# Math 570: Topology I

## Colorado State University, Fall 2017

Email: henry dot adams at colostate dot edu
Office: Weber 125
Office Hours: Tuesday 12-1pm and Wednesday 10-11am in Weber 125, or by appointment.

Lectures: MWF 9:00-9:50am in Weber 202
Textbook: Introduction to Topological Manifolds by John Lee (Second Edition).
An electronic copy of this book is freely available to CSU students here.

Overview: We plan to spend the first three weeks of the class on point-set topology (topologies, continuity, quotient spaces, connectedness, compactness). The goal for the rest of the course will be to introduce algebraic topology, while returning to topics from point-set topology as they arise. During the first third of the course, we hope to introduce the very basics of homotopy equivalences, the fundamental group, higher homotopy groups, and category theory. The second third of the course will focus on homology, with simplicial homology before singular homology. In the final third of the course we return to the fundamental group (Seifert-Van Kampen Theorem), covering maps, and other selected topics (some skipped along the way).

Though I hope to cover many of the topics in our textbook, I will cover some of them (such as portions of point-set topology, and the classification of compact surfaces) in a very casual way: I will provide intuition for the main theorems instead of giving complete proofs. Other topics, such as homology, will be emphasized and presented in detail.

Syllabus: Here is a course syllabus.

## Homework

Homework 1 (LaTeX Source) is due Friday, August 25.
Homework 2 (LaTeX source) is due Friday, September 1.
Homework 3 (LaTeX source) is due Friday, September 8.
Homework 4 (LaTeX source) is due Friday, September 15.
Homework 5 (LaTeX source) is due Friday, September 22.
Homework 6 (LaTeX source) is due Friday, October 6.
Homework 7 (LaTeX source) is due Friday, October 13.
Homework 8 (LaTeX source) is due Friday, October 20.
Homework 9 (LaTeX source) is due Friday, October 27.
Homework 10 (LaTeX source) is due Friday, November 10.
Homework 11 (LaTeX source) is due Friday, November 17.
Homework 12 (LaTeX source) is due Friday, December 1.

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 Practice Midterm 1.
Here is Practice Midterm 2.
Here is a Practice Final.

## Notes

Scans of Henry's lecture notes.

## Preparation Materials

One way to prepare for this course is by using the materials available at my class webpage Math 472, Fall 2016. Class notes, homework problems, and practice exams are available there. If you email me, I can also send you homework solutions and practice exam solutions. Not all of this material is assumed for Math 570, but we will be covering much of it at a fairly quick pace.

## Schedule

 Date Topic Remark Aug 21 Course overview & Chp 1 Aug 23 Chp 2: Topologies, Convergence and Continuity Aug 25 Chp 2: Hausdorff Spaces, Bases and Countability Homework 1 due Aug 28 Class cancelled :( Aug 30 Chp 2: Manifolds Sep 1 Chp 3: Subspaces and Product Spaces Homework 2 due Sep 4 No class Sep 6 Chp 3: Disjoint Union Spaces, Chp 7: Categorical Products Last day to drop or change grading option Sep 8 Chp 7: Categorical Coproducts, Chp 3: Quotient Spaces Homework 3 due Sep 11 Chp 4: Connectedness Sep 13 Chp 4: Compactness Sep 15 Chp 4: Compactness Homework 4 due Sep 18 Chp 7: Homotopy Sep 20 Chp 7: The Fundamental Group Sep 22 Chp 7: The Fundamental Group Homework 5 due Sep 25 Chp 7: The Fundamental Group Sep 27 Review Sep 29 Midterm #1 **starting at 8:30am** Oct 2 Chp 7: Homomorphisms Induced by Continuous Maps Oct 4 Chp 7: Homomorphisms Induced by Continuous Maps Oct 6 Chp 7: Homomorphisms Induced by Continuous Maps Homework 6 due Oct 9 Chp 7: Categories and Functors Oct 11 Chp 7: Homotopy Equivalence Oct 13 Chp 7: Higher Homotopy Groups Homework 7 due Oct 16 Chp 5: Cell Complexes and CW Complexes End of course withdrawal period Oct 18 Chp 5: Simplicial Complexes Oct 20 Class cancelled; read Theorem 7.40 & Lemma 7.45 Homework 8 due Oct 23 Chp 13: Simplicial Homology Oct 25 Chp 13: Simplicial Homology Oct 27 Chp 13: Simplicial Homology Homework 9 due Oct 30 Chp 13: Singular Homology Nov 1 Review Nov 3 Midterm #2 **starting at 8:00am** Nov 6 Chp 13: Singular Homology Nov 8 Midterm 2 and Homework 10 comments Nov 10 Chp 13: Singular Homology Homework 10 due Nov 13 Chp 13: Functoriality of simplicial homology Nov 15 Chp 13: Difference between homotopy groups and homology Nov 17 Chp 13: The Mayer-Vietoris Theorem Homework 11 due Fall Recess, Nov 20-24 Nov 27 Chp 13: Homotopy invariance Nov 29 Chp 13: Homotopy invariance Dec 1 Chp 13: Mayer-Vietoris example Homework 12 due Dec 4 Chp 13: Degree theory for spheres Dec 6 Chp 13: Degree theory for spheres Dec 8 Review Final Exam, Monday December 11 7:30-9:30am in Weber 202