# MA3403 Algebraic Topology I - Fall 2013

## Messages

**27.11**The exam dates are fixed. Three students will have the exam on Monday 9 Dec.,and the other ones on Tuesday or Wednesday 17-18 Dec. Letters of information will be sent to the students, telling you where to meet and when.**14.11**The two last lectures will be next week, 19 and 21 November (as before).Then we shall finish the covering space theory, and also discuss a little two other famous results at the end of chapter 4, namely the Hurewicz homomorphism, and Van Kampens theorem about the fundamental group, which is an analogue of Mayer-Vietoris in homology.**13.11**The**exam dates**will be Tuesday and Wednesday, 17-18 December. More later.**5.11**Please fill out the following form for when you can or cannot attend the exam. Deadline for response:**11.11****22.10**We have now gone through Chapter 3 as much as planned. On Thursday 24 Oct. we start with Chapter 4, on the theory of covering spaces. We will do various things differently, in particular, we shall focus more on group theory and how the covevering space theory can be nicely expressed in terms of the concept of transformation groups.**16.10**Tomorrow Thursday 17 Oct. we start with Chapter 3, called the Eilenberg-Steenrod Axioms. You will see why! But we shall not go into details with the constructions of the tensor product, the functor Hom, and the derived functors Tor and Ext. We shall assume a little familiarity with this and only discuss the basic or general constructions. Much of the material belongs to "homological algebra"**10.10**Tomorrow we start from page 48 (Vick's book), with Prop. 2.12, and will try to go through several propositions and theorems, without detailed proofs. Then we shall try to finish Chapter 2 next week. After that we shall spend a week or so on some of the basic things in Chapter3, before we shall spend the rest of the time we have on Chapter 4 (covering spaces).**1.10**We are in Chap.2, about how to build spaces using attaching maps, about CW-complexes (a special type of spaces built up by attaching cells),and also the construction of the long exact homology sequence for a pair (X,A) of spaces, where A is a subspace of X.**16.9**In the text book we are now on page 24, and will soon finish with chapter 1.Here is an*exercise:*Use the Mayer -Vietoris sequence to calculate the homology of the Klein bottle. Regard the space as the union of two Moebius bands glued together by a homeomorphism between their boundary circles.(First of all, what would be the spaces U,V and their intersection, when the union of the open sets U and V is the Klein bottle? Consider their homotopy type to determine what are their homology groups)

**11.9**On Thursday 12.9 we shall first of all focus on the Mayer-Vietoris sequence (check!) and some applications of this, such as calculating the homology of all spheres.**29.8**We are working with Chap.1 of the text book of Vick. So far we have have gone through basic definitions, and our first aim is to determine the homology of a convex set in n-space (Theorem 1.8), and to show that two homotopic maps f,g : X→Y induce the same homomorhism f* = g* in homology (Theorem 1.10). The first 16 pages of the book is devoted to this (heavy reading?). So far it has been difficult to give you exercises to test yourself on, but quite soon we have enough to start up. In the meantime, try very hard to learn and practice on the basic concepts and understand what the lemmas, propositions, theorems are saying. Check the simplest cases you can imagine, just to get some intuition.**15.8**We start up on Tuesday 20 August, 12.15-14, room F3, as announced.

## Lecture hours

- Tuesday 12.15-14.00, Auditorium F3
- Thursday 10:15 - 12:00 , Auditorium F4

## 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

**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 in December 2013. Time will be determined later.