Homotopy Theory
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Description
Wednesday 11.00-13.00
This is an introduction to homotopy theory. It can be thought of as "Algebraic Topology II".
We will learn methods to compute homotopy groups of spaces, in particular homotopy groups of spheres. A key topic is that of spectral sequences appear, which are an extremely useful tool.
The course will follow the notes of Oscar Randal-Williams' Part III course. These are available at:
dpmms.cam.ac.uk/~or257/teaching/notes/HomotopyTheory.pdf
1. Homotopy groups, relative homotopy groups, long exact sequence of a pair.
2. CW complexes, Whitehead's theorem, cellular approximation.
3. Hurewicz theorem, Eilenberg-MacLane spaces, and cohomology.
4. Fibrations, LES of a fibration, function spaces, Moore-Postnikov towers.
5. Spectral sequences and the Serre spectral sequence.
6. The Serre spectral sequence in cohomology, and examples.
7. Mod C theory, the transgression, Freudenthal's suspension theorem.
8. Steenrod squares, Vector fields on spheres.
9. Wu and Stiefel-Whitney classes, constructing the Steenrod squares.
10. Examples of computations and student talks.
ASSESSMENT
Five written homework assignments, due in at the end of weeks 3, 5, 7, 8, and 11. (short, 2-3 questions).
That is, they will be due on:
24th October 2025
7th November 2025
21st November 2025
28th November 2025
19th December 2025
One 10-20 minute talk (depending on numbers) on a topic in homotopy theory, during the last lecture.
Best 5 out of 6 to count, 20% each.
Each lecturer will set and grade one homework.
REFERENCES (for Homotopy Theory (Semester 1) & Stable Homotopy Theory (Semester 2)
References
Frank Adams, Stable homotopy theory and generalised homology, Chicago Lectures in Mathematics, 1974.
Jim F. Davis and Paul Kirk. Lecture notes in algebraic topology.
Grad. Stud. Math., 35, American Mathematical Society, Providence, RI, 2001. xvi+367 pp.
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii+544 pp.
Allen Hatcher, Spectral sequences. Unfinished book project, available at pi.math.cornell.edu/~hatcher/AT/SSpage.html.
Peter May, A concise course in algebraic topology, Chicago Lectures in Math.
University of Chicago Press, Chicago, IL, 1999. x+243 pp.
Robert E. Mosher and Martin C. Tangora. Cohomology operations and applications in homotopy theory. Harper & Row, Publishers, New York-London, 1968. x+214 pp.
Yuli Rudyak, On Thom spectra, orientability, and cobordism. Springer Monogr. Math.
Springer-Verlag, Berlin, 1998. xii+587 pp.
Stefan Schwede, Symmetric spectra. Unpublished lecture notes, available at https://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf.
Edwin H. Spanier, Algebraic topology.
McGraw-Hill Book Co., New York-Toronto-London, 1966. xiv+528 pp.
Robert M. Switzer, Algebraic topology -- homology and homotopy. Die Grundlehren der mathematischen Wissenschaften, Band 212 Springer-Verlag, New York-Heidelberg, 1975. xii+526 pp.
George Whitehead, Elements of homotopy theory.
Grad. Texts in Math., 61 Springer-Verlag, New York-Berlin, 1978. xxi+744 pp.