Suggested Problems
Ph.D. Exam in Topology
August ?, 1995
Instructions: For a passing grade you must work at least five of the following problems.
1. Suppose S = {z ∈ C : |z| = 1} is the unit circle in the complex plane, and define the
α
relation z ∼ w iff either w = e2πiα z or z = e2πiα w. You need not verify that this is an
equivalence relation. Let Sα denote the set of equivalence classes [x] in S for this relation
equipped with the quotient topology: a subset A of S α is open iff the set {x : [x] ∈ A} is open
in S.
Show that Sα is homeomorphic to the circle iff α is a rational number. Partial Hint: When α
is irrational, show that points {[x]} are not closed in S α .
2.
3. Let f : (0, 1] → [0, 1] denote a continuous function with the property that
f
1
2n
=0
and f
1
2n − 1
= 1,
for all n = 1, 2, 3, . . . . Prove that the set
X = Graph(f ) ∪ ({0} × [0, 1])
is connected but not path connected.
4. Let X be path connected. Prove that the following are equivalent:
a) X is simply connected.
b) Every map of the unit circle S1 into X extends to a map from the closed unit disc into X.
c) If f and g are paths in X such that f (0) = g(0) and f (1) = g(1) then f ' g (endpoint
homotopic).
5. Suppose that A is a subset of the topological space X. Suppose that x ∈ A and y lies in the
set complement X\A of A. If ϕ denotes a path in X joining x = ϕ(0) to y = ϕ(1), show that
there exists a “time” t∗ at which ϕ(t∗ ) lies in the boundary ∂A.
6. Prove that every continuous open mapping from R to R is a homeomorphism onto its
image. Recall that an open mapping f from X to Y has the property that if V is an open set
in X, then f (V ) is an open set in Y .
7. In this problem you may use the fact that the homotopy group of the figure eight, denoted
“8”, in the plane is the free group F2 (α, β) on two symbols. Let f : 8 → S be the map which
folds the figure eight at the intersection point to make a circle. Determine exactly the induced
homotopy map f∗ : π1 (8) → π1 (S1 ). Identify the kernel of this homomorphism.