Part I: Elementary homotopy theory. This will cover things like Whitehead and Hurewicz theorem, cellular approximation, fibrations, cofibrations,.. We'll roughly follow Hatcher: "Algebraic Topology" Ch 4.1-4.3
Part II: Homotopy groups and spectral sequences. This will set up the machinery of spectral sequences, concentrating on the Serre Spectral Sequence, and use it to compute, amongst other things, some homotopy groups of spheres. We'll roughly follow Hatcher: "Spectral Sequences in Algebraic Topology" Ch 1. (The book is not finished yet, but Ch 1 is fairly complete, and available from Hatcher's homepage.)
(We're not assuming that people have read Ch 3 of Hatcher---we'll pick up material from there as needed, and perhaps the people who have already taken Alg Top II will give lectures Tuesday afternoon on this, for the people who haven't.)There will be weekly homework due Friday and handed back Tuesday. Students are expected to solve all problems in the weekly homework (they are part of the official curriculum), but only selected problems are handed in and graded. You are strongly encouraged to work on homework in groups but everyone has to write their own solution. During the Tuesday problem session old homework will be discussed, and there will be time to work on the homework for the following Friday.
The curriculum for the course is by definition the material covered in class together with the weekly homework. Since we are not following a textbook strictly you are strongly encouraged to take notes, since otherwise it will be impossible for you to solve homework and prepare for in-class comprehension tests. There will be two in-class tests one half-way through the course and one at the end of the course. (Note: Occationally we will use the exercise session to have a more advanced student explain background material, such as things from cohomology. This material will also become curriculum to the extend that it is relied on for the material covered in class.)
The course is passed by 1) participating actively in class & exercise sessions 2) handing in and receiving a passing score on the weekly homework and 3) passing the two in-class tests.
| Dates | Material to cover/covered |
|---|---|
| Week 1 +2 (Nov 13-Nov 24) | We plan to cover Hatcher § 4.1. On Tuesday Nov 21 during the exercise session, and Friday Nov 24 during class there will be student presentations on cohomology. |
| Week 3+4 (Nov 27-Dec 8) | Hatcher § 4.2 (excluding the general Hurewicz theorem). |
| Dec 12 (afternoon) | TEST. Material: Hatcher § 4.1-4.2 and homework 1,2,3 and 4. |
| Week 5+1/2(6) (Dec 12 - Dec 19) | Hatcher § 4.3 |
| Jan 5-23 | Hatcher Spectal sequence notes Ch 1 + stuff on classifying spaces. More precisely: 1-2) Spectral sequences & exact couples. Definition of Serre spectral sequence. Examples and calculations! sketch of proof. 3) Serre classes, spectral sequence comparison, Kudo transgression theorem, spectral sequence in cohomology. Examples and calculations! Rational homotopy of spheres, first appearance of p-torsion in homotopy groups of spheres etc. 4-5) Classifying spaces and fibrations. Cohomology of grassmannians and classifying spaces of compact Lie groups. characteristic classes. group cohomology of finite dihedral and quaternionic groups. 6) Cohomology of Eilenberg-Mac Lane spaces and pi_{n+1}(S^n) and pi_{n+2}(S^n). |
| Jan 26 | TEST (FINAL) |
| Dates | Problems |
|---|---|
| Nov 17 | Hatcher § 4.1: problems 1,2,3,4,5,8,9,10. Hand in problems 4,5 + the following: a) Compute pi_1(S1 v S^2) b) Show that the identity map S^2 -> S^2 generates an infinite cyclic subgroup of pi_2(S^2). (Hint: Can you use what you know about homology?) c) Show that oplus_{\Z} \Z \subseteq pi_2(S^1 \vee S^2). (Hint: provide explicit generators, and prove that they are independent, by using what you know about the homotopy groups of a product.) d) Explicitly determine the action of \pi_1(S^1 \vee S^2) on the subgroup of pi_2 from above. (Actually, the subgroup of pi_2 is equal to the whole pi_2, which we will see later in the course.) e) Write one (1) paragraph what your background is (what year of study?, what courses?), and what you expect to get out of this course in terms of your future studies. |
| Nov 24 | Hatcher § 4.1: problems 11,12,13,14 (also find example showing assumption no n+1-cells necessary),15,17,18,20. Hand in: 15,20. |
| Dec 1 | Hatcher § 4.1: problems 21,22. § 4.2: problems 1,2. Hand in: 21 and 1. |
| Dec 8 | Hatcher § 4.2: problems 6,7,8,9,12,13,18. Hand in: 8,9,13. |
| Dec 15 | Hatcher § 4.2: problems 19,22,31,32,33,36. Hand in: nothing :-) |
| Jan 5 | Hatcher § 4.2: problems 26,28,38. § 4.3 problems: 1,3,4,5,6,7. + correction of the Midterm. Hand in: Midterm, 4.2 38 and 4.3 1. |
| Jan 12 | Hatcher § 4.3: problems 8,9,12,16. Hatcher SSnotes § 1.1: problems 1, 2. Hand in: 4.3: 12 and 1.1: 2. |
| Jan 19 |
Hatcher SSnotes § 1.2: problems 1,2,3.
4) Describe the spectral sequence associated with the exact couple H_*(X;Z) -p-> H_*(X;Z) -> H_*(X;F_p) -> H_{*-1}(X;Z) ... (The Bockstein Spectral Sequence.) 5) Calculate H^*(S^3<3>;Z) and H^*(S^3<3>;F_p) as rings. 6) Calculate H^*(Z/2^n;F_2) as a ring. 7) Calculate H^*(\Omega S^n;Z) as a ring. 8) Suppose S^n -> X -> B is a fibration with pi_1(B) acting trivially on H_n(S^n). Show using the Serre spectral sequence that there is a long exact sequence relating the homology of B and of X. (The Gysin Sequence.) Hand in: Problems 1,2,3. |