Algebraic Topology
Homework 13: Due Friday, December 3
Problem 1. We saw in class how RP 2 is a CW complex with one 2-cell, one 1cell, and one 0-cell, with e(2) = 2e(1) and e(1) = 0, which implies that H2 (RP 2 ) = 0,
H1 = Z2 and H0 = Z. Find a CW decompo
Algebraic Topology Solutions
Homework 1: Due September 2
Problem 1. Find an example of a topological space X that is not Haussdorf such that
each point in X has a neighborhood homeomorphic to the open interval (1, 1).
Consider the space (cfw_1, 2 IR)/ , w
Algebraic Topology
Homework 2: Due September 9
Problem 1. Consider a hexagon with opposite edges identied, so that the pattern of edges
around the boundary is abca1 b1 c1 . Which vertices are identied? Using orientability
and Euler characteristic, identif
Algebraic Topology
Homework 3: Due September 16
1. Page 42, problem 3.2
It doesnt matter if 1 is abelian. It does matter if 1 isnt. The dierence between
the map induced by two paths , from x to y is conjugation by 1 . Specically,
if f, g : 1 (X, x) 1 (X,
Algebraic Topology
Homework 4 Solutions
1. Page 53, problem 7.3. Note that innite product means with the product topology. Except where specically noted, innite products always have the product topology.
Let X = Xi , and let i : X Xi be the projection on
Algebraic Topology
Homework 5: Due September 30
1. Page 74, problems 4.1, 4.3, 4.4, 4.9.
4.1: Let a G1 and b G2 be nontrivial elements. ab and ba are both in reduced form,
and are dierent words, so the free group is non-Abelian. Also, (ab)n = abab . . . a
Algebraic Topology
Solutions to Homework 6: Due October 7
1. Page 94, problems 3.1 and 3.2.
3.1: The only inclusions are W Vi . Since 1 (W ) is trivial, any map 1 (Vi ) H can
be made into a commutative triangle via the trivial maps 1 (W ) 1 (Vi ) and 1 (W
Algebraic Topology
Homework 7: Due Wednesday, October 14
Recall that a meridian of a solid torus D S 1 is a curve that bounds D(point in
S 1 ), while a longitude is a curve of the form (point in D) S 1 .
ab
be an integer matrix. The linear transformation
Algebraic Topology
Homework 8: Due Wednesday, October 21
Problem 1 Back in early September, we proved that the fundamental group of a circle
was innite cyclic using Lebesgue numbers and an argument about angles. I want you to
take that keep track of angle
Algebraic Topology
Homework 9: Due Wednesday, November 4
Most of this weeks problem set is about simplicial homology, rst of triangulated surfaces, and then of
simplicial complexes in general.
Given a triangulated surface, we can order the vertices and de
Algebraic Topology
Homework 10: Due Wednesday, November 11
Cone operators. Let X be a star-like subset of Euclidean space, meaning that if
x X , then tx X for every t [0, 1].
As usual, let Cn (X ) be generated by the singular n-chains on X , modulo the de
Algebraic Topology
Homework 11: Due Wednesday, November 18
This weeks problem set is all about using Mayer-Vietoris to compute homology, and
hopefully getting a better feel for what homology means in the process.
A useful piece of terminology is of the co
Algebraic Topology
Homework 12: Due Wednesday, November 25
(If youre going out of town for Thanksgiving, then please drop o your homework
before you go.)
Let X1 , X2 , . . . be a sequence of path-connected topological spaces, and suppose that
xi Xi for ea
Algebraic Topology
Homework 13: Due Friday, December 4
Problem 1. We saw in class how RP 2 is a CW complex with one 2-cell, one 1cell, and one 0-cell, with e(2) = 2e(1) and e(1) = 0, which implies that H2 (RP 2 ) = 0,
H1 = Z2 and H0 = Z. Find a CW decompo
Algebraic Topology Midterm Exam, October 26, 2009
1) Covering spaces. (a) Find all compact topological spaces that are covers
of the genus-2 surface X = T 2 #T 2 (where T 2 = S 1 S 1 is the usual torus).
For each covering space, show how it can cover X .
Algebraic Topology Midterm Exam Solutions, October 15, 2010
1. (30 points) (a) Let X be a punctured torus and let be a loop around the
puncture, as depicted on the blackboard. Let x0 be a point on . Compute
1 (X, x0 ) and express the class of in terms of
Algebraic Topology
Homework 12: Due Wednesday, November 24
Let X1 , X2 , . . . be a sequence of path-connected topological spaces, and suppose that
xi Xi for each i. The wedge of the X s, denoted i Xi , is the quotient space of the disjoint
union i Xi by
Algebraic Topology
Homework 11: Due Wednesday, November 17
Problem 1. Compute the homology of the Lens space L(p, q ).
L(p, q ) is obtained from two solid tori, with the meridian of each torus
identied with p times the longitude, plus or minus q or q time
Algebraic Topology
Solutions to homework 10: Due Wednesday, November 10
Problem 1. As we dened in class, a short exact sequence 0 A B C 0,
with maps i : A B and j : B C splits if there is an isomorphism : B A C with
i being inclusion in the rst factor an
Algebraic Topology
Homework 9: Due Wednesday, November 3
Problem 1. Page 163, problem 2.1
The rational numbers with their usual topology and a countable discrete set are the
same when it comes to homology, since in either case there are countably many pat
Algebraic Topology
Homework 8 Solutions: Due Wednesday, October 27
Problem 1. Page 135, problem 7.2 For example 2.4, you can restrict attention to
the cover IR2 T 2 .
For example 2.2, our groups are G = Z and G0 = nZ . The normalizer of G0 is all of
G, an
Algebraic Topology
Homework 7: Due Wednesday, October 20
Problem 1 Back in early September, we proved that the fundamental group of a circle
was innite cyclic using Lebesgue numbers and an argument about angles. I want you to
take that keep track of angle
Algebraic Topology
Solutions to Homework 6: Due October 8
1. Page 94, problems 3.1 and 3.2. Both of these were essentially done in class, but
spell them out anyway.
3.1: The only inclusions are W Vi . Since 1 (W ) is trivial, any map 1 (Vi ) H can
be made
Algebraic Topology
Homework 5: Due October 1
1. Page 74, problems 4.1, 4.3, 4.4, 4.9.
4.1: Let a G1 and b G2 be nontrivial elements. ab and ba are both in reduced form,
and are dierent words, so the free group is non-Abelian. Also, (ab)n = abab . . . ab =
Algebraic Topology
Homework 4 Solutions
1. Page 53, problem 7.3. Note that innite product means with the product topology. Except where specically noted, innite products always have the product topology.
Let X = Xi , and let i : X Xi be the projection on
Algebraic Topology
Homework 3: Due September 15
1. Page 42, problem 3.2
The path from x to y doesnt matter if 1 is Abelian. The path does matter if 1 is
non-Abelian. To see this, note that the dierence between the map induced by two paths
, from x to y is
Algebraic Topology
Homework 2: Due September 9
Problem 1. Consider a hexagon with opposite edges identied, so that the pattern of edges
around the boundary is abca1 b1 c1 . Which vertices are identied? Using orientability
and Euler characteristic, identif
Algebraic Topology Solutions
Homework 1: Due September 1
Problem 1. Find an example of a topological space X that is not Haussdorf such that
each point in X has a neighborhood homeomorphic to the open interval (1, 1).
Consider the space (cfw_1, 2 IR)/ , w
Preliminary Examination in Topology: January 2011
Algebraic Topology portion
Instructions: If possible, answer all three questions on both sides of this sheet. If this
is not possible, then two complete solutions is better than three partial solutions.
Ti
Algebraic Topology Midterm Exam, October 26, 2009
1) Covering spaces. (a) Find all compact topological spaces that are covers
of the genus-2 surface X = T 2 #T 2 (where T 2 = S 1 S 1 is the usual torus).
For each covering space, show how it can cover X .