Download I. (25 points) Let p : E → B be a covering map, with E path connected

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.