# 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.

## Notes

Scans of Henry's lecture notes.## 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.

## 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 |