Math 215B: Complex Analysis, Geometry, and Topology - Winter 2013

Tuesday, Thursday 12:50-2:05pm in 380-380D

Teaching Staff

Instructor Sheel Ganatra Course Assistant Cary Malkiewich
Office 380-382F Office 380-381M
e-mail ganatra (at) math (dot) stanford (dot) edu e-mail carym (at) math (dot) stanford (dot) edu
Office Hours This week: Tuesday, 2.10-3:40pm, Thursday, 11.15am-12.45pm Office Hours Monday and Wednesday, 12.30-3.30pm

Course Description

Math 215B will initiate the study of algebraic invariants of topological spaces up to homotopy equivalence . Through this quarter we will construct, prove properties about, and study applications of three such invariants:

  1. Fundamental Group. Covering spaces, the fundamental group of the circle, Van Kampen's theorem, the relation between covering spaces and the fundamental group.
  2. Homology. Simplicial, singular, and cellular homology. Equivalence of definitions. Excision and the Mayer-Vietoris sequence. Classical applications (Jordan curve separation, Borsuk-Ulam, Brouwer-fixed point, etc.).
  3. Cohomology. Cup product. The Universal Coefficient Theorem, and the Kunneth formula. Poincaré duality.

As a necessary ingredient, we will develop techniques in homological algebra. A basic class of topological spaces we will apply our discussion to is the class of CW complexes.


Allen Hatcher's Algebraic Topology, available for free download here. Our course will primarily use Chapters 0, 1, 2, and 3.


In addition to formal prerequisites, we will use a number of notions and concepts without much explanation. Topologically: you should be intimately familiar with point-set topology, in particular various constructions on spaces, the product and quotient topologies, continuity, compactness. Algebraically: groups, rings, homomorphisms, equivalence relations, quotient sets, quotients of groups by normal subgroups.

Course Grade

The course grade will be based on the following:

Homework Assignments

Homework will be posted here on an ongoing basis (roughly a week before they are due) and will be due at 5pm on the date listed, in the Course Assistant's mailbox. Late homeworks will not be accepted.

You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions.

Due date Assignment
Jan 18 Homework 1. Solutions.
Jan 25 Homework 2. Solutions.
Feb 1 Homework 3. Solutions.
Feb 8 Homework 4. Solutions.
Feb 29 Homework 5. Solutions.
March 8 Homework 6. Solutions.
N/A Homework 7 (not for credit: suggested exercises).

Lecture Plan

Lecture topics by day will be posted on an ongoing basis below.

Date Lecture topics Book chapters Remarks
Jan 8 Introduction: Spaces, maps, and homotopies. The fundamental group. Dependence on base point. Chapters 0 (first subsection), 1.1
Jan 10 Covering spaces and lifting properties. The fundamental group of the circle. Functoriality. Chapters 1.1, 1.3 (first few pages)
Jan 15 Fundamental groups of spheres. Van Kampen's theorem. Chapters 1.1, 1.2
Jan 17 Fundamental groups of CW complexes. Chapter 0 (for definitions of CW complexes), 1.2
Jan 22 Covering spaces I: Lifting properties, the universal cover, the correspondence theorem. Chapter 1.3
Jan 24 Covering spaces II: The correspondence theorem, Deck transformations, examples. Graphs and sub-groups of free groups. Chapters, 1.3, 1.A
Jan 29 Singular and simplicial homology---first definitions. Functoriality. Chapter 2, beginning.
Jan 31 The fundamental group and first homology group. Relative homology. Exact sequences. Chapter 2.A, 2.1
Feb 5 Short exact sequences and long exact sequences, and applications to Relative homology. Homotopy invariance. Chapter 2.1, Sections on "Homotopy Invariance" and "Exact Sequences and Excision".
Feb 7 The 5 Lemma. Reduced homology. Excision via Barycentric subdivision. The homology of spheres. Chapter 2.1 Additional notes completing the construction of the (Barycentric) subdivision proposition.
Feb 12 The Mayer Vietoris sequence. Homology of good pairs. Singular homology = simplicial homology. Degrees of maps between spheres. Chapter 2.1, 2.2
Feb 14 More on the degree of a map. Classical applications (Brouwer's fixed point theorem, Jordan-Brouwer curve separation). Towards the homology of CW complexes. Chapter 2.2, 2.B
Feb 19 Cellular (CW) homology groups. Immediate applications, including the homology of complex projective spaces. The cellular boundary formula, and applications to real projective space. Chapter 2.2
Feb 21 The homology of real projective space. Homology with coefficients. Co-chain complexes and cohomology. Chapters 2.2, 3
Feb 26 Singular cohomology groups and the Universal Coefficient Theorem. Chapter 3.1
Feb 28 The Universal Coefficient Theorem continued. Cup product, and the cohomology ring. Statement of the cohomology ring of projective spaces, an application to the Borsuk-Ulam theorem. Chapters 3.1, 3.2
Mar 5 Cup product in the relative setting. Calculation of the cup product on projective spaces, via a reduction to Euclidean spaces. Chapter 3.2
Mar 7 The Künneth Theorem. An introduction to manifolds and orientations. Chapters 3.2, 3.3
Mar 12 Manifolds and orientations in homology. The fundamental class. Chapter 3.3
Mar 14 Orientations in homology continued. Poincaré duality and its applications. Chapter 3.3

Midterm and Final Exam

The take-home midterm is here. It was assigned February 11th night and was due in class, February 19th. Solutions are now posted here.

The final exam was held in room 380D on Thursday, March 21st from 7-10PM. Solutions are now posted here.