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Algebraic Topology Problem Set 3
Please hand in questions 2 and 3 on Tuesday 11th March.
Q1:
In this problem we give an alternative way of understanding the map q : S n RP n .
Suppose that A Mn+1 R and A2 = A. Put U = cfw_x Rn+1 | Ax = x and V = cfw_x
Rn

Algebraic Topology Problem Set 4
No questions to be handed in this week.
Q1: Put X = cfw_(x, y) R2 | y 2 = x3 x. (This is an example of an elliptic curve; such curves
are important in number theory and certain other areas of mathematics.)
Sketch the graph

Algebraic Topology Problem Set 1
Please hand in questions 1 and 3 on Tuesday 25th February.
Q1:
Which of the following rules gives a well-dened, continuous function f : X Y ? Justify your
answer with either a geometric or an algebraic argument.
(a) X is t

Algebraic Topology Problem Set 7
Please hand in questions 2, 3 and 4 on Tuesday 6th May.
Q1: Let X be a metric space, and let f, g : X S 1 be continuous maps. Suppose that f is
linearly homotopic to g. What can you deduce?
Q2: Put
X = cfw_z C | 1 < |z| <

Algebraic Topology Problem Set 6
Please hand in questions 1,2 and 3 on Tuesday April 1st.
Q1: Let X and Y be spaces such that 0 X and 0 Y are nite sets. Suppose that f : X Y and
g : Y X are continuous maps such that gf is homotopic to 1X .
(a) Show that f

GUIDANCE ON THE EXAM
.
1. Examples
This list covers most of the examples that you should know about. The exam may ask you to
prove that two spaces are (or are not) homeomorphic (or homotopy equivalent) to each other. The
spaces involved will either be tak

PMA333 EXAMINATION SPRING 2001 SOLUTIONS
N. P. STRICKLAND
(1)
(2)
(i) A metric space X is compact if for every sequence (xn ) in X there is a subsequence
(xnk ) and a point x X such that xnk x.
(ii) Let f : X Y be a continuous surjective map, and suppose

Algebraic Topology Problem Set 5
Please hand in questions 1,2 and 3 on Tuesday 25th March.
Q1: Let f : X R \ cfw_0 be a continuous map. Show that f is homotopic to a map g such that
g(x)2 = 1 for all x X.
Q2: Recall that for any space X and any x X we hav