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.
HomeworkHomework 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.
ExamsThe 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.
NotesScans of Henry's lecture notes.
|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|
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|
|May 4||Final exam|