# MA3403 Algebraic Topology I - Fall 2014

## Messages

**19.11**Today and tomorrow (Thursday 20 Nov.) are the last lectures. Today we discussed van Kampens Theorem concerning the calculation of the fundamental group of a space written as the union of two open subsets. Thus we considered the amalgamated free product of two groups. We also considered examples where the theorem is quite easy to apply.On Thursday we shall continue and finish the discussion of covering spaces, illuminate the basic properties, without going into the proofs in detail.**10.11**There will be lectures only for two more weeks. We are now working with Chap.4, where the topics are first of all covering spaces and basics about the fundamental group of a space.This is an important part of the curriculum, but not so easy to read about in the textbook. So I simply "warn" the students about this.**6.11**The*exam dates*will be Tuesday 9 or Wednesday 10 December for most of the students. A few students, for which both of the two dates are inappropriate, will have exam on Friday 5 December. Each student will get an e-mail letter very soon, with information about the date, time, and where to meet.**4.11**This week we start from the beginning of Chap. 4, about covering spaces. After reading (and contemplating on) the pages 85-87, you should do*EXERCISE 1 on page 87*.**29-10**Several students have so far answered in the DOODLE (see message on 26.10 ) with regard to possible exam date. The student should mark (all) those days which are OK for him, so that the other days (in light red colour) are the ones which are bad for the student.

**26.10**We are now at the end of Chap.2, to be finished on Wednesday 29 Oct. Then we shall have a brief look at Chap.3,including how to set up an axiom system for homology theory in general. As a consequence, if you have two homology theories (such as the singular homology theory), and they agree on the space consisting of one single point, then they will agree on all CW-complexes. Then we shall continue with Chap.4 where the topic is "covering spaces", a nice theory which is seen to involve a bit group theory, namely (discrete or finite) transformation groups, including the so-called fundamental group of a space.

*Exercise* : Use Prop. 2.24 to do Exercise 8 on page 64.
*Exercise* : see exalgtop.pdf

**26.10****EXAM DATES**- We shall determine two appropriate dates for the oral exam in December. Please participate in the following Doodle : http://doodle.com/t2qv4b9zf4ecdr6r
- and select those dates which are OK for you.
- ——————-
**15.10**In Chapter 2, we have done many calculations with the Mayer-Vietoris sequence, to calculate the homology of spaces which have a cell decomposition, especially when one attaches one single cell at a time. We shall now introduce "relative homology", and a central result is Theorem 2.11(Excision theorem).

*Exercise* : Do Exercise 3 page 41, and Exercise 4 page 46 (The five lemma), in the special case that the four vertical maps (as explained) are isomorphisms.

**7.10**We are now studying Chap.2, about building spaces using attaching maps. The building blocks are cells (homeomorphic to open disks) glued together along their boundaries, which can be more or less "collapsed" depending on the glueing maps.

*Exercise* : By studying the pages 35-37 in the textbook, do Exercise 1 and 2 on page 37, which are really problems in general topology. Study also the Example on page 38, about the real projective plane P^2, and how one can calculate its homology using the Mayer-vietoris sequence. This is a good example to test yourself. If you dont understand the arguments, ask yourself why, and figure out what you need to understand better.

**30.9**We did not finish Chap.1 last week, so at the beginning on Wednesday we shall finish this chapter. After this we start on Chap.2, about cell complexes (more precisely CW-complexes), which is a theory about how topological spaces can be build by successively attaching cells (homeomorphic to an open disk) of increasing dimension. Most of the spaces of interest for us, for example manifolds, are of this kind, in fact.

*Exercise* : Find a vector field on the 3-sphere (unit sphere in Euclidean 4-space) which is never zero (read page 29 for definitions). Moreover, try this: the 3-sphere has three vector fields which are linearly independent at each point.

**24.9**This week we shall finish what we want to take from Chapter 1. It is important to become familiar with the Mayer-Vietoris sequence! See page 22. We shall calculate homology groups using this technique, in special cases. For example, we shall recall on Wednesday how to calculate the homology groups of the circle (you also find a proof in the textbook). Now, I give you the following*Exercise 1 :*Assume the homology of the circle is already known, use the Mayer-Vietoris sequence to calculate the homology of the 2-sphere.

*Exercise 2* : Repeat the argument used in Exercise 1 to calculate the homology of the n-sphere for all n, by induction on n.

**11.9**This week we have gone through the proof of Theorem 1.8 and 1.10 in great detail.So continuing in the book p.16 and onward, we shall draw various interesting consequences about singular homology. We have discussed the concepts, for functions: retraction, deformation retraction, and for subsets of spaces: retract and deformation retract. Make sure you understand these very well.Here is one*exercise*about this: Show that the unit circle centered at the origin in the xy-plane is a deformation retract of the plane minus the origin. You should read the book to page 19, so that you can start working on Exercise 4 on page 19. But we shall continue with these topics next week.

**2.9**We continue with the study of singular homology, where a basic result is Theorem 1.10 (in Vick's book) saying that homology is a homotopy functor, that is, homotopic continuous maps induce the same homomorphism between the homology groups. So we will define the concept "homotopy" for maps, spaces, and also for chain complexes. In fact, the latter is an algebraic construction which we shall need in order to handle the topological case.*Exercise*Make sure you understand Exercise 1.7 p. 10, namely the proof of Theorem 1.7. The result follows from the functorial properties of the homology group construction, as explained somewhere at the bottom of page 6 (just before the calculation of the homology of X = point.)**26.8**We started last week on Chapter 1 (see Vick's book) on singular homology theory, which will continue to be our main focus this week and also later. You need some prerequisite knowledge from general topology and algebra, which you may not find in the textbook, such as topological spaces, continuous functions, free abelian groups, finitely generated abelian groups, fields, rings,….

*As an exercise,* you should try (very hard) to solve Exercise 2 (page 5). This is a basic result in homological algebra in general, saying that the composition of two consecutive boundary operators is zero.

**11.8**The first lecture is on Wednesday 20 August, 14.15-16, room R41, as scheduled.

## Lecture hours

- Wednesday 14.15-16.00, Auditorium R 41
- Thursday 10:15 - 12:00 , Auditorium R 59

## Lecturer

- Room 1250, Sentralbygg 2
- Phone: 73 59 66 83

## Course material

- James W. Vick:
*Homology Theory - An Introduction to Algebraic Topology*, 2nd edition, Graduate Texts in Mathematics, vol. 145, Springer Verlag, 1994

### Supplementary texts

- M. Greenberg, J.R. Harper: Algebraic topology. A first course. Benjamin/Cummings Publ. Co. 1981
- M. Greenberg: Lectures on Algebraic Topology. Benjamin 1966,1971 ,..?
- A. Hatcher:
*Algebraic topology*http://www.math.cornell.edu/~hatcher/AT/AT.pdf

- A note on covering spaces (supplement to chap.4 in Vick's book): Covering spaces
- A brief survey of basic concepts and results, etc.A brief survey

## Syllabus (preliminary)

**From the book of James Vick :**- 1.
*Chapter 1*: Singular Homology Theory - 2.
*Chapter 2*: Attaching Spaces with Maps - 3.
*From Chapter 3*: Only a) page 74-75, on the definition of a homology theory by axioms, and - b) page 77-78, how to define the singular cohomology theory.
- 4.
*Chapter 4*: Covering Spaces. - Here the proofs of Prop. 4.21 (Hurewicz homomorphism) and 4.26 (Van Kampen Theorem) can be skipped, but
- the definitions and usage of the results are required.

## Final exam

- Oral exam on 5,9, and 10 December.For most of the students the dates will be 9-10 December. The registered students will get a message (e-mail) about the exam, where and when the exam takes place.