Josiah Oh
Topology
Assignment 2
22 April 2015
52.4 Let A X; suppose r : X A is a continuous map such that r(a) = a for
each a A. (The map r is called a retraction of X onto A). If a0 A, show
that
r :
Josiah Oh
Topology
Assignment 1
8 April 2015
51.1 Show that if h, h : X Y are homotopic and k, k : Y Z are homotopic,
then k h and k h are homotopic.
Proof. Let F be a homotopy between f and f and let
Josiah Oh
Topology
Assignment 8
27 May 2015
79.3 Let p : E B be a covering map; let p(e0 ) = b0 . Show that H0 = p (1 (E, e0 )
is a normal subgroup of 1 (B, b0 ) if and only if for every pair of point
Josiah Oh
Topology
Assignment 1
8 April 2015
29.11 Prove the following:
(a) Lemma. If p : X Y is a quotient map and if Z is a locally compact
Hausdor space, then the map
= p iZ : X Z Y Z
is a quotien
Josiah Oh
Topology
Assignment 6
15 May 2015
59.1 Let X be the union of two copies of S 2 having a single point in common. What
is the fundamental group of X? [Be careful! The union of two simply conne
Josiah Oh
Topology
Assignment 7
20 May 2015
60.4 The space P 1 and the covering map p : S 1 P 1 are familiar ones. What are
they?
Proof. Let p : S 1 S 1 be the squaring map. Since p(z) = p(z), it iden
Josiah Oh
Topology
Assignment 4
1 May 2015
54.7 Generalize the proof of Theorem 54.5 to show that the fundamental group of
the torus is isomorphic to the group Z Z.
Proof. Let p : R S 1 be the coverin
Josiah Oh
Topology
Assignment 5
8 Friday 2015
57.3 Let h : S 1 S 1 be continuous and antipode-preserving with h(b0 ) = b0 . Show
that h carries a generator of 1 (S 1 , b0 ) to an odd power of itself.