# MA3403 Algebraic Topology - Fall 2018

Welcome to Algebraic Topology!

• The Lecture Notes are now avaialable as one file: Lecture Notes
• The exam results will be sent out via individual emails by Tuesday evening.
Schedule Room
Lectures: Tuesday 12.15-14.00 S21 Sentralbygg 2
Friday 12.15-14.00 734 Sentralbygg 2
Exam: oral exams Exam schedule 634 Sentralbygg 2
Lecturer
Gereon Quick
Office: 1246 Sentralbygg 2
Email: gereon [dot] quick [at] ntnu [dot] no

## What this course is about

To study geometric objects by associating algebraic invariants to them is a powerful idea which influenced many areas of mathematics. For example, deciding about the existence of a map between spaces (often a difficult task) may be translated into deciding whether an algebraic equation has a solution (often a piece of cake). One of the birth places of this idea is Algebraic Topology. The goal of the course is to introduce the most important examples of such invariants such as singular homology and cohomology and homotopy groups. Along the way we are going to calculate many examples and see applications.

## What you need to know before this course

You should have seen the definition of topological spaces and continuous maps. You have seen examples of topological spaces as subspaces of $\mathbb{R}^n$, but you should also know that there are lots of other topological spaces. Ideally you have taken the courses MA3002 General Topology and/or TMA4190 Introduction to Topology in the spring. If you havn't, no problem! You could refresh your knowledge by looking at the books by Jänich [J] or Munkres [Mu2]. Some familiarity with algebraic notions and concepts will also be assumed. For example, you should know what a group is and preferably also what rings and modules are. In addition, it is recommended to take the course MA3204 Homological Algebra this semester.

If you have any questions, just contact me!

## Lecture Plan

Week Topic Exercises Notes
34.1 Introduction Lecture 1
34.2 Cell complexes and homotopy Set 1 Lecture 2
35.1 Singular chains Lecture 3
35.2 Singular homology, functoriality and $H_0$ Lecture 4
36.1 Relative chains and long exact sequences Lecture 5
36.2 Eilenberg-Steenrod axioms and homology of the sphere Set 2 Lecture 6
37.1 Generators of $H_n(S^n)$ and first applications Set 3 Lecture 7
37.2 No Lecture
38.1 Calculating degrees Lecture 8
38.2 Local vs global degree and open questions Set 4 Lecture 9
39.1 No Lecture
39.2 No Lecture Set 5
40.1 Homotopies of chain complexes Lecture 10
40.2 Homotopy invariance of singular homology Lecture 11
41.1 Locality and the Mayer-Vietoris sequence Set 6 Lecture 12
41.2 Cell complexes Set 7 Lecture 13
42.1 Homology of cell complexes Lecture 14
42.2 Computations of cell homologies and Euler characteristic Set 8 Lecture 15
43.1 Moore spaces and Homology with coefficients Lecture 16
43.2 Tensor, Tor and the UCT Set 9 Lecture 17
44.1 Singular cohomology Lecture 18
44.2 Hom, Ext and the UCT for cohomology Lecture 19
45.1 Cup products in cohomology Lecture 20
45.2 Some applications of cup products Set 10 Lecture 21
46.1 Poincaré duality and intersection form Lecture 22
46.2 Classification of surfaces Lecture 23
47.1 More on Poincaré duality Lecture 24
47.2 Q&A

## Reference Group

• 1st reference group meeting: Friday, September 21, 2018.
• 2nd reference group meeting: Tuesday, November 20, 2018.
• 3rd reference group meeting: Thursday, December 13, 2018.

## Course material

Some interesting books:

• [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
• [Mu] J.R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.
• [F] W. Fulton, Algebraic Topology - A First Course, Springer-Verlag, 1995.
• [V] J.W. Vick, Homology Theory - An Introduction to Algebraic Topology, Springer, 1994.
• [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.
• [MS] J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, 1974.

Some books on general topology:

• [J] K. Jänich, Topology, Springer, 1984.
• [Mu2] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.