Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

MIDTERM II

DECEMBER 23, 2000

Problem 1. (15 pts) Formulate a definition of (1) a chain complex of abelian groups,

(2) the Betti numbers,

(3) a simplicial complex,

(4) a homotopy between chain maps f, g: C → C′.

Problem 2. (15 pts) Find the homology groups of (1) a punctured sphere,

(2) a punctured torus,

(3) a torus punctured in two points.

Problem 3. (20 pts) X = S^{2}/{p_{1}, p_{2}, p_{3}}, i.e., X is obtained by collapsing a tree-point set on a sphere.

(1) Describe a cellular decomposition of X.

(2) Describe the corresponding cellular chain complex (groups C_{n} and the differentials d_{n}).

(3) Compute the cellular homology groups, H_{∗}(X).

Problem 4. (20 pts) Consider a set X ⊂ S^{4} homeomorphic to T^{2}/{p_{1}, p_{2}} (the quotient of a torus by a two-point set). Determine the groups

(1) H_{∗}(X),

(2) X_{∗}(S^{4}/X).

Problem 5. (20 pts) A cubic curve A ⊂ ℂP^{2} is known to be a torus, such that the generator [A] ∈ H_{2}(A) ≅ ℤ maps into 3 ⊂ ℤ ≅ H_{2}(ℂP^{2}) under the inclusion homomorphism i_{∗}: H_{2}(A) → H_{2}(ℂP^{2}). Find the groups H_{∗}(ℂP^{2}, A).

Problem 6. (20 pts) Determine the groups H_{∗}(S ∖ K), where K ⊂ S^{3} is a knot (i.e., K ≅ S^{1}), using a Mayer-Vietoris exact sequence. Hint: use a splitting S^{3} = T ∪ N, where T is a solid torus compact neighborhood of K and N = Cl(S^{3} ∖ T).

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