MAT 530 Topology, Geometry I, Fall 2015.
 Instructor: Olga Plamenevskaya, office 2112 Math Tower,
email: olga@math.stonybrook.edu
 Office hours: W 1:002:00pm in P143
, M 1:002:00pm and Fri 11:0011:55am in my office 2112, or by appointment.
 Grader: Chandrika Sadanand, office 2105 , email:
chandrika@math.stonybrook.edu,
office hours: Fri 4:005:00pm
 Class meetings: Monday and Wednesday, 10:0011:20am, Earth and Space Building, Rm 181.
References :
 O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev,
Elementary Topology. Problem Textbook.
A version of this book is available for free download from Viro's webpage.
(Caution: the link goes directly to a large pdf file.) This is an unusual textbook: all theorems
are presented in stepbystep questions that you have to figure out yourself.
Note that this online
version does not have any solutions; you can buy a complete book, but I don't recommend using solutions
too much. (You will need to be able to produce detailed proofs in quizzes! So please work through
the questions carefully.)
The first few weeks of the course will be "flipped": you will be responsible
for learning/reviewing the theory via the homework problems; in class, we will address questions and
discuss some of the finer points.
 Allen Hatcher, Algebraic Topology.
This book is available for free download from Hatcher's webpage. (This book will mostly be used for
the fundamental group & coverings part. This is Chapter 1 of Hatcher.)
Other useful books:
James R. Munkres, Topology.
William S. Massey, Algebraic Topology: An Introduction, GTM
56. (The first few chapters are also contained in another book by Massey,
GTM 127).
Exams: Final Exam will be on Monday, Dec 14, 10am12:30pm, in P131.
Homework: weekly assignments will be posted on
this page. Homework will make a significant part of your course grade. There will also
be regular quizzes (base on homework material), especially during the first part of the course
(pointset topology).
Important: Please
write up your solutions neatly, be sure to put your name on them and staple all pages.
Illegible homework will not be graded. Late homework will not be accepted.
You are welcome to collaborate with others and even to consult books,
but your solutions should be written up in your own words,
and all your
collaborators and sources should be listed.
 There will be no
quizzes until further notice.
 Homework 13: pdf, due Wednesday, Dec 2.
Please read about deck transformations and group actions in Hatcher p.7074; we'll
go over this topic on Monday.
 Homework 12: pdf, due Wednesday, Nov 18.
 Homework 11: Hatcher p. 53 questions 3, 4, 8, 9. Also, compute the
fundamental group of the projective plane and of the Klein bottle by using van Kampen theorem: represent the
projective plane as a Mobius band with its boundary glued up by a disk, the Klein bottle as the union of two
Mobius bands glued along their boundary circles. Hand in this part, due Wednesday, November 11.
Go through (at least the first part of) 37'4x of Viro's book to learn about winding number.
Cellular spaces: read 42'1 in Viro's book, do 42.1, 42.2, 42.G, 42.H (do not hand in). Do 43.Ax, 43.Bx, 43.Ex, 43.Fx,
43.Hx, 43.Ix, 43.Jx; you may assume that all the cellular complexes are finite (except for the questions such as 43.Ix that become vacuous
if you assume finiteness). Hand in 43.Ax, 43.Bx, 43.Ix only
 Homework 10: Hatcher p. 39 question 16 (except (f), which we did in class);
p. 79 question 4 (guess what the covering should look like, and prove that it works);
Viro 34.M, 34.21, 34.24, 35.7, 40.A. Hand in on Wednesday, Nov 4.
Please also go through Viro 34'1 34'5, especially 34'3, which we haven't discussed in class. (Do not hand in anything
except the questions listed above, feel free to skip questions such as 34.9 and 34.23.)
 Homework 9: Hatcher p. 18 questions 1, 10; p. 39 questions
10,14, 15, and two more questions pdf . Hand in on Wednesday,Oct 28.
Optional: question 6 p. 18; you will need question 5 for the proof (do not hand in).
 Homework 8: section 1.1 of Hatcher's book, questions
3, 5, 7, 13 on p. 38 (to hand in on Wed, Oct 21). Caution: questions 5 and 7 deal with the general
notion of homotopy, not homotopy of paths. This material is contained in Chapter 0 of Hatcher and will be
discussed in class on Monday.
 Reading: "Homotopy and homotopy type" in Hatcher Chapter 0;
section 1.1 of Hatcher EXCEPT "The fundamental group of the circle". Sections 3033 in Viro's book
are also useful (but are not a mandatory assignment).
Please also do takehome makeup, due Monday, Oct 19. Please carefully
write the solutions you need to improve/redo on separate paper (do not correct the original exam!)
Hand in your makeup solutions together with your exam. While reworking your solutions,
you are allowed to use books and notes to
better understand the material.
 Homework 7:
Part I (do not hand in) Elem. Top. Problems:
23 AH, IL (basic material on projective spaces) to be done by Monday
25'1x25'4x (lettered quesions only, skip numbers; skip 25.Hx if unfamiliar with complete metric spaces)
Section 25 is about various topologies on spaces for maps X > Y. Some of it will be familiar from analysis;
compactopen topology is important in topology and geometry.
Part II (hand in, due Wednesday, October 14) 25.Cx, 25.Ix, 25.Lx;
22.26. 22.O. For 22.O, you might need a lemma: any homeomorphism of the sphere S^{n1} can be
extended to a homeomorphism of the entire disk D^{n}
(we are thinking of the sphere as the boundary of the disk). Prove this lemma first (sometimes it's called the Alexander trick).
Note that the lemma is breaks down if you want to get a smooth (i.e. differentiable) map;
in fact, Milnor's famous exotic 7dimensional spheres are obtained by gluing two copies of D^{7}.
The resulting manifolds are homeomorphic but not diffeomorphic to the standard 7sphere!
 Homework 6:
Part I (do not hand in) Elem. Top. Problems:
19'1x (19.1x, 19.2x, 19.Dx, 19.Ex, 19.Fx only); 19'2x;
20'320'5 (settheoretic summaries 20'1 and 20'2 may be useful!); 20'6 (20.2620.29 only); 20'7
(do letters NU, skip numbers 20.3720.42); 20.8;
21'121'3, 21'421'5 (only as much as you need for 22'1), 22'122'3.
Also, take a look at further examples of quotient spaces in 22'5, 22'6, 22'8, 22'922'13
(no need to prove them carefully).
We will be discussing the remaining questions on compactness + product spaces on Monday,
quotient spaces on Wednesday.
Part II (hand in, due Monday, October 5) pdf
 Homework 5:
Part I (do not hand in) Elem. Top. Problems:
17'1, 17'317'5, 17'6 (skip 17.11, 17.12), 17'7, 17'8 (skip 17.1717.23), 17'9 (X,Y,Z only);
16'2 (skip 16.11), 16'316'6;
18'1, 18'2 (skip 18.2)
We will discuss this material in the same order (section 17 first, then section 16, then section 18).
Section 16 requires familiarity with countable sets; 16'1 contains everything you need to know.
For a quick and fun refresher, you can watch
a video about the Infinite Hotel and
a video about countable and uncountable sets.
Part II (hand in, due Wed 9/23) 18.3x18.5x.
(If you are not familiar with normed spaces, just think about the corresponding metric
d(x, y) = xy = supx_{n}  y_{n}. By definition, this will give the same topological space.)
 Homework 4:
Part I (do not hand in) Elem. Top. Problems:
13'113'2, 13'3x (13.6x and 13.7x only);
14'114'7 to be done by Monday, Sept 14, and 15'115'5, 15'715'8
to be done by Wednesday, Sept 16. (We will be discussing (path)connectedness in Monday class and separation axioms
on Wednesday.) There will be a quiz on Wednesday as usual.
Part II (hand in, due Wed 9/16) 14.27x (1), (2) only; 14.28x (1),(2) only.
 Homework 3:
Part I (do not hand in) Elem. Top. Problems: 12'1 12'8 (except question 12.31),
to be done by Wed 9/9.
Part II (hand in, due Wed 9/9)
11.23 (please try to write a reasonably careful proof, although no formulas are required)
10.6 (please also give an example where the inclusion is strict)
3'5 (3.9 and 3.10) Use topology to show that there are infinitely many primes.
(Give an "honest" new topological proof as suggested in 3'5; rewriting Euclid's proof in topological terms
will be worth little credit.) Arithmetic progression is a sequence of the form {am+b}, m=0,1,2,...,
where a and b are positive integers.
Hint: suppose that the set of primes is finite. Show that {1} would then be open.
Please also do these questions about the Cantor set, a remarkable nowhere dense set.
 Homework 2:
Part I (do not hand in) Elem. Top. Problems:
3'6, 4'11, 4'12, 4'14, 5'15'3, 6'16'7, 6'12 (6.M, 6.N, 6.32, 6.33 only),
6'13 (6.O, 6.376.40, 6.42 only); 6'14, 10'110'8, 10'10 (10.O, 10.P only),
to be (mostly) done by Monday, Aug 31. Don't forget to do both "theorems"
(labeled by letters, eg 6.B), and "examples" (labeled by numbers, eg 6.19).
Added on 08/31, to be done by 09/2:
11'111'9: please do all the lettered questions 11.A11.Y, numbered questions
11.111.8, 11.10, 11.19, 11.20 only, and (optional) any of questions 11.2130
that seem interesting or difficult to you.
We'll be discussing this material
in class on Monday and Wednesday. (There will be a quiz on Wednesday, Sept 2.)
Part II (hand in, due Wed 9/2)
2'Ix (note that this about a disjoint union  the question is not trivial),
4'15x (4.Mx4.Px only). Please write complete proofs.
 Homework 1: Elem. Top. Problems: 2'12'10; 3'13'3;
4'14'5; 4'94'10, to be done by Wednesday, Aug 26.
Syllabus: we will follow the basic outline from
the graduate core course requirements (see below), not necessarily in the same order, with
additional topics
as time permits.

Basic point set topology
 Metric Spaces
 Topological spaces and continuous maps
 Comparison of topologies
 Separation axioms and limits
 Countability axioms, the Urysohn metrization theorem
 Compactness and paracompactness, the Tychonoff theorem
 Connectedness
 Product spaces
 Function spaces and their topologies, Ascoli's theorem
 Introduction to algebraic topology
 Fundamental group
 Fundamental group of
S^{n};
examples of fundamental groups of surfaces
 Seifertvan Kampen theorem
 Classification of covering spaces, universal covering spaces; examples
 Homotopy; essential and inessential maps
Students with Disabilities: If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 6326748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.