Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

MIDTERM I

NOVEMBER 18, 2000

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Problem 1. (5 pts) Prove that a mapping f: X → S^{4}, which is not surjective, must be null-homotopic.

Problem 2. (10 pts) How many cells contain (1) ℝP^{2} × ℝP^{3},

(2) ℝP^{2} ∨ ℝP^{3},

(3) ℝP^{2} ∧ ℝP^{3},

(4) ℝP^{2} ∗ ℝP^{3}. Determine the Euler characteristic of the above spaces.

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Problem 3. (10 pts) Consider a three-points set A ⊂ S^{2}. What is π_{1}(S^{2}/A) ? Problem 4. (10 pts) Determine π_{1}(X), if X is a union of two hyperplanes in ℝP^{3}.

Problem 5. (10 pts) Prove that (1) ℤ/3 cannot act freely on ℂP^{3} × S^{2};

(2) Prove that ℤ/m cannot act freely on ℝP^{2n}, n ⩾ 0, m ⩾ 2.

Problem 6. (10 pts)

(1) Which surfaces may cover F_{3} (a genus 3 oriented closed surface) ? Can it be, for instance, F_{6} ?

(2) Which surfaces may cover a Klein bottle ?

(3) Which surfaces may be covered by S^{2} ?

Problem 7. (10 pts) Prove that π_{2}(ℝP^{3}) = 0 and π_{n}(S^{1}) = 0 for n ⩾ 2. Problem 8. (10 pts) Assume that a 3-manifold M with a finite group π_{1}(M) has ℝ^{3} as its universal covering. Prove that M is non-compact.

Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

Midterm I. Part II: Take-Home Problems

Problem 9. (10 pts) Prove that (1) ℝP^{2} cannot be retracted to ℝP^{1}

(2) a Möbius band cannot be retracted to its boundary.

Problem 10. (10 pts) Assume that (X, A) is a CW pair, where both X and A are connected. Knowing generators and relations of π_{1}(X) and π_{1}(A), determine π_{1}(X/A). In particular, use this result to

(1) find π_{1}(T^{2}/C), where C ⊂ T^{2} is a curve realizing element [C] = (2, 3) ∈ π_{1}(T^{2}) = Z ⊕ ℤ;

(2) show that X/A is simply connected if X is;

(3) prove that X ∨ Y is simply connected if X and Y are path connected.

Problem 11. (10 pts) (1) Prove that any index 2 subgroup of ℤ ∗ ℤ is isomorphic to ℤ ∗ ℤ ∗ ℤ.

(2) Prove that any finite index subgroup of a free group of finite rank is itself a free group of finite rank.

(3) Determine the rank of an index p subgroup of ℤ ∗ ℤ ∗ ℤ.

(4) Prove that any finite index subgroup of a finitely presented group is itself finitely presented.