This is a course on Homotopy Theory. It is a continuation of the courses Algebraic Topology and Algebraic Topology 1,5.

Content:
The first part (ca 5 weeks) will focus on elementary homotopy theory covering things like fibrations, cofibrations, cellular approximation, Whitehead and Hurewicz theorem. We will roughly follow Hatcher: Algebraic Topology Ch 4.1-4.3.
The second part (ca 4 weeks) will focus on spectral sequences in homotopy theory. We will set up the Serre spectral sequence and use this to calculate homology and homotopy groups of a number of interesting spaces. We will roughly follow Hatcher: Spectral Sequences in Algebraic Topology Ch 1, available for download on his homepage. We will also cover material from Chapter 3 of Hatcher: Algebraic Topology as needed. Students who wish an added experience are encouraged to read this chapter as self-study.

The curriculum is by definition what is covered in class, and participants are strongly
encouraged to take their own notes.

Textbooks:
Hatcher: Algebraic Topology
Hatcher: Spectral Sequences in Algebraic Topology
(Supplementary material: May: A Concise Course in Algebraic Topology, Selick: An
Introduction to Homotopy Theory.)

Time and place:
Lectures: Mondays 10:15-12 in Aud 9, Universitetsparken 5, HCØ and Fridays 10:15-12 in Aud 9, Universitetsparken 5, HCØ (by Jesper Grodal and Nathalie Wahl). (Special schedule on weeks with holiday --- see below.)

Exercise session: Tuesdays 13:15-16 in Aud 8, Universitetsparken 5, HCØ (by Robin Sroka).
Final: Friday June 21nd

Homework:
There will be weekly homework. Homework problems will be uploaded to Absalon around Friday, and should be handed in at the start of Friday's lecture the following week. They will then be discussed in the following exercise class on Tuesday. Late submissions of homework assignments will not be accepted. The lowest scoring homework assignment will be disregarded when computing the grade. Please write in your solutions which people you've discussed the problems with.

Evaluation:
There will be weekly homework and a final "in-class assignment". The grade will be determined by homework (50%) and the final (50%).

The following is a plan for the lecture. Note that this plan is tenative and will periodically be updated during the course!

Date	Material covered /planned covered
Week 1A	No class monday due to due to Easter.
Week 1B	Tuesday class (~2hr) in exercise time slot. Higher homotopy groups, proof that they are abelian. Basepoint dependence, action of pi_1 on higher homotopy. Relative homotopy groups, long exact sequence in relative homotopy. (Briefly: mapping spaces, compact-open topology, adjunction between based loop functor and suspension.)
Week 2A	Cellular approximation theorem. Whitehead theorem.
Week 2B	CW approximation theorem., etc.
Week 3A	CW approximation wrapped up. Excision theorem and consequences: Freudenthal suspension theorem, $π_{i} (X, A) ≅ π_{i} (X / A)$ in a range. Calculation that $π_{n} (S^{n}) ≅ Z$ .
Week 3B	Homotopy groups of a wedge and Whitehead products. Existence and uniqueness of K(G,n) spaces. Hurewicz theorem. For simply connected spaces, homology iso implies weak equivalence (Whitehead).
Week 4A	[3B roundup (Hurewich grp homo+whitehead)+] Homotopy lifting and extension property: Serre fibrations, Hurewicz fibrations. Long exact sequence of homotopy groups of a Serre fibration. Fiber bundles, lots of examples: Möbius strip, Klein bottle, projective spaces, Hopf maps, covering maps.
Week 4B	Tuesday class, 1hr 13:15-14, since Friday off. (Exercise class will be 14:15-17.) More examples of fiber bundles. Proof that fiber bundles are Serre fibrations.
Week 5A	Properties of fibrations. Turning maps into fibrations. Homotopy fibers.
Week 5B	The Puppe and dual Puppe sequences (Eckmann-Hilton duality). Spectra and Brown representability. Eilenberg-Mac Lane spaces represent ordinary cohomology. Cohomology operations (Yoneda Lemma). Principal fibrations. Postnikov towers. Obstruction theory.
Week 6A	[20 min roundup of obstruction theory+] Generalities on spectral sequences: definitions, convergence, filtrations. The spectral sequence for the homology of a filtered space. The Serre spectral sequence. Example calculation: Hopf bundle. Loop space of a sphere.
Week 6B	Spectral sequences via exact couple. Construction of the Serre spectral sequence.
Week 7A	Serre spectral sequence (end of the construction). Cohomological Serre spectral sequence, with multiplicativity. Example: complex projective spaces. Further generalizations. Comparison of spectral sequences.
Week 7B	Transgressions in Serre spectral sequence. Serre classes and Hurewicz mod a Serre class.
Week 8A	Tuesday class (1hr) since Monday off. Proof of Hurewicz theorem mod Serre class. The first p-torsion of pi_*(S^3).
Week 8B	Rational homotopy groups of spheres. Cohomology of Eilenberg-Mac Lane spaces mod 2 and the Steenrod algebra.
Week 9A	Cohomology of Eilenberg-Mac Lane spaces mod 2 and the Steenrod algebra (end). Other spectral sequences...
Week 9B	Final "in-class assignment".