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MATH 751 MIDTERM EXAM (due Nov 6) FALL 2009 NAME: You must hand this sheet in with your exam in order to receive a grade. You must show all your work in order to receive full credit. You must obey the usual principles of academic integrity. I. (25 points) Let p : E → B be a covering map, with E path connected. Show that if B is simply-connected, then p is a homeomorphism. II. (50 points) Let Z6 act on S 3 = {(z, w) ∈ C2 , |z|2 + |w|2 = 1} via (z, w) 7→ (z, w) , where is a primitive sixth root of unity. Denote by L the quotient space S 3 /Z6 . (1) What is the fundamental group of L? (2) Describe all coverings of L. (3) Show that any continuous map L → S 1 is nullhomotopic. III. (25 points) Let A be a real 3 × 3 matrix, with all entries positive. Show that A has a positive real eigenvalue. IV. (25 points) Show that any continuous map f : S 2 → S 1 × S 1 is nullhomotopic. Can you construct a continuos map f : S 1 × S 1 → S 2 which is not homotopic to a constant map? V. (25 points) Let X be the topological space obtained by identifying by parallel translation the opposite edges of a solid regular hexagon. Calculate the fundamental group of X. VI. (25 points) Show that RP3 and RP2 ∨ S 3 have the same fundamental group. Are they homeomorphic? VII. (25 points) (1) Is there a deformation restract from X = D2 ∨ D2 to its boundary? Explain. (2) Let X be the space obtained from D2 by identifying two distinct points on its boundary. Is there a retract from X to its boundary? Explain.