Middle East Technical University Fall 2000 Math 537 Algebraic Topology I

HOMEWORK II. FUNDAMENTAL

GROUPS AND COVERING SPACES

Problems from the textbook:

page 37: 8

page 53–54: 10, 11, 14, 21, 2, 5, 8 page 78: 9, 10, 12, 14

Problem 1. Prove that ℂP^{2} \ ℝP^{2} is simply connected.

Problem 2. Consider a surface F and a pair of simple closed curves, C_{1}, C_{2} on C. Consider X obtained from F after gluing curves C_{1} and C_{2} via a homeomorphism f: C_{1} → C_{2}. Calculate π_{1}(X) if

(1) F = T^{2} (a torus) and C_{1} intersects C_{2} transversally at a point (i.e., a longitude is glued with a meridian).

(2) F = F_{2} (closed oriented genus 2 surface) and C_{1} ∩ C_{2} = ∅.

Problem 3. Find π_{1}(T_{f}), where T_{f} = X × I/(x, 0) ∼ (f(x), 1) is the mapping torus of a mapping f: X → X), where

(1) X = S^{1} ∨ S^{1} and the restriction of f to the first copy of S^{1} is z ↦ z^{2}, and the restriction to the second copy is z ↦ z^{3};

(2) X = T^{2} and f(x, y) = (y, x).

Problem 4. Which groups may act freely on S^{2} × S^{2} × S^{2} ? Give examples of possible actions and prove non-existence for the other group actions.

Problem 5. Determine for which n, m there exists a covering (1) of F_{n} by F_{m},

(2) of N_{n} by N_{m},

(3) of N_{n} by F_{m} Prove that an oriented surface cannot be covered by a non-orientable surface.

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