Introduction to Algebraic Topology

The homework of week 12 has been graded and can be found in an envelope in my (Stefan's) mailbox. Make sure to collect it and bring all your graded homework to the exam. can be collected in my (Stefan's) office. I can send you an overview of the homework grades (including the average); send me an e-mail including your name and student number and I'll reply with the overview as soon as possible.

Organization

Teacher: Gabriel Zalamansky, office 225, g.s.zalamansky(at)umail.leidenuniv.nl

Teaching assistant: Stefan van der Lugt, office 242 238, stefanvdlugt+iat(at)gmail.com

Lectures: 15:45 - 17:45 on Tuesday from 15 September to 15 December. There will be no lecture on 3 November.

Grading: there will be a homework assignment every week (12 assignments in total), and an oral exam at the end of the course. The exam counts for 60% towards the final grade, and the homework for 40%. The lowest homework grade will not count; that way it is possible to miss one assignment (because of absence, illness, a busy schedule, ...) and still get a maximal grade.

Exam

The oral exams will be on 25 and 26 January. The full schedule can be found here. The exams are in office number 240.

It is a 30mn oral. We will begin with (sometimes deliberately vague) questions about the lectures and the homeworks. You are expected to have a decent stack of examples (ie the most classical spaces : spheres, projective spaces, torus, Mobius band, Klein bottle, bouquet of circles...).

You should have an idea of which theorems are the most important and what hypothesis are needed.You will not be asked to reproduce the full proofs of the theorems but you are expected to be able to give an outline of them, the ideas that are used etc..

By default, the definitions and conventions in Lee's book stand. You should be aware of when and why this is needed. But at some point (eg for covering spaces and Galois theory) we relaxed them. You should be able to identify exactly when and what it implies.

All the homeworks are considered to be part of the Lectures. You should bring them with you so we can discuss them.

If time permits we will move on to a short exercise.

Literature

• A. Hatcher: Algebraic topology, available at http://www.math.cornell.edu/~hatcher/AT/ATpage.html
• J. M. Lee: Introduction to topological manifolds, Springer GTM 202. Available at this webpage (only works on university network)
• T. Szamuely: Galois groups and fundamental groups, Cambridge University Press
• V. Runde: A Taste of Topology, Springer
• W. Fulton: Algebraic Topology: A First Course, Springer
Sebastian Lucic shared a pdf-file of the lecture notes he has written in LaTeX. You can find it here.

Lectures

 Lecture 1 Reminder on the fundamental group (definitions, functoriality, examples) Ref: Runde chap. 5.1, Lee chap 7. Homotopy equivalence and deformation retractions. Ref: Lee chap 7.4 (especially prop 7.46) Lecture 2 Pushouts in categories. Ref: Lee problem 10.21, Link 1, Link 2 Pushouts of groups, a.k.a amalgamated free products. Ref: Link 1, Lee chap 9-10.1, Link 2 The Van Kampen theorem. Ref: Lee chap 10, Hatcher chap 1.2 Lecture 3 Application of the Van Kampen theorem to wedge sums. Ref: Lee th. 10.7 Reminder on covering spaces. Ref: Szamuely chap 2.1, Runde chap 5.2, Hatcher chap 1.3, Lee chap 11 Lifting criterion. Ref: Hatcher prop 1.33, Lee th. 11.18 Lecture 4 Reminder on path and path homotopy lifting. Ref: Runde 5.2.3-5.2.4 Proof of the general lifting criterion. Ref: Lee th. 11.18, Hatcher prop 1.33 Lecture 5 We treated the chapter "covering homomorphisms" from Lee's book. This is also proposition 1.37 in Hatcher. Lecture 6 Reminder on sets with transitive actions : Lee 11.23 to 11.28, application to isomorphism criterion : Lee ths. 11.37,11.40 and 12.4, Hatcher prop 1.37 Automorphism groups of coverings and their actions : Lee 12.1 to 12.5 The "Main theorem" relating fundamental groups and automorphism groups + consequences: Lee 12.7 to 12.9 ; Hatcher prop 1.39 Existence of the universal covering : Lee 11.41-11.43, Hatcher p.63 to 65 NB : Hatcher calls covering automorphisms Deck transformations. It has the exact same meaning. In th 11.43, Lee considers only locally path connected spaces. This is not quite necessary but sufficient for most purposes. Lecture 7 Reminder and complements on covering space actions : Lee 12.14, Hatcher p.72-73 Galois theory of covering spaces : Lee th. 12.8-12.9, Hatcher th. 1.38 Lecture 8 Galois theory of coverings, II. Application to the Van Kampen theorem. Ref : Fulton 14 & 16 Lecture 9 Singular homology and chain complexes, I Ref : Lee ch. 13 Lecture 10 Singular Homology II : Homotopy invariance (Lee th. 13.8) Lecture 11 Singular Homology III : The Hurewicz homomorphism (Lee th. 13.14) and the Mayer-Vietoris sequence (Lee th.13.16) Lecture 12 Homology of spheres, degree theory for maps of spheres (Lee 13.23 to 13.32) Lecture 13 Homology with coefficients (Hatcher 3.A) and the universal coefficient theorem (Don't worry too much about Tor, you are not expected to know its definition. Just know that if X is a space with torsion-free homology then its homology with coefficients in an abelian group G is indeed its homology with integral coefficients tensored with G.Be aware that it is not the case in general. In particular the top homology of a sphere with coefficients in Z/pZ is Z/pZ.) Application to the Borsuk-Ulam theorem. Link

Homework

Students are encouraged to work on solving the homework problems together, but everybody has to write down the solutions individually. Copying another student's solutions is not acceptable and may result in a lower grade.

Homework has to be handed in on the Tuesday of the next lecture, either in class or in the mailbox of Stefan van der Lugt found in the common room at the second floor of the Snellius building. It is also possible to mail the homework, but please send your homework to Stefan's address found at the beginning of this page. If you send the homework by mail please send a (single) PDF file typeset in LaTeX. Scans will not be accepted. If you decide to hand in handwritten homework then make sure everything is legible; if it takes too long to decypher your handwriting your homework will not be graded. LaTeX is always fine!

Homework that is handed in too late will not be graded.

 Homework Deadline Remarks Homework 1 22 September Exercise 3: in the hint: replace 'positive' by 'non-negative'. Homework 2 29 September You may freely use the result of exercise 3 in exercise 2. Homework 3 6 October Exercise 3: $$G$$ is a topological group and the action of $$G$$ on $$X$$ is continuous. You may assume that $$X$$ is Hausdorff. Homework 4 13 October Exercise 3: $$p$$ is surjective (some authors do not include this in the definition of a covering map). Also: replace 'compact' by 'compact and Hausdorff' everywhere (the French word 'compact' means 'compact and Hausdorff'; the English 'compact' translates to 'quasi-compact' in French). Homework 5 20 October Only 3 exercises this week; please do them all for a maximum grade! In exercise 2 the covering spaces are assumed to be connected. Homework 6 27 October This week there are only two exercises, and you have to do both for a maximum grade. Homework 7 10 November Again, hand in all 3 exercises. Homework 8 17 November Again, hand in all 3 exercises. Homework 9 24 November Again, hand in all 3 exercises. Homework 10 1 December Only 2 exercises this week. Homework 11 8 December Hint for exercise 2: use Mayer-Vietoris! You are allowed to use the fact that the homology of $$S^1$$ is given by $H_n(S^1) = \left\{\begin{array}{ll}\mathbb{Z} & n=0,1\\0 & n\geq 2. \end{array}\right.$ Homework 12 15 December You are allowed to use the fact that the Mayer-Vietoris sequence is natural. See Wikipedia for more details. Homework 13 - Some exercises involving theory from the last lecture. These exercises are not graded but feel free to ask any questions you have.