Introduction to algebraic topology
MA 572- Spring 2021, Prof. Manuel Rivera, Purdue University
This course will be an introduction to algebraic topology. The main topics to be covered will be:
1) singular homology
2) computational methods based on cellular homology of CW complexes
3) the Eilenberg-Zilber and Alexander-Whitney maps, Universal Coefficient Theorem, and Kunneth Theorem
4) cohomology groups and cup product
5) Poincaré duality of manifolds
Along the way will introduce some basic category theory, simplicial techniques, and develop the necessary homological algebra. These tools were born in the context of algebraic topology but are widely used throughout modern mathematics.
Instructor: Prof. Manuel Rivera (manuelr at purdue dot edu)
Textbooks: The lectures will not strictly follow any textbook. They will be roughly based on these notes written by Haynes Miller:
However, it is highly recommended to complement the lectures by reading the following textbooks:
1) Algebraic Topology, by A. Hatcher
2) Elements of Algebraic Topology, by J. Munkres,
3) Topology and Geometry, G. Bredon
4) Peter May, A Concise Course in Algebraic Topology,
Pre-requisites: We will assume familiarity with basic point set topology, covering spaces, and the fundamental group, as discussed in MA571. We will also assume some basic results in algebra such as the classification theorem of finitely generated abelian groups.
Grading: Homework 75 %, one exam 25%
Grader/TA: The TA for this course is Daniel Tolosa: dtolosav at purdue dot edu
Office Hours: Tuesdays after class at 10:30am or by appointment
Special accommodations: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: email@example.com or by phone: 765-494-1247.
In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php
Due Firday January 29: 1) Exercises 1.8, 2.2., 2.3, 3.7,3.8 from HM's notes 2) Prove that the first singular homology group of a path-connected space is isomorphic to the abelianization of its fundamental group at any point.
Due Friday February 5: Exercises 4.10, 5.15, and 6.3 from HM's notes.
Due Friday Februrary 26: Exercises 7.3, 8.8, 9.8, 9.10 from HM's notes as well as the following:
Problem 1: Let A —> B —> C —> D —> E be an exact sequence of abelian groups. Prove that C=0 if and only if the map A—> B is surjective and D—>E is injective. Deduce that for all pairs of spaces (A,X), the inclusion A —> X induces isomorphisms on all homology groups if and only if H_n(X,A)=0 for all n. Problem 2: Homology does not determine the fundamental group: Give an example of two spaces that have isomorphic homology groups but non-isomorphic fundamental groups.