GEOMETRY AND TOPOLOGY I
ASSIGNMENT 11-12 SOLUTIONS
Problem 0.1 (Assignment 11, Problem 4). Compute the homology groups of K , where
K is a Klein bottle, using Z and Z2 coecients.
Solution. View K as the standard 2-complex of a, b abab1 , and denote the 2-
GEOMETRY AND TOPOLOGY I
ASSIGNMENT 7 SOLUTIONS
Problem 1. Let S be a genus-2 surface with two (distinct!) points removed. Draw some
pictures illustrating a deformation retraction from S to a graph.
Solution. See Figure 1. To get from Picture (1) to Pictur
GEOMETRY AND TOPOLOGY I
ASSIGNMENT 8 SOLUTIONS
Problem 1. Find a covering space of a bouquet of 2 circles whose automorphism group
is A4 .
Solution. A4 is the subgroup of S4 consisting of the even permutations. Let a = (123) =
(12)(23), b = (12)(34), and
GEOMETRY AND TOPOLOGY I
ASSIGNMENT 9-10 SOLUTIONS
Problem 1 (Problem 3, Assignment 9). Let X denote the standard 2-complex of
a, b, c, daba1 b1 cdc1 d1 ,
so that X is a genus 2 surface. Draw a tiling of the plane by (combinatorial) pentagons
with four aro
GEOMETRY AND TOPOLOGY I
ASSIGNMENT 6 SOLUTIONS
Problem 54.5. Consider the covering map R R S 1 S 1 given by
(s, t) = (cos(2s), sin(2s), (cos(2t), sin(2t) .
Consider the path
f (t) = (cos(2t), sin(2t), (cos(4t), sin(4t)
in S 1 S 1 . Sketch f and sketch a l
22.2
(a) For any y Y , we have that p( f (y) = ( p f )(y) = y, so p is surjective.
Since p is continuous, then if U Y is open, then p1 (U ) is open. Suppose
p1 (U ) X is open. Then, f 1 p1 (U ) = ( p f )1 (U ) = U . Since f is
continuous, then f 1 p1 (U )
SOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS BY OLIVIER MARTIN
13
#5. Let T be the topology generated by A on X. We want to show
T = J B J where B is the set of all topologies J on X with A J .
This amounts to showing that if A J then T J . This is automa
SOLUTIONS TO ASSIGNMENT 2 MATH 576
SOLUTIONS BY OLIVIER MARTIN
18
2) No. Consider f : R cfw_0 where cfw_0 is given the only possible topology and R the
standard topology. Let A = (0, 1) then 1 is a limit point of A but f (1) = 0 is not a limit
point of f
SOLUTIONS TO ASSIGNMENT 3 MATH 576
23
5. Let X be a topological space with the discrete topology. Let A X be a non-singleton
non-empty subset i.e. a subset containing more than one element of X . Let a A then
A = cfw_a (A cfw_a) is a separation of A since