Introduction to Algebraic Topology

Organization

Teacher: Gabriel Zalamansky, office 238, mail address: click here

Teaching assistant: Stefan van der Lugt, office 227a, mail address: click here

Lectures: 11:15 - 13:00 on Monday from 6 February to 22 May. There will be no lecture on 13 March, 3 April, 17 April.

Grading: there will be three homework assignments, and an (oral/written) exam at the end of the course.

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

Lectures

Lecture 1	Reminder on the fundamental group : Definition, basic properties. No proofs were given. Please read Lee's book, chapter 7 or better yet do them on your own as exercises. Deformation retracts and homotopy equivalence : Lee prop 7.46. Please read the proof p. 206-207-208.
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	Covering maps. Ref: Lee, Chapter 11, covered up to theorem 11.8
Lecture 4	Proof of path lifting and the general lifting criterion. Ref: Lee th. 11.18, Hatcher prop 1.33
Lecture 5	Monodromy action (transitivity,stabilisers ...) Lee 11.22, 11.29 to 11.35.
Lecture 6	Covering homomorphisms : Lee 11.36 to 11.40 Covering Automophisms : Lee 12.1 to 12.6
Lecture 7	covering automorohism structure theorem (Lee 12.7) + the universal covering (Hatcher p.62-64)
Lecture 8	existence of the universal covering (Hatcher 62-64)
Lecture 9	Classification theorem (Lee 12.18) + covering space action (Lee 12.14-12.15)
Lecture 10	Finished lecture 9 material; started G-sets and coverings (Fulton 16d)
Lecture 11	Equivalence of categories between coverings and \(\pi_1\)-sets (Fulton 16d) Galois coverings and group homomorphisms (Fulton 16e) Proof of Van Kampen theorem (Fulton 14c) Further readings (not required for exam): Grothendieck SGA1 chap. V.4-V.5-V.6; Douady & Douady Algebre et theories Galoisiennes chap. 4 & 5.

Practice exercises

On the webpage of the previous edition of this course you'll find lots of exercises to practice with. The following exercises are also nice to look at.

Practice sheet	Remarks/Errata
Practice sheet 1

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 Monday 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.

The exercises will be made available at least a week before the deadline.

Homework	Deadline	Remarks/Errata
Homework 1	27 February	In exercise 3a, the definition of the points \(p,q,r\) is wrong. It should say that \(q\in M\setminus i_1(\mathbb{B}')\) and \(r\in N\setminus i_2(\mathbb{B}')\) are points that are identified by the relation \(\mathcal{R}\) and map to the same point \(p\) under the quotient map.
Homework 2	10 April	In exercise 1(ii) you should assume that \(G'\) is connected!
Homework 3	22 May 29 May	A Galois covering is the same as a normal (or regular) covering. In 2.3, depending on the construction of the functors in 2.1 and 2.2, the diagram need not commute. The two functors obtained by composing the functors in the diagram need not be equal. Instead, prove that these functions are isomorphic, that is: show that there exists a natural isomorphism from one functor to the other.