# Math 571: Topology II

## Colorado State University, Spring 2018

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