22 1. THE TROPICS τ Στ Figure 16. A fan structure given by the map 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 := σ P s.t. τ σ Int(σ). A fan structure along τ P is a continuous map : Rk where k = dim B dim τ, satisfying (1) −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 ). Then Στ := {Kτ,σ | τ σ P} is a fan with |Στ| convex. Two fan structures Sτ, are considered equivalent if = α for some α GLk(Z).
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