Algebraic Topology Fall 2010

What is this course about?
Is the complement of two points in ordinary 3-space homeomorphic to the complement of a single point? We learn in this course a technique that helps us answer such questions by converting them into an isomorphism problem for abelian groups. To be a little more concrete, we attach to each topological space a sequence of abelian groups H0(X), H1(X), H2(X),... , called the homology groups of X, with the property that a continuous map f : X→Y of topological spaces gives rise to group homomorphisms Hk(X)→Hk(Y). This passage from spaces behaves in a reasonable manner, for instance the identity map of X yields the identity map in each Hk(X) and if f is a homeomorphism, then the maps Hk(X)→Hk(Y) must be group isomorphisms. This fact alone already implies that if for some k, Hk(X) and Hk(Y) are not isomorphic as abelian groups, then X and Y cannot be homeomorphic. This would be not much more than an interesting philosophy, were it not that these homology groups are relatively easy to compute. For instance, for X the complement of N distinct points in ordinary 3-space, we find that H2(X) is isomorphic to ZN .
The term Algebraic Topology refers in general to a collection of techniques which attach to a topological space (or something somewhat more involved) an algebraic structure and what we just sketched is the most basic example.

Contents of the course
We begin with defining homology groups, a discussion that has its place in commutative algebra rather than in topology.This is followed by a number of applications, one of which is a proof that if Rn and Rm are homeomorphic, then n=m. We then develop the topological notion of a cellular structure, which aids us to compute these groups. This leads to another set of applications in fixed point theory. We then explain the relation between the homology group of a product X×Y with the homology groups of the factors as embodied in the Künneth theorem. If time permits we also discuss cohomology theory.

Prerequisites
Basic topology and familiarity with abelian groups and homomorphisms.

Literature
Lecture Notes in PDF format will be produced while the course is in progress (essentially a translation into English of part of the Dutch version). I advise you to consult some books as well. Here is a sample:

E. Bredon: Topology and Geometry.
Graduate volume 139 of Graduate Texts in Mathematics. Springer, 1993.

I.M. Singer and J.M. Thorpe: Lecture notes on elementary topology and geometry.
Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976.

Spanier: Algebraic topology.
Springer-Verlag, New York, 1966, ISBN: 0-387-94426-5.

P.J. Hilton and S. Wylie: Homology theory: An introduction to algebraic topology.
Cambridge University Press, New York 1960.

R.M. Switzer: Algebraic topology---homotopy and homology.
Grundlehren der math. Wissenschaften, Band 212. Springer-Verlag, New York-Heidelberg, 1975.

Dold: Lectures on algebraic topology.
Grundlehren der math. Wissenschaften, Band 200. Springer-Verlag, New York-Berlin, 1972.

M.J. Greenberg and J.R. Harper: Algebraic topology. A first course.
Mathematics Lecture Note Series, 58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981 (comes with a separate errata sheet). ISBN: 0-8053-3558-7; 0-8053-3557-9.

Dieudonné: A history of algebraic and differential topology. 1900--1960.
Birkhäuser Boston, Inc., Boston, MA, 1989. ISBN: 0-8176-3388-X

The Algebraic Topology book project of Allen Hatcher (the PDF of which comprises about 550 pages) is downloadable from his webpages.

Place and time of the course
On Wednesdays 14:00-16:45, starting September 15, 2010. Since the interest in this course turned out to be much greater than anticipated, it was necessary to move it to another location. However, this late in the game, it was no longer possible to keep the same room throughout (for which I apologize on behalf of the Faculty of Sciences). The meeting places are now as follows:

Week 37 (Sept 15) Buys Ballot Lab BBL 001
Week 38 (Sept 22) Buys Ballot Lab BBL 283
Week 39 (Sept 29) Buys Ballot Lab BBL 001
Week 40 (Oct 6) Went Building N022
Week 41 (Oct 13) Buys Ballot Lab BBL 001
Week 42 (Oct 20) Unnik Building, Room 211
Week 43 (Oct 27) Buys Ballot Lab BBL 001
Week 44 (Nov 3) Went Building N022
Week 45 (Nov 10) Buys Ballot Lab BBL 001
Week 46 (Nov 17) Buys Ballot Lab BBL 83
Week 47 (Nov 24) Buys Ballot Lab BBL 001
Week 48 (Dec 1) Mathematics Building 611ab
Week 49 (Dec 8) Buys Ballot Lab BBL 001
Week 50 (Dec 15) Went Building N022

As there is hardly a pattern, I recommend you to copy this information in your diary. The location of these buildings can be found here. If you come from Utrecht Central Station and need to go to the Buys Ballot Building or the Mathematics Building, then, take Bus 11 and get off at Botanische Tuinen. For the Unnik Building and Went Building you may take Bus 11 or 12(s) and get off at the stop Kruytgebouw or Bestuursgebouw.

Exam: date, time and place
The written exam is on January 26, 13:30-16:30 in the Green Room (Groene Zaal) of the F.C. Went Building.

Grading
Exercises with an asterisk will be graded (with grade ‘sufficient’, ‘in between’ or ‘insufficient’) and must for that purpose be handed in within a week after the mentioned date (by mail or hard copy). These grades can contribute towards your final grade as follows: the grade (named H) for your homework is determined by taking the average of all the weekly grades except for the two worst ones, using the rule: sufficient=10, in between=7, insufficient=5 and with failure to hand in a starred (*) homework exercise resulting in a zero grade. Besides, you may miss at most two hand ins. If you get grade W for the written exam, then your final grade is the computed according the rule 0.4H+0.6W or W (rounded off appropriately), whichever is highest, provided W is at least 6. Assistence with the exercise sessions course will be provided by Jeroen Goudsmit J.P.Goudsmit@students.uu.nl. Our records of the grades of the homework assignments can be accessed via the pair of data consisting of affiliation and student number.

Material covered and exercises
To be updated weekly, the starred exercises will be graded and this count towards your final grade. Jeroen has volunteered to supply solutions of the graded problems.

Week 37: Sect. 1 until chain maps; exerc. 1* and add as part (b)*: Prove that if X is arcwise connected, then any map f: X → X induces the identity in H0(X), 2, 3a.
Week 38: Sect. 1 until page 8 (Homotopy); exerc. 5 and 6*.
Week 39: Sect. 2 until Theorem 2.11; exerc. 8, 9 and 10*.
Week 40: End of Sect. 2; exerc. 11*, 13 and 14.
Week 41: Until Lemma 3.10; exerc. 15*, 16, 17.
Week 42: An extra week is granted for submitting exerc. 15*; for next week: 17 has acquired an asterisk: 17*, 18, 19. Todays lecturer is Dr van der Kallen.
Week 43: Until Lemma 4.1; exerc. 18, 19*, 23.
Week 44: Until Lemma 4.14; exerc. 24*, 26a.
Week 45: Finished section 4; exerc. 26bcd, 27*,28.
Week 46: Until Lemma 5.7; exerc. 29, 30*,31 (Warning: in exercise 30 it must be assumed that A is nonempty. Alternatively, you may show that the homology of the pair (X,A) is that of (X/A, A/A).
Week 47: Until Lemma 6.2; exerc. 32, 33*.
Week 48: Until Cor. 6.12; exerc. 34 (except the `Ext’ part), 35*.
Week 49: Finished section 6; exerc. 37*. The class in week 50 will be devoted to discussing problems and exercises.