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.

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.

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: Two criteria for homotopy equivalence Homework 1 due

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

Jan 29 Section 1.2: Van Kampen's theorem, Example 1.23
Jan 31 Section 1.2: Applications to CW complexes Last day to drop or change grading option
Feb 2 Section 1.3: Covering spaces Homework 3 due

Feb 5 Section 1.3: Lifting properties
Feb 7 Section 1.3: The classification of covering spaces
Feb 9 Section 1.3: The classification of covering spaces Homework 4 due

Feb 12 Section 1.3: Deck transformations and group actions
Feb 14 Section 1.3: Deck transformations and group actions
Feb 16 Section 1.3: Cayley complexes, Section 1.A Homework 5 due

Feb 19 Section 2.1: Simplicial homology of delta complexes
Feb 21 Section 2.1: Singular homology
Feb 23 Section 2.1: Exact sequences and excision Homework 6 due

Feb 26 Section 2.1: Relative homology
Feb 28 Section 2.1: Short exact sequences of chain complexes
Mar 2 Section 2.1: Equivalence of simplicial and singular homology

Mar 5
Mar 7 Midterm
Mar 9

Spring Break, Mar 12-16
Mar 19 Section 2.2: Cellular homology End of course withdrawal period
Mar 21 Section 2.2: Cellular homology
Mar 23 Section 2.2: Cellular homology Homework 7 due

Mar 26 Section 2.2: Euler characteristic, split exact sequences
Mar 28 Section 2.2: Mayer-Vietoris
Mar 30 Section 2.2: Mayer-Vietoris Homework 8 due

Apr 2 Section 2.2: Homology with coefficients
Apr 4 Section 2.2: Homology with coefficients
Apr 6 Section 2.3: The formal viewpoint Homework 9 due

Apr 9 Section 3.1: Cohomology groups
Apr 11 Section 3.1: Cohomology groups
Apr 13 Section 3.1: Cohomology groups Homework 10 due

Apr 16 Section 3.2: Cup products
Apr 18 Section 3.2: Cup products
Apr 20 Section 3.2: Kunneth formula Homework 11 due

Apr 23 Section 3.3: Poincaré duality
Apr 25 Section 3.3: Poincaré duality
Apr 27 Section 3.3: Other dualities Homework 12 due

Apr 30
May 2
May 4 Final exam