# MA3403 Algebraic Topology - Fall 2019

**NOTE CHANGE OF TIME AND PLACE!**

Schedule | Room | ||
---|---|---|---|

Lectures: | Wednesday | | |

14.15-16.00 | S734 (Sentralbygg 2) |
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Thursday | | ||

14.15-16.00 | S734 (Sentralbygg 2) |
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Exam: | Oral exams | 25-26 November | |

Lecturer | |||

Rune Haugseng | |||

Office: | 1250 Sentralbygg 2 | ||

Email: | rune [dot] haugseng [at] ntnu [dot] no |

## What this course is about

Studying geometric objects by associating algebraic invariants to them is a powerful idea which has influenced many areas of mathematics. For example, determining the existence of a map between spaces (often a difficult task) may be translated into deciding whether an algebraic equation has a solution (sometimes a piece of cake). One of the birthplaces 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 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 definitions of topological spaces and continuous maps. You will 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 already taken the courses MA3002 General Topology and/or TMA4190 Introduction to Topology in the spring. If you haven'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

The first lectures will be 21 and 22 August.

Lecture | Date | Topic | Notes | References |
---|---|---|---|---|

1.1 | 21/08 | Review of topological spaces, homotopies | 1.1-1.5 | [Q], lect. 2 |

1.2 | 22/08 | Definition of singular homology | 2.1-2.5 | [Q], lect. 3 |

2.1 | 28/08 | H_{0}, disjoint unions, orientations, functoriality | 2.6-2.9 | [Q], lect. 4 |

2.2 | 29/08 | Δ-complexes, simplicial homology | 3.1-3.3 | [H], p. 102-107 |

3.1 | 04/09 | Relative homology, exact sequences, singular and simplicial homology | 4.1-4.3 | [Q], lect. 5, [H], p.138-130 |

3.2 | 05/09 | Eilenberg-Steenrod axioms/excision, exercises week 1 | 4.4-4.5 | [Q], lect. 6 |

4.1 | 11/09 | Homology of spheres, applications | 4.6-4.8 | [Q], lect. 6,7 |

4.2 | 12/09 | Mayer-Vietoris, exercises 2 | 4.9 | [G], lect. I.6 |

5.1 | 18/09 | Cell complexes | 5.1-5.3 | [Q], lect. 13,14 |

5.2 | 19/09 | Cellular homology, exercise 3 | 5.4-5.5 | [Q], lect. 14,15 |

6.1 | 25/09 | Degrees, homology of real projective space | 5.6-5.7 | [Q], lect. 7,15 |

6.2 | 26/09 | Exterior product, homotopy invariance, exercises 4 | 6.1-6.3 | [Q], lect. 10,11, [G], lect. I.5 |

7.1 | 02/10 | Excision, locality, barycentric subdivision | 6.4-6.5 | [Q], lect. 12, [G], lect. I.7 |

7.2 | 03/10 | Tensor products, exercises 5 | 7.1 | [Q], lect. 17 |

8.1 | 09/10 | Homology with coefficients | 7.2-7.3 | [Q], lect. 16,17 |

8.2 | 10/10 | Tor, Universal Coefficient Theorem | 7.4-7.5 | [Q], lect. 17 |

9.1 | 16/10 | Hom, singular cohomology | 8.1-8.2 | [Q], lect. 18 |

9.2 | 17/10 | Ext, Universal Coefficient Theorem for cohomology, exercises 6+7 | 8.3-8.4 | [Q], lect. 19 |

10.1 | 23/10 | Cellular cohomology, tensor product of chain complexes | 8.5, 9.1 | [G], lect. II.4, II.5 |

10.2 | 24/10 | Eilenberg-Zilber and Künneth theorems, exercises 8 | 9.2 | [G], lect. II.5 |

11.1 | 30/10 | Variants of Eilenberg-Zilber and Künneth, cup and cross products | 9.3-10.1 | [Q], lect. 20 |

11.2 | 31/10 | Ring structures on cochains and cohomology, exercises 9 | 10.2-10.3 | [G], lect. II.6 |

12.1 | 06/11 | Cohomology rings of projective spaces | 10.4 | [H], p. 212-214 |

12.2 | 07/11 | Manifolds, exercises 10 | 11.1 | [H], p. 230-239 |

13.1 | 13/11 | Orientations, fundamental classes, cap products and Poincaré duality | 11.2-3 | [H], p. 233-241 |

13.2 | 14/11 | Poincaré duality and cup products, exercises 11 | 11.4 | [H], p. 249-250 |

14.1 | 20/11 | Cohomology with compact support, proof of Poincaré duality (*) | 11.5-6 | [H], p. 242-248 |

14.2 | 21/11 | Review |

(*) Not examinable

## Course material

## References

We will not follow any particular textbook. Last year's lecture notes will give you a good idea of the content of the course:

- [Q] G. Quick,
*2018 Notes*

Other good lecture notes:

- [Mi] H. Miller, Lectures on Algebraic Topology I
- [G] M. Groth, Algebraic Topology I and II

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.

## Reference Group

- Lukas Anzeletti, lukas.anzeletti@gmx.net
- Ole Andreas Berre, oaberre@stud.ntnu.no
- Håkon Johnstuen, hsjohnst@stud.ntnu.no