1.4. AFFINE MANIFOLDS WITH SINGULARITIES 21 (b, c) (a + b, b + c) (0, 0) (a, b) Figure 15. A tropical curve of genus two in R2/Γ, where Γ is generated by (a, b) and (b, c). The dotted lines give a fundamental domain, and the three external vertices of the curve are in fact identified, so these vertices represent a single trivalent vertex. example, obtained by cutting an angular sector out of a piece of paper and then gluing together the two edges of the sector. (However, in fact we will not ultimately allow this particular example.) A priori, the singularities of the aﬃne structure can be arbitrarily complicated, and there are many reasonable examples which we shall not wish to consider. In addition, it is often convenient to consider restrictions on the nature of the boundary of B. To control the singularities and the boundary, we introduce a refined notion which arises from the following construction. Construction 1.25. Let B be a topological manifold (possibly with bound- ary) equipped with a polyhedral decomposition P, i.e., B = σ∈P σ where • σ ∈ P is a subset of B equipped with a homeomorphism to a (not nec- essarily compact) polyhedron in MR with faces of rational slope and at least one vertex. Thus in particular any σ ∈ P has a set of faces: these faces are inverse images of faces of the polyhedron in MR. • If σ ∈ P and τ ⊆ σ is a face, then τ ∈ P. • If σ1,σ2 ∈ P, σ1 ∩ σ2 = ∅, then σ1 ∩ σ2 is a face of σ1 and σ2. For each σ ∈ P, viewing σ ⊆ MR yields a tangent space Λσ,R ⊆ MR to σ, and we can set Λσ := Λσ,R ∩ M. The assumption that σ has faces of rational slope implies in particular that if σ is of codimension at least one in MR, then the aﬃne space spanned by σ has rational slope. Thus Λσ generates Λσ,R as an R-vector space. Now the interior of each σ ∈ P carries a natural aﬃne structure. Indeed, σ is equipped with a homeomorphism with a polyhedron in MR, which is embedded in the aﬃne space it spans in MR. This gives an aﬃne coordinate chart on Int(σ). However, this doesn’t define an aﬃne structure on B, but only an aﬃne structure

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