22 1. THE TROPICS τ Sτ Στ Figure 16. A fan structure given by the map Sτ in a neighbour- hood of the one-dimensional cell τ. This can be viewed as describ- ing an aﬃne structure in a direction transverse to τ. on the subset of B given by σ∈Pmax Int(σ), where Pmax is the set of maximal cells in P. This is insuﬃcient for giving a structure of tropical aﬃne manifold with singularities to B, so we need to extend this aﬃne structure. To do so requires the choice of some extra structure, known as a fan structure. Definition 1.26. Let τ ∈ P. The open star of τ is Uτ := σ ∈ P s.t. τ ⊆ σ Int(σ). A fan structure along τ ∈ P is a continuous map Sτ : Uτ → Rk where k = dim B − dim τ, satisfying (1) Sτ −1 (0) = Int(τ). (2) If τ ⊆ σ, then Sτ|Int(σ) is an integral aﬃne submersion onto its image, with dim Sτ(σ) = dim σ − dim τ. By integral aﬃne submersion we mean the following. We can think of σ as a lattice polytope in Λσ,R. Then the map Sτ|σ is induced by a surjective aﬃne map Λσ → W ∩ Zk, for some vector subspace W ⊆ Rk of codimension equal to the codimension of σ in B. (3) For τ ⊆ σ, define Kτ,σ to be the cone generated by Sτ(σ ∩ Uτ ). Then Στ := {Kτ,σ | τ ⊆ σ ∈ P} is a fan with |Στ| convex. Two fan structures Sτ, Sτ are considered equivalent if Sτ = α ◦ Sτ for some α ∈ GLk(Z).

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.