the posets). In particular, this ring is defined by the poset structure on X (the “combinatorics” of the arrangement) and the dimensions of its elements.

7. Geometric lattices

In this section, we suppose that X (augmented by 0) is a geometric lattice, i.e., every element is the join of a set of atoms, for every A ∈ X all maximal flags ending on A have the same length (that is the number of elements minus 1) denoted by rkA and rk(A ∨ B) + rk(A ∧ B) ≤ rkA + rkB for every A, B ∈ X. In this case we are able to give a more explicit description of H^{∗}(C(A)) (all cohomology and homology in this and the following section are over Ql ).

By Folkman’s theorem [7] H̃_{p}(0, A) ≠ 0 only for p = rkA − 2.

The following observation will be helpful.

Lemma 7.1. For every A, B ∈ X such that A∩B = 0 we have rk(A∨B) = rkA + rkB.

Proof. Put k = rkA, ℓ = rkB and fix two flags in X

0 = A_{0} < A_{1} < A_{2} < ⋯ < A_{k} = A, 0 = B_{0} < B_{1} < B_{2} < ⋯ < B_{ℓ} = B.

For any i, 0 ≤ i ≤ ℓ, we have A ∩ B_{i} ⊆ A ∩ B = 0 whence dim(A + B_{i}) = dim A + dim B_{i}. Thus we have the flag in X

0 < A_{1} < A_{2} < ⋯ < A_{k} = A < A + B_{1} < A + B_{2} < ⋯ < A + B_{ℓ} = A + B

which implies rk(A ∨ B) ≥ rkA + rkB. The opposite inequality follows immediately from the inequality for rank in geometric lattices. □

Thus the multiplication ν(A, B), which is 0 if A ∩ B ≠ 0, is a map

ν(A, B) : H̃_{rkA−2}(0, A) ⊗ H̃_{rkB−2}(0, B) → H̃_{rk(A∨B)−2}(0, A ∨ B) (7.1)

otherwise.

To describe multiplication (7.1) more precisely and give a presentation of H^{∗}(C(A)) we use the well known basis of the homology of geometric lattices (e.g., see [13], p.144). Recall that a set σ of atoms of a geometric lattice is called independent if rk ⋁(σ) = |σ|. Clearly every subset of an independent set is again independent. Suppose A ∈ X with rkA = p. Fix an arbitrary linear order on the set of atoms A(X). Then for any independent subset σ = {A_{1}, …, A_{p}} ⊆ A(A) the cycle ∂(σ) = ∑_{i=1}^{p}(−1)^{i}σ \ {A_{i}} of Δ(A) defines a non-zero element ζ_{σ} ∈ H̃_{p−2}(0, A) and all these elements generate the space. If a set σ is dependent we will use the symbol ζ_{σ} sometimes too but its value will be always 0. In general the elements ζ_{σ} are not linearly independent over ZZ even for independent sets σ. Linear relations among them are generated by

∑_{j=1}^{p+1}(−1)^{j}ζ_{τ\{B_{j}}}  (7.2)

for every dependent set τ = {B_{1}, …, B_{p+1}} ⊆ A(A). Lemma 7.1 implies that if A ∩ B = 0 and σ ⊆ A(A) and τ ⊆ A(B) are independent sets then the set σ ∪ τ ⊆ A(A ∨ B) is again independent.

Now Theorem 6.6 implies the following result.

Proposition 7.2. Suppose X is a geometric lattice. Then the algebra H^{∗}(C(A)) is generated by ζ_{σ} ∈ H^{q}(C(A)) where σ runs through all the independent sets of atoms of X and q = 2 dim ⋁(σ) − |σ|. Besides the linear relations (7.2), only other generating relations are

ζ_{σ}ζ_{τ} = signϵ(σ, τ)ζ_{σ∪τ}

if ⋁(σ) ∩ ⋁(τ) = 0 (where ϵ(σ, τ) is the permutation of σ ∪ τ putting all the elements of τ after elements of σ) and

ζ_{σ}ζ_{τ} = 0

otherwise.

Remark 7.3. (i) In general, relations (7.2) are not linearly independent. One can easily construct a linear resolution of the space of generators ζ_{σ} and recover Folkman’s formula for the dimension of H_{∗}(0, A) from [7].

(ii) Proposition 7.2 shows in particular that there exists a quasi-iso-morphism H^{∗}(C(A)) → M (of graded differential algebras with d = 0 on H^{∗}(C(A))) whence the space C(A) is formal in this case.

(iii) In the case when X is not only geometric but Boolean all the homologies in (7.1) are 1-dimensional and the multiplication, if nontrivial, is clearly an isomorphism (cf. [6]). In general the multiplication is not an isomorphism.

8. Cohomology rings of “k-equal” manifolds

In this section, for each pair (n, k) of integers with 2 ≤ k ≤ n and for each k-subset ω ⊆ {1, 2, …, n} we consider the subspace V (ω) ⊂ C^{n} = {(x_{1}, …, x_{n})} defined by the system of equations ∑_{i∈ω} x_{i} = 0 and x_{j} = 0 for j ∉ ω. The set of all V (ω) (for fixed n and k) is denoted by A_{n,k}^{∘}. Notice that the annihilator V (ω)^{0} of V (ω) is defined in (C^{n})^{∗} = {(z_{1}, …, z_{n})} by the equations z_{i} = z_{j} for i, j ∈ ω. That justifies the name k-equal manifold introduced in [3] for M_{n,k} = C(A_{n,k}^{∘}).

The linear structure of H^{∗}(M_{n,k}) has been studied in detail in [3]. Completing A_{n,k}^{∘} by the sums of its elements one obtains the lattice Π_{n,k} that is (naturally isomorphic to) the lattice of all partitions of the set {1, …, n} whose blocks consist of either 1 element (trivial blocks) or at least k elements (non-trivial blocks). In the rest of the paper, we will regard elements of Π_{n,k} both as subspaces and partitions. Theorem 1.5 of [3] asserts that H̃_{p}(Π_{n,k}) ≠ 0 if and only if p = n − 3 − t(k − 2) for some integer t such that 1 ≤ t ≤ ⌊n/k⌋.

Although the lattice Π_{n,k} is not geometric in general, it has a similar property; there exist generators of its homology groups represented each by a simplicial sphere in the atomic complex. To prove that we need to use a recursive construction from [3] hence to include into consideration more general lattices introduced there. For any non-negative integers n, k, and ℓ such that n, k ≥ 2 and either k ≤ n or ℓ > 0, define Π_{n,k}(ℓ) as the family of all partitions of the set {1…. , n} such that every block b satisfies at least one of the following requirements:

(i) |b| = 1,

(ii) |b| ≥ k,

(iii) b ∩ {1, …, ℓ} ≠ ∅. Ordered by refinement Π_{n,k}(ℓ) becomes a lattice and Π_{n,k}(0) = Π_{n,k}.

First we need to recall the recursive construction used in [3] for the homology groups of Π_{n,k}(ℓ). Suppose n ≥ 3 and k, ℓ ≤ n − 1. Let T be a coatom of Π_{n,k}(ℓ) having the non-trivial block b = {1, …, n − 1}. Clearly there are two types of complements S of T, both are atoms whose unique non-trivial blocks we denote by b_{1}. Either |b_{1}| = k, n ∈ b_{1}, and b_{1} ∩ {1, …, ℓ} = ∅ or b_{1} = {i, n} where i ≤ ℓ. In the former case the poset [S, 1] is isomorphic to Π_{n−k+1,k}(ℓ + 1), in the latter case it is isomorphic to Π_{n−1,k}(ℓ). The subposet (Π_{n,k}(ℓ))_{0} = (0, 1) of Π_{n,k}(ℓ) is homotopy equivalent to the wedge of suspensions of the posets (S, 1) where S runs over C(T) (see details and references in [3]). In particular, taking a basis of homology for each [S, 1] and applying suspension one obtains a basis of homology of Π_{n,k}(ℓ). Notice that for each choice of S the homotopy type of the suspension of (S, 1) can be realized as a subposet of (0, 1). For instance, one can take as such a realization (S, 1) ∪ {S, T } to which at least one atom A_{0} ≤ T ∧ A for each A ∈ (S, 1) is added. We will call any of these (homotopy equivalent to each other) posets the suspension of (S, 1) with vertices S and T. We will also use the same term for the suspension of homology classes of (S, 1).

The result of the above recursive construction for Π = Π_{n,k} can be described as follows. Fix a maximal flag 0 = S_{0} < S_{1} < S_{2} < ⋯ < S_{p+1} < 1 in Π and a sequence of elements T_{1}, T_{2}, …, T_{p} such that for every i = 1, 2, …, p the element T_{i} has a non-trivial block of size n − 1, T_{i} ∨ S_{i} = 1 and T_{i}∧S_{i} = S_{i−1}. A pair (S′, S_{p+1}), where S′ is any atom of [S_{p}, 1] different from S_{p+1}, defines an element α_{p} ∈ H̃_{0}(S_{p}, 1). Applying recursively suspensions with vertices S_{i} and T_{i} one obtains a sequence of α_{i} ∈ H̃_{p−i}(S_{i}, 1). Then elements α_{0} taken for all the above data generate H̃_{p}(Π). The condition α_{0} ≠ 0 puts restrictions on S_{p} and S_{p+1}. In fact, the variety of possibilities reduces to the following four cases (cf. Lemma 4.4 from [3]).

(a) n = k + 1 and p = 0;

(b) S_{p} has only one non-trivial block of n − k elements and S_{p+1} has besides the complementary block (of k elements);

(c) S_{p} has three blocks all together exactly one of which is non-trivial (k ≥ 3);

(d) S_{p} has three blocks all together at least two of which are non-trivial. Now we are ready to study the multiplication on local homology of Π.

For every p consider the homomorphism

ξ_{p} : ⊕_{{A,B}}H̃_{p}(Π_{A,B}) → H̃_{p}(Π)

induced by embeddings where {A, B} runs through all the subsets of Π such that A and B are one-block partitions, A ∧ B = 1 and A ∩ B = 0. The last two conditions can be substituted by b_{A} ∪ b_{B} = {1, …, n} and |b_{A} ∩ b_{B}| = 0 where b_{A} and b_{B} are the non-trivial blocks of A and B respectively. Recall that the maximal p with H̃_{p}(Π) ≠ 0 is p = n − k − 1. It is easy to see that ξ_{n−k−1} = 0 since its domain is 0. On the other hand we have the following.

Lemma 8.1. For p < n − k − 1 the homomorphism ξ is surjective.

Proof. It suffices to fix generators of H̃_{p}(Π) (p < n − k − 1) and for each α of them find A and B with required properties and such that α ∈ Imξ.

To fix an element α of a generating set we fix sequences (S_{i}) and (T_{i}) as above and, using suspension, construct recursively α_{i} ∈ H̃_{p−i}(S_{i}, 1) starting from α_{p} and finishing at α_{0} = α. The condition p < n − k − 1 restricts possibilities for S_{p} to (b) and (d) only. In case (b), we take as S′ any atom of [S_{p}, 1] different from S_{p+1} (α_{p} is essentially unique in this case). In case (d), we are a little more careful. Each S_{i} has a block that contains the nontrivial block of S_{1}. We call this block of S_{i} distinguished. While choosing an atom S′ we avoid repetition of the distinguished block of S_{p}, i.e if this block is a proper subset of another block in S_{p+1} then in S′ this block is not a proper subset of any block. Clearly such elements α_{p} generate H̃_{0}(S_{p}, 1).

Now we denote by A_{p} the element of the pair {S_{p+1}, S′} that contains the distinguished block of S_{p} as a proper subset of another block and by B_{p} the other element of the pair. Our goal is to construct recursively a sequence (A_{i}, B_{i}) (i = p, p − 1, …, 0) with the following properties.

(i) A_{i}, B_{i} > S_{i} and each has only one non-trivial block (b′_{i} and b′′_{i} respectively) with respect to S_{i};

(ii) b′_{i} ∩ b′′_{i} is a block of S_{i};

(iii) A_{i} ∨ B_{i} = 1;

(iv) For i > 0 only A_{i} (and not B_{i}) has a block that contains the distinguished block of S_{i} as its proper subset;

(v) α_{i} is in the image of H̃_{i}([S_{i}, 1]_{A_{i},B_{i}}) in H̃_{i}(S_{i}, 1).

Notice that A_{p} and B_{p} satisfy the conditions (i)-(v). Now suppose that for some i, 0 < i ≤ p, A_{i} and B_{i} satisfying (i)-(v) are constructed. The construction of A_{i−1} and B_{i−1} depends on b′_{i} ∩ b′′_{i}. Denote by b_{i} the unique block of S_{i} that is non-trivial with respect to S_{i−1} and let b_{i} = ∪_{j}b_{ij} be its partition into blocks of S_{i−1}. If b′_{i} ∩ b′′_{i} ≠ b_{i} then define A_{i−1} and B_{i−1} as having the non-trivial blocks b′_{i−1} = b′_{i} and b′′_{i−1} = b′′_{i} respectively, relative to S_{i−1}. If b′_{i} ∩ b′′_{i} = b_{i} then define b′′_{i−1} as in the previous case and put b′_{i−1} = (b′_{i} \ b_{i}) ∪ b_{ij} for some j such that b_{ij} is a subset of the non-trivial block of T_{i}. The property (iv) of A_{i} and B_{i} guaranties that A_{i−1} ∈ Π. In any case the conditions (i)-(iv) for i − 1 are satisfied by construction and we only need to check (v).

To simplify notation put P_{i} = [S_{i}, 1]_{A_{i},B_{i}} for every i and denote the element not contained in the non-trivial block of T_{i} by a_{i}. First notice that by construction either A_{i−1} or B_{i−1} ≤ T_{i}. Suppose that A_{i−1} ≤ T_{i} since the other case can be handled similarly (even more easily). This implies that b_{i} ⊂ b′′_{i−1}. Then let us prove that T_{i} ∈ P_{i−1}. Since Π is atomic we have A_{i−1}, B_{i−1} ∈ P_{i−1}. Define B′ as the one-block partition with the nontrivial block b′′_{i−1} \ {a_{i}}. Since b_{i} ⊂ b′′_{i−1} we have B′ ∈ Π whence B′ ∈ P_{i−1}. Since A_{i−1} ∨ B′ = T_{i} we have T_{i} ∈ P_{i−1}.

Our second observation is that P_{i} ⊂ P_{i−1}. To see that it suffices to prove that any atom C of [S_{i}, 1] such that C ≤ A_{i} or C ≤ B_{i} is in P_{i−1}. Denote the only non-trivial with respect to S_{i} block of C by b_{C}. If b_{C} ⊅ b_{i} then the atom C_{0} of [S_{i−1}, 1] having only non-trivial block b_{C} with respect to S_{i−1} is in P_{i−1} and C_{0} ∨S_{i} = C. Suppose b_{C} ⊃ b_{i} and C ≤ A_{i}. Then C_{0} = C ∧A_{i−1} is an atom of P_{i−1} and again C_{0} ∨ S_{i} = C. The case C ≤ B_{i} is similar. In any case we have C ∈ P_{i−1}.

Summing up our above observations we see that P_{i−1} contains the suspension of P_{i} with vertices S_{i} and T_{i} (cf. [2] or Proposition 2.3 and p.290 of [3]). Since by the inductive hypothesis (v) holds for i we obtain it for i − 1.

Putting A = A_{0} and B = B_{0} we complete the proof. □

In order to exhibit more explicit generators of H^{∗}(M_{n,k}) we use the atomic complexes as in the previous section. We will always assume that the atoms are linearly ordered and all sets of atoms are provided with the induced ordering. By C(L) we denote the chain complex of the abstract simplicial complex formed by all sets of atoms in L; the differential in C(L) is denoted by ∂.

Definition 8.2. A set σ of atoms of a lattice L is called independent if ⋁(τ) ≠ ⋁(σ) for every proper subset τ of σ. It is called essential if it is not a simplex in the atomic complex Δ(L), i.e. if ⋁(σ) = 1. An element ω of C(L) is called a relater in L if ∂ω can be represented as a linear combination of independent essential sets and simplexes of the atomic complex, i.e., sets bounded by elements less than 1.

Any set σ = {A_{1}, A_{2}, …, A_{p}} of atoms of L, that is either a simplex in Δ(L) or independent, defines the cycle

∂σ = ∑_{i=1}^{p}(−1)^{i}σ \ {A_{i}} ∈ C_{p−2}(Δ(L)).

Denote its homology class by ζ_{σ}. If σ is a simplex of Δ(L) then ζ_{σ} = 0. Any relater ω with ∂ω = ∑_{i} a_{i}ω_{i} + ∑_{j} b_{j}ω′_{j} where a_{i}, b_{j} ∈ ZZ, ω_{i} are independent essential sets and ω′_{j} are simplexes in Δ(L) defines the linear relation ∑_{i} a_{i}ζ_{ω_{i}} = 0. This motivates the following. Denote by G_{p} the linear space generated by symbols η_{σ} where σ is either a simplex of Δ(L) or an independent essential set of atoms of L and |σ| = p + 2. Denote by R_{p} the subspace of G_{p} generated by the elements η_{σ} where σ is a simplex in Δ(L), and let r_{ω} = ∑_{i} a_{i}η_{ω_{i}} where ω is a relater as above and |ω| = p + 3.

Fix n, k, ℓ such that 2 ≤ k ≤ n and ℓ ≥ 0 and put Π = Π_{n,k}(ℓ).

Lemma 8.3. (i) For every atom S of Π and a coatom T with only one nontrivial block such that T ∈ C(S) there exists an injective map θ : A([S, 1]) → A(Π) \ {S} such that ⋁(Imθ) ≤ T and θ(τ) ∪ {S} is independent (essential) in Π if and only if τ is independent (resp. essential) in [S, 1].

Proof. Let S and T be as in the statement of the lemma. Consider the sets W_{A} = A(T ∧ A) for A ∈ A([S, 1]). It is easy to see that for every B ≤ T ∧ A we have

B ∨ S = A.

This implies that the sets W_{A} are pairwise disjoint. Choosing arbitrarily θ(A) ∈ W_{A} we obtain an injective map θ : A([S, 1]) → A(Π) \ {S}.

First let us prove the following property of θ. For every τ ⊆ A([S, 1])

⋁(θ(τ) ∪ {S}) = ⋁(τ).  (8.1)

Indeed put τ′ = θ(τ ∪ {S}). If D ≥ B for every B ∈ τ then D > S and D > (T ∧ B) whence D > A for every A ∈ τ′. Conversely if D ≥ A for every A ∈ τ′ then D ≥ (θ(B) ∨ S) = B for every B ∈ τ.

Now it is easy to see that all the required properties of θ follow from construction and (8.1). □

Remark 8.4. The construction of a map θ in Lemma 8.3 uses an atom S and a coatom T. We will say that θ is constructed with S and T. Also it is clear from the construction that for every σ ⊆ A([S, 1]) the class ζ_{θ(σ)} is the suspension of ζ_{σ} with vertices S and T.

To make the notation easier put Π(ℓ) = Π_{n,k}(ℓ) for the length of the next proposition and denote the set of all essential independent subsets of atoms of Π(ℓ) by I(Π(ℓ)).

Proposition 8.5. For every p, the linear map ι : G_{p}/R_{p} → H̃_{p}(Π(ℓ)) defined by ι(η_{σ}) = ζ_{σ} for all σ ∈ I(Π(ℓ)), |σ| = p + 2, is an isomorphism.

Proof. First we prove that ι is surjective. We apply induction on p (for all n, k and ℓ at the same time). As the base consider the case of p = −1. An essential 1-set σ exists if and only if (0, 1) = ∅ (i.e., n = k, ℓ = 0 or n = 2, ℓ ≥ 0) and then it is unique and independent. This is precisely the case when H̃_{−1}(Π(ℓ)) ≠ 0 and ζ_{σ} is a generator of this space.

Now suppose p ≥ 0. If k = n then the result is trivial. If ℓ ≥ n or n = 2 then Π(ℓ) is a geometric lattice and the result follows from Proposition 7.2. Thus we can assume that n ≥ 3, k, ℓ ≤ n − 1 and apply the above recursive construction for H_{p}(Π(ℓ)). Fix a coatom T with only non-trivial block of n − 1 elements and atom S such that S ∈ C(T). Recall that (S, 1) is isomorphic to a poset Π(ℓ)_{n′,k′}(ℓ′) for some value of the parameters n′, k′ and ℓ′. Thus by the inductive hypothesis, one can find a basis of H̃_{p−1}(S, 1) consisting of classes ζ_{σ} where σ runs through independent essential p+1-sets of atoms of (S, 1). Using Lemma 8.3 (i) fix a map θ constructed with S and T and having property from that lemma. In particular, if σ′ = θ(σ) then σ′ is independent and essential. By Remark 8.4 ζ_{σ′} is the suspension of ζ_{σ} with the vertices S and T. Since the suspensions of generators of all spaces H_{p−1}(S, 1) when S runs through C(T) form a generating set of H_{p}(Π(ℓ)) the proof is complete.

Now we prove that ι is an isomorphism. For that consider the factor complex D = C(Π(ℓ))/C(Δ(Π(ℓ))). Notice that D_{p} is generated by the essential sets of atoms of cardinality p+1 and the differential on D is defined by

d(σ) = ∑(−1)^{i}σ \ {A_{i}}

where σ = (A_{1}, …, A_{p+1}) and the summation is taken over all i such that

⋁(σ \ {A_{i}}) = 1.

In particular, an essential set is a cycle in D if and only if it is independent. Applying the exact homological sequence of pair we see that the homomorphism δ : H̃_{p+1}(D) → H̃_{p}(C(Δ)) given by δ(z) = ∂z for any cycle z ∈ D_{p} is an isomorphism. On the class [σ] of an independent essential set σ we have δ([σ]) = ζ_{σ}. Thus the first part of the proof can be restated as follows: the classes of all the independent essential sets generate H_{∗}(D). On the other hand, R_{p} is naturally isomorphic to the intersection d(D_{p+2})∩ < I > in D_{p+1} and the result follows immediately. □

The next lemma follows from Theorem 6.6 (ii) and [3], p.296.

Lemma 8.6. Suppose U ∈ Π_{n,k} and has precisely s non-trivial blocks b_{1}, …, b_{s}. Denote by U_{i} the element of Π_{n,k} having only one non-trivial block b_{i}. Then for every p the repeated multiplication induces the isomorphism

⊕_{p_{1}+···+p_{s}=p−2(s−1)}H̃_{p_{1}}(0, U_{1}) ⊗ ⋯ ⊗ H̃_{p_{s}}(0, U_{s}) ≅ H̃_{p}(0, U).  (8.2)

Definition 8.7. Let σ be a set of one-block partitions (e.g. atoms) from Π_{n,k}. Consider the collection D of all the subsets τ of σ such that the nontrivial blocks of elements of τ are pairwise disjoint. Then the rank of σ (rk(σ)) is the maximal cardinality of elements of D.

Here is the main result of the section.

Theorem 8.8. (i) The space H^{∗}(M_{n,k}) is generated by the classes ζ_{σ} ∈ H^{2 dim U−|σ|}(M_{n,k}) where σ runs through all the independent sets of atoms of Π_{n,k} and U = ⋁(σ). The linear relations among ζ_{σ} are generated by r_{ω} for the relaters ω of all [0, A] (A ∈ X).

(ii) The product ζ_{σ}ζ_{τ} = 0 if ⋁(σ)∩⋁(τ) = 0 (equivalently dim ⋁(σ ∪τ) ≠ dim ⋁(σ) + dim ⋁(τ)) and ζ_{σ}ζ_{τ} = signϵ(σ, τ)ζ_{σ∪τ} otherwise where ϵ(σ, τ) is as above the shuffle of σ ∪ τ putting all the elements of τ after elements of σ.

(iii) The linear relations from (i) imply that ζ_{σ} = 0 unless

|σ| = n(σ) − rk(σ)(k − 2) − s,  (8.3)

where s is the number of non-trivial blocks in the partition ⋁(σ) and n(σ) is the number of points in all these blocks. In particular, ζ_{σ}ζ_{τ} = 0 unless rk(σ ∪ τ) = rk(σ) + rk(τ).

(iv) As an algebra, H^{∗}(M_{n,k}) is generated by the classes ζ_{σ} for independent sets of atoms of rank 1.

Proof. (i) and (ii) follow immediately from Proposition 8.5 and Theorem 6.6.

(iii) Let σ be a independent set of atoms, r = rk(σ), U = ⋁(σ) and U_{1}, …, U_{s} as in Lemma 8.6. Then σ is a disjoint union of σ_{i} (i = 1, …, s) where σ_{i} is independent and ⋁(σ_{i}) = U_{i}. Since r = ∑_{i=1}^{s} rk(σ_{i}) and ζ_{σ} = ζ_{σ_{1}} ⊗⋯⊗ζ_{σ_{s}} under identification (8.2) it suffices to consider the case where s = 1, i.e. U has only one non-trivial block of size n(σ). It is easy to compute that

n(σ) ≥ |σ| + r(k − 2) + 1.  (8.4)

Now the recursive construction before Lemma 8.1 shows that we can assume without any loss of generality that there exists a flag in the support of f_{[0,U]}(σ) that is maximal in (0, U]. If this maximal flag has an element with t non-trivial blocks and no elements with t + 1 ones then its length is

t + (t − 1) + (n(σ) − tk) = n(σ) − t(k − 2) − 1

(t steps to “create” new blocks, t − 1 steps to glue blocks to each other and n(σ) − tk steps to add a point to a block). Since for flags in the support of f_{[0,U]}(σ) the maximal possible t is r we have

|σ| ≥ n(σ) − r(k − 2) − 1.  (8.5)

Comparing (8.4) and (8.5) we obtain (8.3).

The second part of (iii) follows since the equality dim ⋁(σ) + dim ⋁(τ) = dim ⋁(σ ∪ τ) implies n(σ ∪ τ) = n(σ) + n(τ) − 1 for σ and τ such that ⋁(σ) and ⋁(τ) have only one non-trivial block each.

(iv) Fix a independent set σ of atoms of rank r > 1 and let again s be the number of non-trivial blocks of U = ⋁(σ). Using Lemma 8.6 again we can assume that s = 1. Suppose ζ_{σ} ≠ 0. Since r > 1 we have from (iii) that ζ_{σ} ∈ H̃_{p}(0, U) with

p = |σ| − 2 = n(σ) − r(k − 2) − 3 < n(σ) − k − 1.

Now due to Lemma 8.1 and Theorem 6.6 (ii), ζ_{σ} is a linear combination of classes ζ_{τ} ζ_{ω} = ±ζ_{τ∪ω} where for each pair (τ, ω) there exist A and B ∈ [0, U] such that A ∨ B = U, A ∩ B = 0, ζ_{τ} ∈ H_{q}(0, A) and ζ_{ω} ∈ H_{p−q}(0, B) for some q. Since ⋁(τ ∪ ω) = U we obtain from (iii) that rk(τ ∪ ω) = r. Using (iii) again we have rk(τ) < r and rk(ω) < r. Now downward induction on r completes the proof. □

Remark 8.9. For any specified values of n and k, the linear relations among ζ_{σ} can be used to find a basis of H̃_{p}(0, ⋁(σ)) (p = |σ| − 2). In the case rk(σ) = 1 a basis is easy to describe in general which gives a generating set for the algebra H^{∗}(M_{n,k}).

Indeed let the only non-trivial block of U = ⋁(σ) be b = {1, 2, …, n(σ)}. Fix a subset a of b_{0} = b \ {1} with |a| = k − 1 and put σ(a) = {A_{i}|i ∈ b \ a} where A_{i} is the atom whose only non-trivial block is a ∪ {i}. Notice that |σ| = n(σ) − k + 1 as it should be. Then the set (of cardinality \binom{n(σ)−1}{k−1} ) of all ζ_{σ(a)} is a basis of the subspace of H̃_{p}(0, U) generated by ζ_{σ} with rk(σ) = 1.

Example.

Let us consider the case where n = 6 and k = 3. The results for the additive structure of H^{∗}(M_{6,3}) are as the following table shows (cf. [3], p. 311).

dim U |σ| p rank(σ) s(σ) n(σ) q dimH̃^{q}(M)

2  1 −1  1  1  3  3  20
3  2 0  1  1  4  4  45
4  3 1  1  1  5  5  36
5  4 2  1  1  6  6  10
4  2 0  2  2  6  6  10
5  3 1  2  1  6  7  10

In this table, U = ⋁(σ), p = |σ| − 2 is the dimension of the homology of [0, U] generated by ζ_{σ} and q is the respective dimension of the cohomology H^{∗}(M) with M = M_{6,3}. Notice that H^{6}(M) is represented as H^{6}(M) = H_{1}^{6}(M)⊕H_{2}^{6}(M) where H_{1}^{6}(M) is generated by ζ_{σ} with |σ| = 4 and rk(σ) = 1 and H_{2}^{6}(M) is generated by ζ_{σ} with |σ| = 2 and rk(σ) = 2.

As a ring, H^{∗}(M) is generated by H^{i}(M), i = 3, 4, 5, and H_{1}^{6}(M). Using Remark 8.9 it is easy to exhibit a concrete set of algebar generators ζ_{σ}. The multiplication ν is not zero only on H^{3,3} = H^{3}(M) ⊗ H^{3}(M) and H^{3,4} = H^{3}(M) ⊗ H^{4}(M). Due to Theorem 8.8, ν(ζ_{σ} ⊗ ζ_{τ}) ≠ 0 on H^{3,3} only if b_{A} ∩b_{B} = ∅ where σ = {A}, τ = {B} and b_{A}, b_{B} are respective non-trivial blocks of the atoms A and B. This and the skew commutativity of ν give 10 linearly independent products ζ_{σ∪τ} in H_{2}^{6}(M) that generate this whole space.

On H^{3,4}, consider the product π = ζ_{σ∪τ} where σ = {A} and τ = {B_{1}, B_{2}} with |b_{B_{1}} ∩b_{B_{2}}| = 2. Then Theorem 8.8 implies that π ≠ 0 only if either b_{B_{1}} or b_{B_{2}} is disjoint with b_{A} and in this case π does not depend on the other atom in τ. Again it is clear that there are 10 linearly independent products and they generate the whole space H^{7}(M).

It may be instructive to exhibit a independent set of atoms σ such that ζ_{σ} = 0. The simplest examples are given by σ = {B_{1}, B_{2}} with |b_{B_{1}} ∩ b_{B_{2}}| = 1. For instance, if σ = {{1, 2, 3}, {1, 4, 5}} where the atoms are given by their non-trivial blocks then ω = σ ∪ {{1, 2, 4}} is a relater of [0, ⋁(σ)] and the relation ∂ω gives ζ_{σ} = 0. For a less obvious example one can take σ = {{1, 2, 3}, {3, 4, 5}, {3, 5, 6}}. Then rk(σ) = 1 and 3 = |σ| ≠ n(σ) − rk(σ)(k − 2) − s(σ) = 4. On the other hand, if ω = σ ∪ {{1, 3, 5}} then it is a relater of Π_{6,3} and the relation r_{ω} gives ζ_{σ} = 0.

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E-mail address: yuz@math.uoregon.edu

University of Oregon, Eugene, OR 97403