Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

FINAL

JANUARY 13, 2000

Problem 1 (25 pts). Determine the homology groups H_{∗}(ℂP^{2} ∧ ℂP^{2}).

Problem 2 (25 pts).

(1) Describe the chain cellular complexes for ℝP^{3} and the lens space L(3, 1). Find also the dual (cohomology) complexes for these spaces.

(2) Determine the homology groups H_{∗}(ℝP^{3}; ℤ/6) and H_{∗}(L(3, 1); ℤ/5).

(3) Determine the cohomology groups H_{∗}(ℝP^{3}; ℤ/6) and H_{∗}(L(3, 1); ℤ/5).

Problem 3 (25 pts). Using the duality, find H_{n}(X ∖ A), where (1) n = 2, X = S^{4} and A ≅ S^{1} (a topological circle embedded in S^{4});

(2) n = 1, X = S^{4}, A ≅ ℝP^{2};

(3) n = 1, X = ℂP^{2} and A is a non-singular algebraic curve of degree d, that is a closed orientable surface of genus g = \frac{1}{2}(d − 1)(d − 2) realizing class d = [A] ∈ H_{2}(ℂP^{2}) ≅ ℤ.

Problem 4 (25 pts).

(1) Prove that if the universal covering over X is S^{2n}, then it is either a homeomorphism, or a double covering.

(2) Prove that if the universal covering over X is S^{2n+1}, then π_{1}(X) is finite.

(3) What can you say about the universal coverings over the closed surfaces F_{g} (oriented, of genus g) and N_{k} (connected sum of k copies of ℝP^{2}) ?

(4) Prove that any covering ℂP^{2} → X is a homeomorphism.

Hint: Think about the Betti numbers of X. Observe that b_{1}(X) = 0 and use the Poincare duality to deduce from this that b_{3}(X) = 0.

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