## Math 8062: Algebraic Topology

### Spring Semester 2022

 Meets: Tue/Thu 3:00-4:20 PM in Wachman Hall, room 527 Instructor: David Futer Office: 1026 Wachman Hall Office Hours: Tuesday 12:30-2:00, Wednesday 10:30-12:00, or by appointment Email: dfuter at temple.edu Phone: (215) 204-7854

Course outline: The fundamental question that we seek to answer in this course is: how can we tell whether two manifolds are homeomorphic? Over the course of the 20th century, mathematicians have developed a number of algebraic tools to help answer this question. The tools that we will study are the fundamental group (including covering spaces and van Kampen's theorem), homology theory, and some cohomology theory.

Although this course is mostly independent of Math 8061, we will periodically see connections. Many of our examples will be manifolds rather than more general topological spaces. We will see how an orientation on a manifold can be seen algebraically as well as smoothly. Similarly, we will see how the degree of a smooth map can be seen both algebraically and smoothly. Toward the end of the semester, all the threads will converge as we prove the de Rham theorem, relating de Rham cohomology to singular (co)homology. We will also look at Poincare duality from a smooth point of view.

Textbook: Agebraic Topology, by Allen Hatcher. We will cover most of Chapters 1 and 2, plus part of Chapter 3. On a few occasions, we will also draw on John Lee's Introduction to Smooth Manifolds in order to emphasize connections between the smooth theory from Math 8061 and algebraic topology.

Prerequisites: Concepts of analysis (Math 5041-42) and a solid grounding in undergraduate abstract algebra. Math 8061 is not a strict prerequisite, but it really helps to have seen it.

The main components of the final grade are homework and a take-home final exam.

Component Date Worth
Homework Thursdays 60%
Final Exam April 29 40%

Homework policy: Homework assignments will be posted on the course webpage, and will typically be due on Thursdays. At the end of the semester, I will drop your lowest homework score. I encourage you to start early and to discuss the problems with other students. By all means come by my office hours if you have trouble with a problem. The only real caveat to group work is that you must write up your own solutions, in your own words.

Final Exam: The take-home final will be handed out around the last class day, and will be due on April 29.

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