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 : 1 (X, a0 ) 1 (A, a0 )
is surjective.
Proof. Let [f ] 1
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 G be a homotopy
between k and k . Dene H : X I Z by
H(
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 points e1 , e2 of
p1 (b0 ), there is an equivalence h : E E
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 quotient map.
Proof. Let 1 : X Z X and 2 : X Z Z be the projec
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 connected
spaces having a point in common is not necessarily
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 identies
antipodes and so we can think of p as mapping into
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 covering map given by p(s) = e2is . Then
p p : R R S 1 S 1 is
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.
Proof. Let b0 = ei0 . Let q : S 1 S 1 be the map q(ei )