Middle East Technical University Spring 1994 Math 471 Elementary Topology and Geometry

Problem List 3

SIMPLICIAL COMPLEXES

Problem 1. (1 pt) Prove that a linear monomorphic map ℝ^{n} → ℝ^{m} sends simplicies to simplicies and preserves barycentric coordinates of their points.

Problem 2. (2 pts) A map ℝ^{n} → ℝ^{n} is called affine, if it is product of a linear map and a parallel transition. Prove that given n-symplicies [s], [t] ⊂ ℝ^{n}, there exist the unique affine map in ℝ^{n}, which sends [s] to [t]. Prove that it preserves barycentric coordinates.

Problem 3. (1 pts) Is there exists a simplicial complex K, which contains precisely 4 simplicies ? The same question, if [K] has to be connected.

Problem 4. (1 pt) Construct a triangulation of sprere S^{2}, which contains the least possible number of simplices.

Problem 5. (1 pt) How many 1-simplicies contains the barycentric subdivision of a 4-simplex ?

Problem 6. (2 pts) Let A is a two points set. What is the topological type of the iterated join A ∗ A ∗ ⋯ ∗ A (n times) ?

Problem 7. (1 pt) Prove that Lk(v) = Cl(St(v)) − St(v) for any vertex v of a simplicial complex K.

Problem 8. (1 pt) Given a simplicial complex K, (s) ∈ K, prove that St(s) =

⋂_{v∈s^{0}} St(v).

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