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3) , it suffices to prove just that foldings preserve type. Every folding f , by definition, fixes pointwise some chamber Co . Let D be the closest chamber to Co so that f might fail to preserve the type of some simplex inside D. Let Co , . . , Cn = D be a minimal gallery connecting Co to D. By hypothesis, f preserves the type of simplices inside Cn−1 . In particular, f preserves the type of all the vertices in the common facet F = Cn−1 ∩ D. Let x be the vertex of D not contained in F . Since λ and λ ◦ f are dimension-preserving simplicial complex maps to the ‘simplex’ (simplex-like poset) of subsets of S with inclusion reversed, neither λx nor λf (x) can lie in λf (F ) = λF .
Thus, the facets of x are the codimension one faces of x. The relations y ⊂ x holding in a simplicial complex are the face relations. For a simplex x ∈ X, write x ¯ for the simplicial complex consisting of the union of x and all faces of x. We may refer to this as the closure of x. Two simplices x, y in a simplicial complex X are adjacent if they have a common facet. A simplex x in a simplicial complex X is maximal if there is no simplex z ∈ X of which x is a proper face. In the rest of this book, we will consider only simplicial complexes in which every simplex is contained in a maximal one.
Next, we show that Y and Y have no chamber in common, so that the two partition the chambers of A. Indeed, if D were a common chamber, then both f and f fix D pointwise. Let γ be a minimal gallery from D to a chamber with face F . Then f γ and f γ still are galleries from D to a chamber with face F . Since γ was already minimal, these galleries cannot stutter.