Simplicial Complexes: Second Lecture
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چکیده
Suppose that K and L are simplicial complexes. Recall that a vertex map between these complexes is a function φ : V ert(K) → V ert(L) such that the vertices of a simplex in K map to the vertices of a simplex in L. Given such a φ, we can create a simplicial map f : |K| → |L| by linearly extending φ over each simplex. On the other hand, suppose we have an arbitrary continuous map g : |K| → |L|. There is no reason to assume that g would be simplicial. On the other hand, we can hope to approximate g by a function f which is itself simplicial and is not “too far away” from f in some sense. That’s the goal today, and we start by defining it rigorously. A simplicial map f : |K| → |L| is a simplicial approximation of g if, for every vertex u ∈ K, g(StK(u)) ⊆ StL(f(u)); in other words, if g maps points “near” v to points “near” f(v), where points are considered “near” if they live in a common simplex. If the simplices in K are reasonably small, it seems likely that we can do this. Our goal now is to make this happen by repeatedly subdividing K.
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