Middle East Technical University Fall 1995 Math 471 Elementary Topology and Geometry

Problem List 1

SIMPLICIAL COMPLEXES AND SIMPLICIAL HOMOLOGIES

Problem 1. Determine which letter (A, B, …) is the most complicated in the sense that its simplest thiangulation contains the maximal number of 1-simplicies.

Problem 2. Triangulate a cube geometrically, i.e., find a simplicial complex K such that [K] is a cube.

Problem 3. Construct a triangulation of a torus T^{2}, which contains less then ten 2-simplices. Try to get the least possible number, 7.

Problem 4. Let K denote the simplicial complex of a 3-simplex. Determine St(s) and Lk(s) for

(1) K, (2) K^{2}, (3) K^{1}, if [s] is

(a) a vertex, (b) an edge, (c) a 2-face.

The answer can be given in form of pictures.

Problem 5. Given a simplicial complex K, (s) ∈ K, prove that St(s) = ⋂_{v∈s^{0}} St(v).

Problem 6. Determine the topology type of S^{n} ∗ S^{m}. Hint: consider first the case n = 0 and then prove by induction.

Problem 7. Determine the homology groups with ℤ/2, interger and real coefficients for the following simplicial complexes.

(1) X ∗ X, where X is a 3 points set,

(2) K^{1}, where K is the complex of a 3-simplex,

(3) K^{1}, where K is the complex of a 4-simplex, Determine their cohomology groups (with the same coefficients).