# MA3403 Algebraic Topology I - Fall 2015

## Messages

**7 December**: To get a brief review, take a look at the "brief survey" (handwritten, dated 2012), see below. The same collection of good advice is relevant today as before.See you tomorrow at the last (supper) meeting in the 13th floor.

**30 November**: I hope you have time for preparations these days. The examen will be on Thursday and Friday 10-11 December. The adm. will soon send an email to each candidate with d 2012) specific information about when and where etc.- Let us have a
*last meeting*(last oil, but no supper) on Tuesday 8 December at 4.15 pm. I propose the Institute's lunch/coffee room in the 13th floor. - Moreover, re
- garding the "Brief survey …." which you can find below, an updated version will be there soon. (The old one is still OK, but …?!?)

**15 November**: Tomorrow Monday 16 Nov. and Wednesday 18 Nov. will be the last lecture days. We shall complete our discussion of covering spaces, and we shall also discuss Van Kampen's Theorem. More information with regard to the course will be given later on these pages.**4 November**: We continue with Chap. 4 for the rest of the time we have. About covering spaces and their properties. We have already this week discussed the fundamental group of a space, as a functor. We also finished Prop. 3 about homotopy lifting etc. We have lectures two more weeks.

**27 October**: On Wednesday 28 October we spend the first hour on Chap.3 and then continue with Chap.4 for the rest of the semester. Dont miss the basic facts about the theory of covering spaces.

**21 October**:On Monday 26 Oct. we shall talk about some basic res ults from Chap. 3, about homology with coefficients in G, how to define cohomology, and the cup product. After that, we are left with Chap. 4 called "covering spaces". What is that really? Well, try to find out yourself, or perhaps go to the lectures? It is not esay to tell what is relevant for the examen !**Note**Based on the wishes of students who are participating at the lectures, the oral exam is chosen to be*Thursday 10 December*. If for some important reason a student will have difficulties with this date, take contact with the lecturer.

**11 October**:The homology of a pair(X,A) is defined in Chap.2, together with many result related to the long exact homology sequence of the pair. One is primarily interested in spaces which can be built as a CW-complex, and then the homology theory also becomes rather simple. In fact, one can calculate the singular homology of a CW-comples from a chain complex much simpler than the singular chain complex. This will be explained and illustrated with examples. After this we are almost ready to go to Chapter 4, where the fundamental group of a space is defined and the notion of covering spaces is introduced, and the basic properties investigated.Note that this is not part of homology theory, but it belongs to homotopy theory, in fact the first taste of this theory. This is also the last part of the present course.

**28 September**: At present we are working on Chap.2, where the primary concept is "attaching spaces". Namely two spaces can be "glued" together according to a specific procedure. Cell complexes are examples of such spaces, being built up successively by attaching cells together. A cell is a topological space homeomorphic to an open disk. We also introduce relative homology groups H(X,A) of a pair (X,A), where A is a subset of X.

**16 September**: We have now used the Mayer-Vietoris sequence to calculate the homology of spheres (see p.22-23), we have proved Brouwer's fixed point theorem, discussed the degree of a map on an n-sphere to itself. Next week shall discuss further applications of these last results, and finish what is relevant for us in Chap.1. Afterwards, we shall turn to Chap. 2 and some of the basic constructions there.

**9 September**: We have now put behind us the fundamental theorem 1.10 and some immediate consequences.We have also discussed Theorem 1.13, where the so-called connecting homomorphism is defined and is used to arrive at the long exact homology sequence associated with a short exact sequence of chain complexes in general.*Exercise*Using "diagram chasing" prove Theorem 1.13 (cf. Exercise 4 page 19)- The next we shall do is to establish the very important "Mayer-Vietoris sequence", see page 22. The remainder of the chapter is some applications of the previous (hard) results which we have discussed. Although Chapter1 is rather short in the text book, many basic but difficult results are established here, and it is the most important chapter for our introduction to algebraic topology.So, be brave and dont be tempted to give up.

**31 August**: We are working on Chap.1, the construction of singular homology of topological spaces and proving some of the basic properties.This is done by associating a chain complex S(X) to any space X. A chain complex has homology groups !! So we work with chain complexes to a large extent. The interesting maps between chain complexes are the*chain maps*, but also more general maps between chain complexes are needed, for example in connection with the notion of "homotopy" between chain complexes.- After som standard "categorical" results comes the first technical result, namely Theorem 1.8, which gives us the homology groups of a convex subset of euclidean n-space.
- Next, homotopy between (continuous ) maps and between topological spaces is defined. The next fundamental result is Theorem 1.10, whose complete proof ends on page 16. We shall discuss the major ideas of the proofs of Theorem 1.6 and 1.8 this week. After this, the remainder of the chapter is devoted to various applications of homology, and also to solve some famous problems.
**Problem**Prove Prop. 1.3 (page 4). This is a good way to test yourself! The proof is rather combinatorial, and hopefully you see that the proof does not involve the topological space X itself. So, where is the "battle" taking place? (at Waterloo? – no).

**16 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.

**7 August**: The first lecture is on Monday 17 August, 12.15-14, room S-24, as scheduled.

## Lecture hours

- Monday 12.15-14.00, Auditorium S 24
- Wednesday 10:15 - 12:00 , Auditorium S 21

## 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 Thursday 10 December and Friday 11 December. Precise information will be sent to each candidate.