MTH 410: Algebraic Topology

Announcements: 1. Solutions to the final exam is uploaded.

Lecture notes will be posted at the end of each week and HW will be posted generally two or three times a week. Download links are to be found at the bottom of this page.

Course Outline:

• • Homotopy, retract, deformation retract, contractible spaces and homotopy type, fundamental group and its properties, The fundamental group of circle, van Kampen theorem (statement without proof), applications of van Kampen theorem.

  • Simplicial complexes, polyhedra and triangulations, barycentric subdivision and simplicial approximation theorem.

  • Orientation of simplicial complexes, simplicial chain complex and homology, properties of integral homology groups, induced homomorphisms. degree of a map from n-sphere to itself and its applications

  • Invariance of simplicial homology groups. Lefschetz fixed point theorem, Borsuk-Ulam theorem.

  • Definition and examples of covering spaces. path lifting and homotopy lifting property.

  • Covering homomorphisms, deck transformations, classification of Coverings, existence of universal covering (statement without proof).

  • Graphs, coverings of graphs and their fundamental groups.

Basically our goal is to learn the material covered in the first three chapters of Hatcher's Algebraic Topology.

Pre-requisites: Students must have done a course on point set topology to take this course. In particular one must know basic concepts like

connectedness, compactness, separation axioms, metric spaces and so on. Also, basic group theory will be assumed.

Textbooks: Algebraic Topology- Allen Hatcher.

References: (1) Shape of Spaces- J. Weeks, (2) Basic Topology- Armstrong, (3) Algebraic Topology- An Introduction- W. Massey, (4) An Introduction to Algebraic Topology- J. Rotman.

Homework: Two homework sets will be assigned every week. It will be posted in this web page. No homework will be collected or graded.

Exams and Quizzes: Quizzes will be held in the class during tutorial, generally twice a month. However, though quizzes will be collected

but they will not be graded. I will return the quizzes after checking them the following Monday in class. There will be two midterms and

one final exam. Date and time will be announced in due course of time.

Grades: Presentation 10%, Midterms 25% each, Final 40%.

Links to lecture notes and homework: See below. However, note that links to lecture notes will be deleted after one week.