Math 215A, Algebraic Topology, Spring 2017

 

Instructor: Peter Teichner

Teaching Assistants:

Bertram Arnold, 864 Evans, bertram.arnold@gmail.com

Daniel Brügmann, 864 Evans, danielbruegmann@berkeley.edu

Eugene Rabinovich, 869 Evans, e.rabin@berkeley.edu


Lectures: Tu/Th 11:10 - 12:30 in 740 Evans

Office Hours: 703 Evans, Tu. 2:30 - 3:30 and by appointment

Prep Talks: Th. 2 - 4 in 7316 Evans

Course Control Number: 32410

Prerequisites: Basics of point-set topology.

Homework: Weekly homework and talks, homework will be submitted in groups of 3 students, each writing up one problem.


Syllabus:

We’ll first introduce interesting spaces, like manifolds (including knot and link complements) and later CW-complexes. Then we’ll discuss the first tools to study qualitatives features of these spaces, namely fundamental group, higher homotopy and homology groups. We’ll end with proving some important consequences, e.g. invariance of dimension, the generalized Jordan curve theorem and the Lefschetz fixed point theorem.


On Thursdays we'll review the Tuesday material and then get sidetracked by interesting topics like classification of surfaces, knot theory, cobordism groups, fibre bundles, de Rham cohomology, simplicial sets, (higher) categories, limits, colimits etc.


References:

Glen Bredon, Topology and Geometry

Allen Hatcher, Algebraic Topology

Peter’s Notes


Tuesday talks:

Jan. 24: Survey of pointset topology (I, 5,7,10,12): Nick, Har’el, Franklin (Peter)

Problem set #1

Jan. 31: Smooth manifolds (II, 1-5): Chris, Will, Alekos (Bertram)

Problem set #2, Solution 3

Feb. 7: Regular values (II, 6-10): Brandon, Tim, Randolf (Daniel)

Problem set #3, Solution 1, Solution 2, Solution 3

Feb. 14: Fundamental group (III, 1-4): Albert, Nikolay, Sung (Eugene)

Problem set #4, Solution 1, Solution 2, Solution 3

Feb. 21: Covering spaces (III, 5-8): Ninad, Indie, Enya (Bertram)

Problem set #5, Solution 1, Solution 2, Solution 3

Feb. 28: Van Kampen’s theorem (III, 9-10): Vagisha, Laurel, Marissa (Daniel)

Problem set #6, Solution 3

Mar. 7: First steps towards homology (IV, 1-5): Eduardo, Alok, Nancy (Eugene)

Problem set #7

Mar. 14: Axioms for homology (IV, 6-7): Mahrud, Gary, Julio (Bertram)

Problem set #8, Solution 1Solution 2

Mar. 21: CW-complexes (IV, 8-11): Edison, Zirui, Yidong (Daniel)

Problem set #9, Solution 1Solution 2

Apr. 4: Sample computations (IV, 12-14): Joseph, Ben (Eugene)

Problem set #10, Solution 1, Solution 2

Apr. 11: Singular homology (IV, 15-18): Jerry, Sameera (Bertram)

Problem set #11

Apr. 18: Jordan curve thm. (IV, 19-20): Onyebuchi, Peter (Daniel)

Problem set #12

Apr. 25: Fixpoint theorems (IV, 21-23): Madeline, Madeleine, Yi (Eugene)

Problem set #13, Solution 1


Organizational group meetings around the week of your talks:

Week -1: Make a minute by minute list of topics, subdivide them into three 20 minute talks, and discuss this outline with your TA on Mo-Wed. On Th. from 2 - 4 in 736, meet your TA for prep talks, and decide on homework problems with him.

Week +1: Meet with your TA for grading.


In case you need more material, here is our homework from 2006:

HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11