# MA3403 Algebraic Topology I - Fall 2012

## Messages

**23.11**A handwritten Brief Survey of the course contents is also available now from the literature list.

GOOD LUCK TO THE STUDENTS ON DECEMBER 4

**22.11**Registered students have now received an e-mail message about the exam.**22.11**A handwritten note about covering spaces is now available online from the literature list.**19.11**On Wednesday and Thursday, the last lectures,we shall still discuss covering spaces, and we shall also discuss van Kampen Theorem (4.26),its proof and a few applications.

A brief survey note on basic concepts and central results of this course will soon appear under Supplementary texts.

**15.11**The last lectures are Wednesday 21.11 and Thursday 22.11.For the final curriculum, see the Syllabus below.**07.11**We are progressing through Chapter 4 on covering spaces and the fundamental group.Hatcher's discussion of this (see section 1.3 of his book) is quite readable, and has many exercises.So you may also benefit from reading this text.**22.10**This week we shall finish Chapter 2, and also take a brief look at Section 3 and introduce cohomology, change of coefficient group, and look at the axiomatic approach to homology. We also start up with Chapter 4, about covering spaces.

**10.10**Two more supplementary text books are listed, they are more elementary than Hatcher's book.**03.10**We are now working with Chapter 2: Attaching spaces with maps, which leads to so-called CW-complexes. A very large and useful category of spaces.In this chapter the relative homology groups of a pair (X,A), are also introduced, and the long exact sequence involving these groups, H(X),H(A),and H(X,A), is also introduced.**19.09**We continue to have Wednesday 16-18 for the first lecture of the week. Observe that Hatcher's book is available as a free electronic version, and can well be used as a supplementary book although it is rather comprehensive.

**19.09**We are still working with Chapter 1, around page 22 and the rest of the chapter has various applications which we shall study. But remember, we still have not proved the basic fact that homotopic maps between spaces induce the the same homology map.This will be completed as the last issue of Chaper 1.**26.08**The lecture on Thursday Aug.30 is cancelled. The students are advised to study themselves the beginning pages of Chap.1, say the first 10-20 pages in the textbook.**26.08**The next lecture is on Tuesday August 28 (as originally announced). But from week 36 we propose the following: Wednesday 16.15-18.00, Room S23, 2.floor in SB II.

**21.08**Because of overlapping with the time schedule of MA3201 (rings and modules),we plan to change the time (and room) for the lecture on Tuesdays. We are tentatively proposing the time 10.15-12, but nothing is decided yet. So, until further information the time is as announced.**21.08**Today we start up with Chapter 1: Singular homology theory, after a brief introduction to the topics of this course.**10.08**The first lecture will take place on**Tuesday August 21 at 12:15**in**Auditorium F2**.

## Lecture hours

- Wednesday 16.15-18.00, room S23 at the 2.floor in SB II
- Thursday 10:15 - 12:00 in 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

- Tuesday , December 4. The candidates will be informed by letter about where and when to meet.