University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
MA40040: ALGEBRAIC TOPOLOGY
Friday 20th January 2012, 16.3018.30
No calculators may be brought in and used.
Full marks will be given for correct answers to THREE questions.
M40: Exercise sheet 3
1. Let f , g Path(X, x, y). Show that f g if and only if f g x .
2. Let x, y Rn . Show that 1 (Rn , x) = cfw_x . Deduce that 1 (Rn , x, y) consists of a
3. Let x, y X where X is path-connected. Recall that 1 (X, x, y)
M40: Christmas exercise sheet
Some revision questions
1. (1994 exam) Let n N. The map p : S 1 S 1 given by p(eit ) = eint is a covering
map with group of deck translations H. Show that
2. (1991 exam) Let f : S 1 S 1 be continuous and not homotop
M40: Exercise sheet 6
1. Show that the retraction dened in the Brouwer Fixed Point Theorem is indeed continuous.
2. Let X be homeomorphic to the closed unit disc in R2 . Show that any continuous map
f : X X has a xed point. (This is an easy application of
M40: Exercise sheet 7
1. Let p : E X be a covering space and consider the action of 1 (X, x0 ) on p1 (cfw_x0 ).
(a) if E is path-connected, this action is transitive: that is, if e1 , e2 p1 cfw_x0 then
there is  1 (X, x0 ) with e1  = e2 .
M40: Exercise sheet 5
(a) Let A X be a deformation retract of X. Show that the inclusion i : A X is
a homotopy equivalence with the retraction as homotopy inverse.
(b) Let A Rn be convex and a A. Show that cfw_a is a deformation retract of A.
M40: Exercise sheet 4
1. Let 1 , 2 : X Y and 1 , 2 : Y Z be continuous maps. If 1 2 and 1 2 ,
show that 1 1
2 2 . Deduce that homotopy equivalence is an equivalence
relation on topological spaces.
2. Let X be a topological space. Recall that the path comp
M40: Exercise sheet 1 (mostly revision)
1. Let X be a topological space and A, B be closed subsets of X such that A B = X.
Let : X Y be a map into a topological space Y such that |A and |B are
continuous with respect to the induced topologies on A and B r
Let 0 , 1 be paths in a topological space X. What does it mean to say that 0 is
based homotopic to 1 : 0 1 ?
Show that is an equivalence relation.
(b) Let x0 , x1 be points in a topological space X and let : I X be a path from x0
to x1 .
M40: Exercise sheet 2
Fun with topological groups
1. Let X, Y be topological spaces and let A X, B Y have the induced topology.
(a) Show that the product topology on A B is the same as the topology on A B
induced by the product topology on X Y .
(b) If f