# 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.

See the study handbook for more information.

## 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.