Introduction to algebraic topology, Spring 2013
Quiz 2, Tuesday April 23
Name:
1. (30 points) For each space below, determine its homology groups
and the Euler characteristic.
(a) Discrete topological space with two points X = cfw_p1 , p2 .
(b) R3 \ cfw_0
Name:
Introduction to algebraic topology. Midterm exam.
March 14, 2013
You can solve problems in any order. Textbooks and notebooks are not allowed on the exam.
1. (15 points) Mark those squares that are followed by
correct statements.
If Y is a retract o
Introduction to algebraic topology, Spring 2013
Quiz 1
Name:
1. Mark the squares that are followed by correct statements.
Two-sphere S 2 admits a triangulation.
Pair ([0, 1], cfw_0, 1) has the homotopy extension property.
Any nite graph is homotopy equiva
Introduction to algebraic topology, Spring 2013
Homework 12, due Thursday, May 2
Read Hatcher Section 2.C (Lefshetz xed point theorem, pages 179
182)
1. (a) Suppose that a map f : S 1 S 1 has winding number 3.
What is the Lefshetz number of f ? Does f nec
Introduction to algebraic topology, Spring 2013
Homework 10, due Tuesday, April 16
Read Hatcher Sections 2.1 and 2.2 (in the latter mostly MayerVietoris sequence).
Exercises for Section 2.1 (pages 131-133): 15, 16, 17a,b, 22a,b,c, 29,
31.
Exercises for Se
Introduction to algebraic topology, Spring 2013
Homework 11, due Tuesday, April 23
Exercises from Hatcher Section 2.2 (pages 155-157) 4, 9ac, 12, 20.
1. (a) We construct a 2-dimensional CW complex X by starting
with the 1-skeleton X 1 = S 1 S 1 a bouquiet
Introduction to algebraic topology, Spring 2013
Homework 9, due Tuesday, April 9
Hatcher, Section 2.1 (page 131) exercises 4, 5 (in exercise 5 Hatcher
is referring to page 102; also look at his examples 2.2-2.5 on page
106), 11, 12.
1. (a) Show that the s
Introduction to algebraic topology, Spring 2013
Homework 8, due Tuesday, April 2
1. (10 points) Suppose that complex C is the direct sum of complexes A and B. Prove that Hn (C) Hn (A) Hn (B) for all n.
=
2. (30 points) Consider the following complexes
0
A
Introduction to algebraic topology, Spring 2013
Homework 7, due Tuesday, March 12
1. Explain why the long exact sequence of a pair (X, A) is exact in
the n (A)-term.
2. What does the long exact sequence of a pair tell you in the case
(a) A = X, (b) A is c
Introduction to algebraic topology, Spring 2013
Homework 6, due Tuesday, March 5
1. List all regular coverings among 14 coverings depicted on page
58 of Hatcher. Which of these coverings have innite degree?
Give an example of a degree 4 covering of S 1 S
Introduction to algebraic topology, Spring 2013
Homework 4, due Tuesday, February 19
1. Which of the following are categories? In each example, composition of morphisms is the obvious one.
(a) Objects are nite sets, morphisms are injective maps of sets.
(
Introduction to algebraic topology, Spring 2013
Homework 5, due Tuesday, February 26
1. Mark the squares that are followed by correct statements.
Any CW-complex is path-connected.
Any topological space has a universal covering space.
Any locally simply-co
Introduction to algebraic topology, Spring 2013
Homework 3, due Tuesday, February 12
1. (20 points) Describe a triangulation of (a) RP2 , (b) Klein bottle,
(c) 3-dimensional cube I 3 = I I I. In each case check whether
your triangulation is a simplicial c
Introduction to algebraic topology. Final exam, due
Friday May 10, 2013.
Name:
I certify that my answers and solutions are my own
work.
Email me if you have any questions. You can solve problems
in any order.
1. (10 points) Mark those squares that are fol
Introduction to algebraic topology, Spring 2013
Homework 2, due Tuesday, February 5
1. Classify all CW-complexes with two 0-cells and two 1-cells up to
(a) homeomorphism, (b) homotopy equivalence.
2. Describe a CW-complex homeomorphic to R2 . Can you gene
Introduction to algebraic topology, Spring 2013
Homework 1, due Tuesday, January 29
1. Show that homotopy is compatible with composition. If f, g :
X Y are homotopic and f , g : Y Z are homotopic, then
f f, g g : X Z are homotopic.
2. Prove that if X is c