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Homotopical Topology
Instructor: Pranav Pandit
Course description: This will be a reading course in algebraic topology, emphasizing the homotopy theoretic point of view. The main topics will be higher homotopy groups, (generalized) homology and cohomology, characteristic classes, and computational tools like spectral sequences. We will aim to cover Chapters 1 - 3 of [FF16]. There will be no lectures; the students will read the material on their own. There will be weekly online office hours, during which students can interact with the instructor.
Prerequisites: Basic point-set topology; it will be helpful if the student has previously taken a course in algebraic topology covering topics like the fundamental group, homology and cohomology.
Textbooks: We will mainly follow [FF16].
Other useful references: Other books that may be useful supplements include [Hat02], [May99], [BT82], [Hov99] and [GJ09].
Evaluation: Homework will be assigned every week, and will usually be due the next week. The final grade will be based on timely completion of homework assignments (60% of the grade) and performance on two exams (40% of the grade). Both exams will carry equal weight.
References:
[BT82] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982.
[FF16] Anatoly Fomenko and Dmitry Fuchs, Homotopical topology, second ed., Graduate Texts in Mathematics, vol. 273, Springer, [Cham], 2016.
[GJ09] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Modern Birkh ̈auser Classics, Birkh ̈auser Verlag, Basel, 2009, Reprint of the 1999 edition.
[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
[Hov99] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.
[May99] J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999. MR 1702278