5310 PRELIM
Introduction to Geometry and Topology
August 2011
Justify all your steps rigorously. You may use any results that
you know, unless the question says otherwise, or unless the ques-
tion asks you to prove essentially the same result.
1. Prove that if a space Xis path-connected, then it is also connected.
2. Let D⊂R2be the closed unit disc. Define an equivalence relation on
Dby
(x, y)∼(¯x, ¯y)⇔x2+y2= ¯x2+ ¯y2.
Prove that the quotient is homeomorphic to the unit interval: D/∼∼
=
[0,1]
3. Let Xbe a topological space, and ∼an equivalence relation on X.
Decide whether the following statements are true:
(a) If Xis compact, then so is X/∼.
(b) If X/∼is compact, then so is X.
(c) If Xis Hausdorff, then so is X/ ∼.
(d) If X/∼is Hausdorff, then so is X.
For each statement, either give a counter-example, or give a proof.
4. (Path lifting property.) Let p: ( ˜
X, ˜x0)→(X , x0) be a base-point
preserving covering map. Given any path γ: [0,1] →Xwith γ(0) =
x0, show that there exists a lift ˜γ: [0,1] →˜
Xwith ˜γ(0) = ˜x0and
p◦˜γ=γ.
5. Let p: ( ˜
X, ˜x0)→(X , x0) be a base-point preserving normal covering.
(Note: “normal coverings” are also sometimes called ”regular cover-
ings”.) Let a, b be two loops in X, based at x0. Let ˜abe the lift of a,
starting at ˜x0; similarly, let ^
bab−1be the lift of bab−1starting at ˜x0.
Show that ˜ais a loop if and only if ^
bab−1is a loop.
6. Let Xbe the plane R2with two points removed: X=R2\{(1,0),(−1,0)}.
Prove that π1(X) = Z⋆Z.