HOMEWORK 3
Problem 1: Let X be a topological space and let x, y, z, w X. Let f, g, h be paths from x to y,
y to z, and z to w, respectively. Show that the paths (f g) h and f (g h) are homotopic relative
endpoints. (Remark: You need to write down an expli
Algebraic Topology M3P21 2015
Homework 3
AC
Imperial College London
a.corti@imperial.ac.uk
16th February 2015
N.B.
Turn in 5 questions by Monday, 9th March, at 12:00 either in
class or in my pigeon-hole in the mail-room on the 6th floor.
0
1
n1
2
n
(1) Le
Algebraic Topology M3P21 2015
Homework 1
AC
Imperial College London
a.corti@imperial.ac.uk
18th January 2015
N.B.
Turn in 5 questions by Monday, 2 February, at 12:00 either in class
or in my pigeon-hole on the 6th floor.
(1) Suppose that f : X Y be a quot
Algebraic Topology M3P21 2015
Homework 2
AC
Imperial College London
a.corti@imperial.ac.uk
2nd February 2015
N.B.
Turn in 5 questions by Monday, 16 February, at 12:00 either in
class or in my pigeon-hole in the mail-room on the 6th floor.
(1) Show that fo
Algebraic Topology Comments on Problem Sheet 1
Andrea Petracci
a.petracci13@imperial.ac.uk
February 2015
Exercise 2. Let me show a nice trick that I have learned from the
solution of some of you. The following diagram sums up the situation.
`
i
/X Y
X
q
p
M3/4/5P21 - Algebraic Topology
Imperial College London
Lecturer: Professor Alessio Corti
Notes typeset by Edoardo Fenati and Tim Westwood
Spring Term 2014
These lecture notes are written to accompany the lecture course of Algebraic Topology in the
Spring
Algebraic Topology M3P21 2015
solutions 1
AC
Imperial College London
a.corti@imperial.ac.uk
9th February 2015
(1)
(a) Quotient maps are continuous, so preimages of closed sets are closed
(preimages of open sets are open, and f 1 (Y \ A) = X \ f 1 (A) for
Algebraic Topology M3P21 2015
solutions 2
AC
Imperial College London
a.corti@imperial.ac.uk
23rd February 2015
A small disclaimer
This document is a bit sketchy and it leaves some to be desired in
several other respects too. I thought it is more useful to
Algebraic Topology Comments on Problem Sheet 2
Andrea Petracci
a.petracci13@imperial.ac.uk
March 2015
Exercise 1. To maintain mental sanity, I will denote points of I (resp.
S 1 , D2 ) by the letters t, s (resp. z, w) and I will use the letters H, G, F fo
HOMEWORK 4
Problem 1: Show that S 1 is a retract of S 1 S 1 , but not a deformation retract.
Solution: There are many retractions r : S 1 S 1 S 1 . E.g. we can keep the rst factor pointwise
xed and collapse the second factor onto the usual basepoint. More
HOMEWORK 6
0
1
2
n1
n
Problem 1: Let 0 V1 V2 . Vn 0 be a complex of vector spaces, meaning that
the Vi are vector spaces and the i are linear maps with i i1 = 0 for i = 1, ., n. In particular,
ker i im i1 , so it makes sense to dene the quotient spaces Hi
HOMEWORK 5
Problem 1: Van Kampens theorem talks about decompositions X = U V , where U, V are open
and path-connected, and U V = is path-connected as well. Show that the assumption that both
U and V are open is necessary for the theorem to hold.
Solution:
HOMEWORK 7
Problem 1: Let X be an arbitrary nonempty set. Compute the singular homology of X equipped
with the trivial topology cfw_, X and of X equipped with the discrete topology.
Solution: Let Xtriv and Xdisc denote the topological space that consists
HOMEWORK 2
Problem 1: Let X, Y be topological spaces, A X a subspace, and f : A Y a quotient map.
Show that X f Y is homeomorphic to the quotient X/, where is the equivalence relation on X
generated by x1 x2 for all x1 , x2 A for which f (x1 ) = f (x2 ).
SOLUTION TO QUESTION 1 OF HOMEWORK 8
As I said in lecture, the questions on this problem sheet are much too dicult. Nevertheless Im
posting a solution to Question 1 here since this should help clarify the Mayer-Vietoris sequence for
mapping tori, for thos
HOMEWORK 1
Problem 1: Let f : X Y be a quotient map of topological spaces.
(a) Show that if Y is Hausdor, then the bers f 1 (y) (y Y ) are closed.
(b) Is Y necessarily Hausdor if all the bers are closed?
Solution: (a) Quotient maps are continuous, so prei
Algebraic Topology M3P21 2015
solutions 3
AC
Imperial College London
a.corti@imperial.ac.uk
11thth March 2015
A small disclaimer
This document is a bit sketchy and it leaves some to be desired in
several other respects too. I thought it is more useful to