Spring 2016
MTH602: Topics in topology (Introduction to Homotopy and Homology)
Instructor: Anandam Banerjee
Email: anandam[at]iisermohali.ac.in
Office: Health Center - Room 34
Office hours: Mon, Tues 2 to 3pm.
Textbook: Introduction to Homotopy Theory (Universitext) by Martin Arkowitz, Springer; 2011 edition.
Course Outline
Week 1: Introduction to some basic notions in homotopy theory, categories and functors, CW complexes. Fundamental group. Notes.
Week 2: H-spaces, co-H-spaces. Notes.
Week 3: Loop space, suspension space, Higher homotopy groups. Notes
Week 4: Singular homology. Computation of H_0. Reduced homology.
Week 5: Homotopy invariance of singular homology.
Week 6: Relative homology and the long exact sequence. Statement of homology excision theorem and proof of Meyer-Vietoris theorem. Some computations.
Week 7: Degree of self-maps of spheres. Higher homotopy groups, Hurewicz Theorem
Week 8: Computation of the fundamental group of the real line, circle, real projective plane, Mobius strip etc. Notes.
Week 9: Cofibrations, fibrations.
Week 10: Homotopy cofiber, homotopy fiber, relative homotopy groups. Notes.
Week 11: Exact sequence of relative homotopy groups, exact and coexact sequences of spaces. Notes.
Week 12: Midsemester Break.
Week 13: HELP Lemma, Whitehead's Theorem.
Week 14: Freudenthal Suspension Theorem, homotopy excision (statement of homotopy excision theorem).
Week 15: Cellular Approximation Theorem, Hurewicz Theorem revisited.
Week 16: Homology Excision Theorem,.
Presentation topics