Tropical Poincaré duality spaces

Definition 2.1

A rational polyhedral cone σ in a lattice N is a set of the form

for vectors v_i ∈ N, such that σ_ℝ = σ⊗_ℤℝ ⊂ N_ℝ is closed and strictly convex, hence has a vertex at the origin. The lattice L_ℤ(σ) is the saturated sublattice of N generated by σ, and the dimension of a cone is the rank of L_ℤ(σ).

Another cone τ is said to be a face of σ if there is some element m ∈ Hom_ℤ(N, ℤ), with m(x) ≥ 0 for all x ∈ σ, i.e. a positive functional, such that τ = {x ∈ σ | m(x) = 0}. Any face can also be exhibited by setting particular coefficients a_i to 0.

For a face τ of σ, the set L_ℤ(τ) ⊂ L_ℤ(σ) is a sublattice. For dim τ = dim σ − 1, we may select a primitive integer vector v_σ/τ ∈ N such that L_ℤ(σ) = L_ℤ(τ) + ℤv_σ/τ.

Definition 2.2

A rational polyhedral fan Σ is a finite collection of rational polyhedral cones in N such that:

• For any cone σ ∈ Σ, if τ is a face of σ, then τ ∈ Σ,

• For σ₁, σ₂ ∈ Σ, the intersection σ₁ ∩ σ₂ is a face of σ₁ and σ₂.

The cones in Σ are also called faces, and the collection of faces of dimension i is denoted by Σⁱ . The dimension of Σ is the supremum of the dimensions of cones of Σ. We write τ ⪯ σ if τ is a face of σ and τ ≺ σ if τ is a proper face, which gives a partial ordering on Σ. We say that a face σ ∈ Σ is maximal if it is maximal with respect to the ordering ⪯. We will require that all fans are pure dimensional in the sense that all maximal by inclusion faces are of equal dimension.

Abusing notation, we also write Σ for the category associated to the partial ordering ⪯, whose objects are the cones σ ∈ Σ, with a morphism τ → σ ∈ Hom_Σ(τ, σ) if and only if τ ⪯ σ.

Note that all cones intersect in a common minimal cell, and since we required each cone to have a vertex, this is the unique vertex v in Σ. Moreover, a rational polyhedral fan corresponds to a cell complex in the sense of [27; 11], when considering the fan as glued abstractly from the interiors of the cones.

Example 2.3

Consider the fan Σ displayed in Figure 1, which consists of the rays τ₁, τ₂, τ₃, τ₄ and the vertex v. The fan is pure dimensional, its maximal faces are the τ_i and it has dimension 1. It consists of the union of the line x = 0 and y = 0 when considered in N_ℝ. We have v ≺ τ_i for each i.

Figure 1

The cross

Example 2.4

Another example of a fan is shown in Figure 2. This fan has one vertex v, three one-dimensional cones τ_i , and three two-dimensional cones σ_i . For instance, the faces of σ₁ are the cones τ₁ and τ₂ as well as the vertex v.

Figure 2

The complete fan

Fans of particular interest in tropical geometry are the Bergman fans of matroids; see [30; 6] for definitions. These serve as the local models of abstract tropical manifolds; see [24, Section 1.6].

Example 2.5

Let M be a matroid on E = {0, . . . , n} with lattice of flats L, and let ℤ{e₀, . . . , e_n} be the lattice of rank n + 1 generated by elements e₀, . . . , e_n. Let N be the quotient defined by

For any subset S ⊆ E, let p_S = Σ_i∈S π(e_i) in N, so that in particular p_E = 0. For any chain F_∙ of flats of the matroid M, say

the cone associated to F_∙ is the non-negative span

The Bergman fan of M is the simplicial fan Σ(M) consisting of the cones σ(F_∙) for all flags of flats F_∙.

The U_3,4 matroid on the set E = {0, . . . , 3} given by the rank function r: 2^E → ℤ_≥0 taking values r(S) = min(|S|, 3) has the lattice of flats given by Figure 3. The Bergman fan of this matroid is shown in Figure 4.

Figure 3

Lattice of flats of the U_3,4 matroid

Figure 4

Bergman fan of the U_3,4 matroid, visualization by [13]

Definition 2.6

The star γ_⪰ at a cone γ ∈ Σ is the rational polyhedral fan with underlying set⋃_γ⪯κ⊂ N, where = {t(x − y) | t ∈ ℤ_≥0, x ∈ κ, y ∈ γ} ⊆ N, subdivided into rational polyhedral cones with a shared vertex.

The cone at a face γ ∈ Σ is the fan γ_⪯ consisting of the faces κ ∈ Σ for each κ ⪯ γ. In particular, the vertex v of Σ is the minimal cell in each cone.

Example 2.7

We give two examples of stars:

1. In the fan from Example 2.4, the cone τ₁ is contained in the cones σ₁ and σ₂. These give rise to the sets σ̃₁ = {(a, b) ∈ N | a ≥ 0} and σ̃₂ = {(a, b) ∈ N | a ≤ 0}, so that the star τ_1⪰ has underlying set equal to the whole of N.

2. In the Bergman fan of the U_3,4 matroid, the star at any of the one-dimensional rays is has underlying set equal to a product of ℤ together with the “tropical line”, i.e. the fan with rays (1, 1), (−1, 0), (0, −1) and a vertex at (0, 0).

In both cases, these sets must then be cut up so as to form a rational polyhedral fan.

An integer weight function on a rational polyhedral fan Σ of dimension d is a function w: Σ^d → ℤ. We are interested in weighted fans satisfying the usual tropical balancing condition. This condition is equivalent to being a Minkowski weight in the sense of [12]. For more on the balancing condition, see for instance [22, Definition 3.3.1] or [10, Definition 5.7].

Definition 2.8

Let Σ be a rational polyhedral fan of dimension d with weights w: Σ^d → ℤ. We say that Σ together with w is balanced at a face β ∈ Σ^d−1 if

using the notation from 2.1. We say Σ together with w is balanced if it is balanced along each face β ∈ Σ^d−1.

Example 2.9

Our previous examples of fans have all been balanced:

1. The fan of dimension one discussed in Example 2.3 and shown in Figure 1 is balanced, for a given weight function w: Σ¹ → ℤ, if and only if w(τ₁) = w(τ₃) and w(τ₂) = w(τ₄).

2. The fan of dimension two in Example 2.4 is also balanced if and only if the weight function w: Σ¹ → ℤ is such that w(σ₁) = w(σ₂) = w(σ₃).

3. It follows from [6, Proposition 2] that the stars γ_⪰ of faces γ in the Bergman fans of a matroid are themselves Bergman fans of matroids. It is shown in [1, Proposition 5.2] that, for the Bergman fan Σ(M) of a matroid M, the only weight functions which satisfy the balancing condition are the constant ones. The uniqueness of such a weight function follows from tropical Poincaré duality in [19, Proposition 5.5] and the earlier [16, Theorem 38]. By our later Definition 3.12, this means that these fans are uniquely ℤ-balanced.

Definition 2.10

Let R be a commutative ring, Σ a rational polyhedral fan. Then:

• A cellular R-sheaf G is a functor G: Σ → Mod_R.

• A cellular R-cosheaf F is a functor F: Σ^op → Mod_R.

A morphism of sheaves or cosheaves is simply a natural transformation of functors or contravariant functors, respectively.

Remark 2.11

The category Σ, when viewed as a set, can be given the Alexandrov topology, such that cellular sheaves and cosheaves in fact are sheaves and cosheaves with respect to this topology. For more on cellular sheaves and cosheaves, see [11].

We have considered the fan Σ as a category with morphisms τ → σ whenever τ is a face of σ, so that a sheaf G induces a map G(τ) → G(σ), and a cosheaf F induces a map F(σ) → F(τ). This convention is in agreement with [11; 27], but reversed from [9; 15] in the sense that their sheaves are our cosheaves, and vice versa.

Example 2.12

Let Σ be a rational polyhedral fan. For a module M over a ring R, the constant cosheaf M^Σ with values in M is the cosheaf defined as a functor M^Σ : Σ^op → Mod_R taking all objects to M and all morphisms to id_M. Similarly, the constant sheaf M_Σ with values in M is the sheaf defined as a functor M_Σ : Σ → Mod_R taking all objects to M and all morphisms to id_M.

Definition 2.13

Given a cellular sheaf G, the cellular cochain groups and cellular cochain groups with compact support are defined, respectively, by

for q ≥ 0, where σ_ℝ is as in Definition 2.1. The cellular cochain maps

are given component-wise for τ ∈ Σ^q and σ ∈ Σ^q+1 with τ ≺ σ by d_τσ : G(τ) → G(σ), where

If τ⋠σ, we let the map d_τσ be 0.

The cohomology groups H^∙(Σ, G) and Hc∙(Σ,G) of these complexes are the cellular sheaf cohomology and cellular sheaf cohomology with compact support with respect to the sheaf G.

Definition 2.14

Given a cellular cosheaf F, the cellular chain group and Borel–Moore cellular chain groups are defined, respectively, by

for q ≥ 0, where σ_ℝ is as in Definition 2.1. The cellular chain maps

are given component-wise for σ ∈ Σ^q and τ ∈ Σ^q−1 by ∂_στ : F(σ) → F(τ), where

If τ ⪯̸ σ, we let the map ∂_στ be 0.

The homology groups H_∙(Σ, F) and H∙BM(Σ,F) of these complexes are the cellular cosheaf homology and cellular Borel–Moore cosheaf homology with respect to F.

Proofs that the cellular (co)chain groups and maps defined above form (co)chain complexes can be found in [11, Definitions 6.2.6–7] and [27, Theorem 1.1.3].

Remark 2.15

The above definitions of cellular cohomology work in the more general setting of polyhedral complexes. Since we are working with pointed polyhedral fans, the unique compact cell is the vertex v. Then, for any sheaf G on a fan Σ, the cellular cochain groups C^q(Σ, G) are trivial for q > 0, and therefore

Similarly, for any cosheaf F, the cellular chain groups C_q(Σ, F) are trivial for q > 0, thus

Example 2.16

Consider the fan from Example 2.3, with orientations chosen so that O(v, τ_i) = 1 for all i. The Borel–Moore homology of the constant cosheaf ℤ^Σ is the homology of the complex

where the matrix ∂₁ is indexed by the τ_i and given by

The Borel–Moore homology becomes H1BMΣ,ZΣ=Z3 and H0BMΣ,ZΣ=0.

Example 3.2

We compute some values of these cosheaves:

1. For Example 2.3, taking the ray τ₁, we have F1Z(τ)=LZ(τ)=〈(1,0)〉Z⊂Z2. For the central vertex v, we have

The cosheaf F0Z is merely the constant cosheaf taking value ℤ, so that F0Zτi=Z and F0Z(v)=Z.

1. For Example 2.4, we have F2Zσ1=∧2LZσ2=〈(1,0)∧(0,1)〉Z≅Z.

Remark 3.3

For any ℤ-module M and commutative ring R, the product M_R := M⊗_ℤR is an R-module. Moreover, by [8, Proposition III.7.5.8] we have ⋀^p M_R ≅ (⋀^p M) ⊗_ℤ R. In particular, for a maximal face σ ∈ Σ of a d-dimensional fan, L_R(σ) := L_ℤ(σ) ⊗_ℤ R is a free R-module of dimension d, and FRp(σ)=⋀pLZ(σ)⊗ZR≅ ∧pLR(σ).

Remark 3.4

Let Σ be a fan of dimension d. For any α ∈ Σ^d, by Remark 3.3 we have FdR(α)=⋀dLR(α)≅R. Given a choice of orientation for α, we can select the unique generator Λα∈FdZ(α)=ΛpLZ(σ) compatible with the chosen orientation, and abusing notation, we let Λα∈FdR(α)≅⋀pLZ(σ)⊗Z R denote the corresponding element Λα⊗1R∈FdR(α).

Example 3.5

In Example 2.4, suppose we choose orientations such that all the one-dimensional rays point outward, with all the two-dimensional cones being oriented clockwise. Choose the standard basis e₁, e₂ for the ambient lattice N. We then have Λσ1=e1∧e2,Λσ2=e1∧e2 and Λσ3=e1∧e2.

Definition 3.6

The cochain complex C∙Σ,FRp,δ from Definition 2.13 has cohomology groups HqΣ,FRp which are called the (cellular) tropical cohomology groups with R-coefficients of Σ. Moreover, the cohomology groups HcqΣ,FRp of the complex Cc∙Σ,FRp,δ are called the (cellular) compact support tropical cohomology groups with R-coefficients of Σ.

Similarly, the chain complex C∙Σ,FpR,∂ from Definition 2.14 has homology groups HqΣ,FpR which are called the (cellular) tropical homology groups with R-coefficients of Σ. Finally, the (cellular) tropical Borel–Moore homology groups with R-coefficients HqBMΣ,FpR are the homology groups of the chain complex C∙BMΣ,FpR,∂.

Proposition 3.7

The tropical cohomology with R-coefficients of any fan Σ is

where v ∈ Σ is the vertex.

Proof. This follows from Remark 2.15. □

Example 3.8

Consider again the 1-dimensional fan from Example 2.3. Since it is of dimension 1, the only FRp sheaves are FR0≅RΣ and FR1. By Proposition 3.7, the only non-zero cohomology groups are H0Σ,FR0=R and H0Σ,FR1=FR1(ν)=R2.

Similarly, the only FpR cosheaves are F0R≅RΣ and F1R. The computation of the homology with the constant cosheaf ℤ^Σ given in Example 2.16 carries through to R^Σ, giving H1BMΣ,F0R=R3 and H0BMΣ,F0R=0. Finally, to compute the Borel–Moore homology for F1R, we have the chain complex

Selecting the ℤ-basis e₁ = (1, 0), e₂ = (0, 1) for N, we can write this complex as

where ∂₁ is now the direct sum of the inclusion maps, and everything is suitably tensored with R. The Borel–Moore homology is then given by H0BMΣ,F1R=0 and H1BMΣ,F1R=〈(a,b,a,b)∣a,b∈R〉≅R2.

Example 3.9

We now show how to perform the above computations using the [21] package for polymake [13], when working with rational coefficients. A code example is given in Figure 5. To compute with polymake, one specifies the rays of a fan, as well as which rays form a cone. The fan must be input in projective coordinates, so that there is a distinct projection point [1, 0, 0], with all rays expressed using an embedding of N into the hyperplane H = {(x₀, x₁, x₂) | x₀ = 0}. Thus the ray τ₁ is [0, 1, 0]. Similarly the cones must all be given as including the projection point [1, 0, 0], so that the one-dimensional ray τ₂ is given as [0, 2].

Figure 5

Code in [13] to compute the tropical cohomology and Borel–Moore homology in Example 2.3

Note also that one may use the command $complex -> VISUAL; to receive a visualisation for two- and three-dimensional fans. The output of the code in Figure 5 is shown in Figure 6.

Figure 6

Output from Figure 5

Recall from Definition 2.6 that one must subdivide the stars γ_⪰ of faces γ ∈ Σ to obtain a fan structure. The next proposition shows that the tropical cohomology of γ_⪰ is determined directly by FpR(γ), and that the tropical Borel–Moore homology can be computed using a simpler complex than the one coming from the subdivision.

Proposition 3.10

Let Σ be a fan and γ ∈ Σ a face of dimension r. Let FR,ΣpandFR,γp denote the p-th multi-tangent sheaves on Σ and γ_⪰ respectively. Then

Similarly, let F p R , Σ a n d F p R , γ denote the p-th multi-tangent cosheaves on Σ and γ _⪰ respectively. Then the Borel–Moore homology H q B M γ ⪰ , F p R , γ is isomorphic to the homology of the complex

where we define ∂qγ=⨁∂στ with the sum taken over all σ, τ ⪰ γ, σ ∈ Σ^q and τ ∈ Σ^q−1.

Proof. First, by Definition 2.6 we must choose a subdivision of the space with support given by the cones= {t(x − y) | t ≥ 0, x ∈ κ, y ∈ γ} for each κ ⪰ γ, and we will have only one compact cell given by the created vertex∈ γ_⪰. By Remark 2.15, we have H0γ⪰,FR,γp=FR,γp(v˜). Then, observe that for each κ ≻ γ, the lattice is unchanged in the sense that L_ℤ() = L_ℤ(κ) as subspaces of N, and each maximal dimensional face α in the subdivision of γ_⪰ is a subspace of a. In particular, this implies that FR,γp(v˜)≅FR,Σp(γ).

Next, observe that the Borel–Moore homology is the equal to the regular homology of the fan when seen as a subset of the one-point compactification N ∪ {∞} of the ambient lattice N. Then every cone σ ∈ γ_⪰ becomes a disk σ_∞ in N ∪ {∞}, and we have a CW complex structure on the underlying set of γ_⪰ ∪ {∞}. Then, similarly to [24, Section 2.2] and [19, Remark 2.8], note that

where the right hand side becomes the cellular homology with coefficients in the local system induced by FpR of the CW complex γ_⪰ ∪ {∞}. This can be computed with an arbitrary CW structure. Therefore, the given complex, which is the local system homology of the CW structure induced on |γ_⪰ ∪{∞}| by taking the non-subdivided cell structure of γ_⪰, will compute the Borel–Moore homology of the FpR cosheaf. □

The above proposition shows that, when working with stars of faces in a fan, the particular cellular structure does not change the tropical (co)homology. This is not the case for general (co)sheaves, see for instance the wave tangent sheaves defined in [24, Section 3].

Definition 3.11

Let R be a ring. An R-weight function w: Σ^d → R on a d-dimensional fan Σ is a function such that for all α ∈ Σ^d, the weight w(α) is not a zero-divisor. A pair (Σ, w) will be called an R-weighted fan.

Definition 3.12

An R-weighted fan (Σ, w) is R-balanced if the fundamental chain Ch(Σ, w) given by

is a cycle, where the Λ_α are chosen as in Remark 3.4. In this case, we have HdBMΣ,FdR≠0, together with an induced fundamental class

If HdBMΣ,FdR=〈[Σ,w]〉≅R, we say that Σ is uniquely R-balanced by w.

Example 3.13

We now compute the fundamental chain in the fan Σ of Example 2.4. Choose orientations such that the elements Λσi are as in Example 3.5. Moreover, we choose a weight function assigning the value 1 to each of the cones σ_i . Then the fundamental chain is:

It is then straightforward to check that, under the boundary map ∂₂, taking into account orientations, this chain is mapped to zero. For instance, for the component of C1BMΣ,F2Z corresponding to τ₁, we have

with the first 2-wedge corresponding to σ₁ and the second to σ₂. Thus the fan is ℤ-balanced, and there is a fundamental class [Σ,w]∈H2BMΣ,F2R. Moreover, it can be checked that this class generated the whole cohomology module, so that this fan is in fact uniquely R-balanced.

The above definition, which is equivalent to the usual balancing condition [10, Definition 5.8], is similar in flavor to the description given by [7, Theorem 2.9]. We illustrate this with the following example.

Example 3.14

In this example, we explicitly relate the above definition of balancing to the one given in Definition 2.8. Let (Σ, w) be a ℤ-balanced fan of dimension d in the sense of Definition 3.12. Then, for each β ∈ Σ^d−1, we pick a generator Λ_β ∈ L_ℤ(β) respecting the orientation. Now for each α ∈ Σ^d with β ≺ α, the vector v_α/β from Definition 2.1 is such that Λ_α = Λ_β ∧ v_α/β.

Looking at the β-component of ∂:CdBMΣ,FdZ→Cd−1BMΣ,FdZ, we have

Therefore, the chain Ch(Σ, w) is closed if and only if all of the faces β are balanced in the sense of Definition 2.8. Thus the definitions are equivalent.

Proposition 3.15

Let (Σ, w) be an R-balanced fan and γ ∈ Σ a face. Then (γ_⪰, w) is R-balanced, where w is understood to be the weight function induced on γ_⪰ by w.

Proof. By Proposition 3.10, we have that HdBMγ⪰,FdR can be viewed as the kernel of the map

Moreover, the class

is a cycle since Ch(Σ, w) is a cycle in CdBMΣ,FdR. Thus (γ_⪰, w) is R-balanced and we have a fundamental class γ⪰,w∈HdBMγ⪰,FdR.

□

Definition 3.16

([8, Section III.11.9]). Let M_R be a rank d free R-module, let MR∗ be the dual module, and let 0 ≤ p₁ ≤ p₂ ≤ d. The interior product or contraction defined by y=y1∧⋯∧yp2∈∧p2MR is the map

which is defined on X=x1∧⋯∧xp∈∧p1MR∗ to be

where the sum is taken over all permutations a ∈ S_d which are increasing on 1, . . . , p and p + 1, . . . , d, and is extended linearly.

Remark 3.17

In [8, Section III.11.10], an explicit formula for this contraction map is given in terms of bases. Letting e₁, . . . , e_m be a basis of M_R, the elements eI:=ei1∧⋯∧eip2∈∧p2MR, for all ordered strictly increasing subsets I = i 1 < ⋯ < i p 2 ⊆ [ m ] of size p, form a basis of ∧p2MR. Letting f₁, . . . , f_m be the dual basis to the e_i for MR∗, the elements f_J with J = j 1 < ⋯ < j p 1 ⊆ [ m ] form a basis of ∧p1MR∗. Then the contraction map defined by e_J is given by

where v is the number of ordered pairs (λ, μ) ∈ K × (J \ K) such that λ > μ.

A proof that these contraction maps are the same as the formulation in terms of compositions given in [5; 2] follows from the arguments given in [2, Lemma 4.1.4].

Definition 3.18

For each facet α ∈ Σ^d of a d-dimensional R-balanced fan Σ, we have chosen a generator Λα∈ FdR(α)=∧dLR(α)≅R by Remark 3.4, and a weight w(α) ∈ R which is not a zero-divisor. The contraction defined by w(α)Λ_α is the map

Since w(α) is not a zero-divisor, Remark 3.17 shows that this map is injective. It is an isomorphism if and only if w(α) is a unit.

Definition 3.19

Let the weighted fan (Σ, w) be an R-balanced fan of dimension d. The cap product ⌢ Ch(Σ, w) with the fundamental chain of Σ is the map given by

where Λ_α is as in Remark 3.4.

Remark 3.20

For any chain σ∈CqBMΣ,FpR, a cap product⌢ σ can be similarly defined. It is noted in [19, p. 13] that the Leibniz formula holds for this cap product, such that ∂(α ⌢ σ) = (−1)^q+1(d(α) ⌢ σ − α ⌢ ∂(σ)). In the case where R = ℝ, the Leibniz formula also follows from [20, Remark 2.2, Definition 4.11]. Therefore the above defined map descends to tropical (co)homology, and we have the cap product with the fundamental class ⌢ [Σ, w]

Example 3.21

In Example 3.14, we computed the fundamental class of the fan Σ from Example 2.4, given the all-one weight function w. We will now compute some examples of the cap product. Let e₁, e₂ ∈ N ≅ ℤ² be the standard basis for the underlying lattice, and e1∗,e2∗ the dual basis for the dual lattice N^∗. Then

Moreover, all other cohomology groups are zero, by 3.7. Next, the Borel–Moore chain complexes are given by

We now compute H2BMΣ,F1Z as an example. We have F1Zτj=e1,e2Z for each j and F1Zσi=e1,e2Z for each i. Taking the direct sum of these bases in⨁i=13F1Zσi, and respecting the orientations, we may express the differential ∂2:⨁i=13F1Zσi→⨁j=13F1Zτj in coordinates as:

with αⁱ and βⁱ corresponding respectively to e₁ and e₂ for σ_i. We compute a basis for the kernel of this map, i.e. a basis for H2BMΣ,F1Z to be

Similar computations show that the remaining Borel–Moore homology groups are given by:

Finally, we now compute an example for the cap product map, in particular⌢[Σ,w]:H0Σ,FZ1→H2BMΣ,F1Z. Working from the definition, we have that

Expanding this using the (αⁱ , βⁱ) basis from above, these contractions are such that e1∗↦(1,0,1,0,1,0), and one can similarly check that e2∗↦(0,1,0,1,0,1). This shows that ⌢ [Σ, w] is in this case an isomorphism.

Proposition 3.22

Let (Σ, w) be an R-balanced fan of dimension d. The cap product with the fundamental class ⌢ [Σ, w] in tropical cohomology

is the 0-map for q ≠ 0.

Proof. By 3.7 we have HqΣ,FpR=0 for q ≠ 0, hence this cap product is non-trivial only when q ≠ 0. □

The above proposition shows that in the fan-case, the only interesting cap products are of the form

for p = 0, . . . , d. Moreover, in 3.23 below, we show that these are injective for any commutative ring R. In the case where R = ℝ, this was shown in [2, Theorem 4.3.1], and for R = ℤ it is stated in [5, Section 3.2.2].

Proposition 3.23

For an R-balanced fan (Σ, w) of dimension d, the map

is injective.

Proof. We have HdBMΣ,Fd−pR=ker∂d and H0Σ,FRp=FRp(v), so that ⌢ [Σ, w] is exactly

where the image lies in HdBMΣ,Fd−pR⊆⨁α∈ΣdFd−pR(α). This is the composition of the map⨁αρv,α:FRp(v)→ ⨁α∈ΣdFRp(α), which is injective since it is dual to the surjection⨁α∈ΣdFpR(α)→FpR(v), and the direct sum of the contractions⨁α∈Σdw(α)Λα, which are injective by Proposition (3.18). Thus this cap product is the composition of injective maps and is therefore injective. □

Proposition 3.24

Let (Σ, w) be an R-balanced fan and γ ∈ Σ a face. Then the cap product map on the star fan

is given by

where we identify H0γ⪰,FRp≅FpR(γ),andHdBMγ⪰,Fd−pR as the kernel of the first map in the complex from 3.10.

Proof. The identifications are justified by Proposition 3.10, and the existence of this fundamental class by Proposition 3.15. It remains to show that the stated formula corresponds to the cap product. Consider a subdivision making γ_⪰ a pointed fan. Each d-cell ã of the subdivision maps to a d-cell α ≻ γ of Σ, similarly as in the proof of Proposition 3.10. The formula then follows from the induced map in homology. □

Definition 4.1

We say that an R-balanced rational polyhedral fan Σ of dimension d with weights w satisfies tropical Poincaré duality over R if the cap product with the fundamental class

is an isomorphism for all p, q = 0, . . . , d.

Example 4.2

Returning again to Example 2.4, one can verify that all the possible cap products are isomorphisms, as we did explicitly in Example 3.21 for the cap product ⌢[Σ,w]:H0Σ,FZ1→H2BMΣ,F1Z, so that this fan satisfies tropical Poincaré duality over ℤ.

Example 4.3

Similarly, explicit computations can be carried out for Example 2.3. Comparing with Example 3.8, we have dimZH1BMΣ,F1R=2 and dimZH1BMΣ,F0Z=3, while dimZH0Σ,F1R=2 and dimZH0Σ,F0R=1. Thus the cap product maps

are not isomorphisms, and the fan does not satisfy tropical Poincaré duality over ℤ.

As mentioned in the introduction, the Bergman fans of matroids satisfy TPD over ℝ and ℤ [20; 19], however these are not the only such fans, as can be seen from the next example.

Example 4.4

Let f₁ := (0, 1, 1, 1), f₂ := (1, 0, −1, 1), f₃ := (1, 1, 0, −1) and f₄ := (1, −1, 1, 0) be vectors in ℝ⁴ and let e₁, e₂, e₃ and e₄ be the standard basis. Consider the fan generated by the cones of vertices connected by an edge in Figure 7, so that for instance the cone of e₁ and f₂ is included. This fan was used in [7] to construct a counter-example to the strongly positive Hodge conjecture. It is not matroidal, since it does not satisfy the Hard Lefschetz property of [1].