MTH602: Topics in topology (Introduction to Homotopy Theory)
Instructor: Anandam Banerjee
Email: anandam[at]iisermohali.ac.in
Office: Academic Block - I 1F12A
Office hours: Mon, Tues 2 to 3pm, Fri 3 to 4pm.
Week 1: Introduction to some basic notions in homotopy theory
Week 2: Definition of fundamental group, and proof that it is a functor from the category of pointed topological spaces to groups. Notes. Assignment 1 (due 20 Jan).
Week 3: Computation of the fundamental group of the real line, circle, real projective plane, Mobius strip etc. and some consequences. Notes. Assignment 2 (due 28 Jan).
Week 4: H-spaces, co-H-spaces. No assignment. Notes
Week 5: Loop space, suspension space, higher homotopy groups. Notes Assignment 3 (due 13 Feb).
Week 6: Higher homotopy groups, CW complexes.
Week 7: Cofibrations, fibrations. Assignment 4 (due 24 Feb)
Week 8: Midsemester Break. Assignment 5 (due 16 Mar).
Week 9: Homotopy cofiber, homotopy fiber, relative homotopy groups. Notes.
Week 10: Exact sequence of relative homotopy groups, exact and coexact sequences of spaces. Notes. Assignment 6 (due 25 Mar).
Week 11: HELP Lemma, Whitehead's Theorem.
Week 12: Freudenthal Suspension Theorem, homotopy excision (statement of homotopy excision theorem). Assignment 7 (due 8th April).
Week 13: Definition of simplicial and singular homology. Computation of H_0. Reduced homology.
Week 14: Homotopy invariance of singular homology. Assignment 8 (due 20th April). Presentations of
Week 15: Relative homology and the long exact sequence. Statement of homology excision theorem and proof of Meyer-Vietoris theorem. Some computations. Assignment 9 (due 30th April).
Week 16: Presentations of