Henry Adams

Math 571: Topology II

                        

Colorado State University, Spring 2018

Instructor: Henry Adams
Email: henry dot adams at colostate dot edu
Office: Weber 125
Office Hours: Tuesdays at 11:00am, Wednesdays at 2:00pm, or by appointment.

Lectures: MWF 10:00-10:50am in Engineering E206
Textbook: Algebraic Topology by Allen Hatcher.
An electronic copy of this book is freely available at https://www.math.cornell.edu/~hatcher/AT/ATpage.html, and paperback copies are also moderately priced.

Overview: This course will be a continuation of algebraic topology, as introduced in Math 570. We will return to the fundamental group in order to discuss Van Kampen's Theorem, covering spaces, and deck transformations and group actions. We will return to homology in order to discuss exact sequences and excision, the equivalence of simplicial and singular homology, cellular homology, Mayer-Vietoris sequences, homology with coefficients, and axioms for homology. Finally, we will introduce cohomology groups, including the cohomology ring and Poincaré duality.

Syllabus: Here is the course syllabus.

Homework

Homework 1 (LaTeX Source) is due Friday, January 19.
Homework 2 (LaTeX source) is due Friday, January 26.
Homework 3 (LaTeX source) is due Friday, February 2.
Homework 4 (LaTeX source) is due Friday, February 9.
Homework 5 (LaTeX source) is due Friday, February 16.
Homework 6 (LaTeX source) is due Friday, February 23.
Homework 7 (LaTeX source) is due Friday, March 23.
Homework 8 (LaTeX source) is due Friday, March 30.
Homework 9 (LaTeX source) is due Friday, April 13.
Homework 10 (LaTeX source) is due Friday, April 20.

We will have weekly homework assignments. All homework is due in class at the beginning of class. Your homework should be legible and stapled.

Exams

The exams will be in-class. You will only be able to use your brain and a pen or pencil - no notes, books, or electronic devices.

Here is a Practice Midterm.
Here is a Practice Final.

Notes

Scans of Henry's lecture notes.

Schedule

Date Topic Remark

Jan 17 Chapter 0: Cell complexes and complex projective space
Jan 19 Chapter 0: Deformation retractions and mapping cylinders Homework 1 due

Jan 22 Section 1.1: Proof of the fundamental group of the circle
Jan 24 Section 1.2: Free products of groups, Van Kampen's theorem
Jan 26 Section 1.2: Proof of Van Kampen's theorem Homework 2 due

Jan 29 Section 1.2: Applications to CW complexes
Jan 31 Section 1.3: Covering spaces and lifting properties Last day to drop or change grading option
Feb 2 Section 1.3: The classification of covering spaces Homework 3 due

Feb 5 Section 1.3: The classification of covering spaces
Feb 7 Section 1.3: The classification of covering spaces
Feb 9 Section 1.3: Deck transformations and group actions Homework 4 due

Feb 12 Section 1.3: Cayley graphs, Introduction to geometric group theory
Feb 14 Section 2.1: Delta complexes
Feb 16 Section 2.1: Simplicial homology of delta complexes Homework 5 due

Feb 19 Section 2.1: Singular homology
Feb 21 Section 2.1: Chain homotopies
Feb 23 Section 2.1: Homotopy invariance of singular homology Homework 6 due

Feb 26 Section 2.1: Exact sequences and excision Dinner at Henry's!
Feb 28 Section 2.1: Relative homology
Mar 2 Section 2.1: The excision theorem

Mar 5 Section 2.1: Equivalence of simplicial and singular homology
Mar 7 Midterm Standard time is 9:00-10:50am in E 206
Mar 9 Class cancelled

Spring Break, Mar 12-16
Mar 19 Section 2.1: Equivalence of simplicial and singular homology End of course withdrawal period
Mar 21 Section 2.2: Degree theory
Mar 23 Section 2.2: Cellular homology Homework 7 due

Mar 26 Section 2.2: Cellular homology
Mar 28 Section 2.2: Euler characteristic
Mar 30 Section 2.2: Mayer-Vietoris Homework 8 due

Apr 2 Section 2.2: Mayer-Vietoris, Homology with coefficients
Apr 4 Section 2.2: Homology with coefficients
Apr 6 Section 3.1: Cohomology groups No homework due

Apr 9 Section 3.1: Cohomology groups
Apr 11 Section 3.1: Cohomology groups
Apr 13 Section 3.1: Cohomology groups Homework 9 due

Apr 16 Section 3.2: Cup products
Apr 18 Section 3.2: Cup products
Apr 20 An introduction to de Rham cohomology Homework 10 due

Apr 23 Section 3.3: Poincaré duality
Apr 25 Section 3.3: Poincaré duality
Apr 27 Section 3.3: Poincaré duality

Apr 30 Class cancelled
May 2 Review Class photo!
May 4 Final exam - no covering spaces Standard time is 9:00-10:50am in Weber 201