# MA3403 Algebraic Topology I - Fall 2016

## Messages

**24 November**:**NB!**Our last meeting (spørretime? siste olje? last supper) will be on Tuesday 6 December,16-18 pm, in room 734 in Sentralbygg 2.

**20 November**: The last two lectures are on Tuesday 22 November and Thursday 24 November. We shall finish our discussion of covering spaces from Chap. 4. We also discuss the Hurewicz homomorphism (page 107-108) and Van Kampen's theorem (at the end of Chap.4). But we shall focus on examples of application rather than the detailed proofs.

**13 November**: On Tuesday 15 November and Thursday 17 November, we shall look at the remaining parts of Chap.2, where Theorem 2.21 is the primary result.

**6 November**:This week, Tuesday 8 Nov. and Thursday 10 Nov. we shall continue with Chap. 4 about covering spaces.

**30 October**: This week there will be lecture only on Tuesday 1 November.We shall continue with Chap.2 on the homology of cell complexes.

**20 October**: Next week (25 and 27 October) we jump again to Chapter 4 and continue with the theory of covering spaces. Make sure you understand the definition of the fundamental group of a space (page 92-95).

**16 October**: On the following Tuesday and Thursday we shall return to Chapter 2 and the study of homology of cell complexes (CW- complexes), in particular we consider the cell decomposition of some familiar spaces. In this chapter we also introduce the relative homology groups H*(X,A) of a pair (X,A), where A is a subset of X, and we shall establish the long exact homology sequence of (X,A).

**8 October**: In the following week (11 and 13 October) we shall jump to Chapter 4 (page 85) in the textbook. The topic is "covering spaces", namely the situation where there is a surjective p: map X →Y with certain properties, in particular it is a local homeomorphism (see the definition p.85). Well known examples are exp: R → S^1, and p: S^n →RP^n. The student should start reading the first 5-6 pages and try to figure out "what is going on", and test his understanding by doing the exercises below :**Exercises**Exercise 1 and Exercise 2 page 87

**27 September**: On Tuesday we shall finish Chap.1, including the last pages from Lemma 1.27 and ending with Theorem 1.30 (the famous Jordan-Brouwer separation theorem). We shall discuss these results, but without going through proofs. The important thing is that the student understands these results.- On Thursday we start with Chapter 2, from the beginning. The topic is the Construction of "Attaching Spaces", by which a space is obtained by "glueing" together two spaces. Of particular interest are the cell complexes (or CW-complexes)obtained by glueing together n-cells of various dimensions.Then we shall apply the Mayer-Vietoris sequence to calculate the homology of a space X obtained from a space Y by attaching an n-cell to it. We shall also continue with this chapter the following week.

**12 September**: We have gone through the proofs of Theorem 1.8 and 1.10, namely a) showing that a convex set has the homology of a single point, and b) two homotopic maps from X to Y induce the same homomorphisms from the homology groups of X to the homology groups of Y. This week we shall deduce many consequences of these basic results. Moreove, we shall discuss the construction of the "long exact homology sequence" associated with a short exact sequence of chain complexes Theorem 1.13),and use it to establish the so-called Mayer-Vietoris sequence (see page 22). Thus we are aiming at finishing most of Chap.1 this week, looking only briefly at the last 5 pages of the chapter.**EXERCISE SET 1**: Exercise 4 (page 19), Exercise 5 (page 23), Exercise 7 (page 25)

**30 August**: We are working with the construction of singular homology, as in Chap.1 of the text book (Vick). Basic concepts are a) a singular q-simplex in X, b) the chain complex of X, c) free abelian groups, d) cycles and boundaries, and the boundary operator, e) homology groups of a chain complex. As a first concrete example in topology, calculate the homology groups of the space consisting of one single point. Note that all this is found within the first 7 pages of the textbook!!

**9 August**: After a general introduction to algebraic topology and what basic knowledge the students are required to have in advance (or learn rather soon), we shall start on the first topic called "homology theory". So we start with Chapter 1 in the textbook, where the classical "singular homology theory" is described in great detail.

**9 August**: The first lecture is on Tuesday 23 August, 12.15-14, room S 22, as scheduled.

## Lecture hours

- Tuesday 12.15-14.00, Auditorium S 22
- Thursday 12:15 - 14:00 , Auditorium S 22

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

## Syllabus

**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 Monday 12 and Tuesday 13 December,both days in room 922, Sentralbygg 2. Precise information will be sent to each candidate.