Math 215B - Complex Analysis, Geometry, Topology - Winter 2009

Homepage http://math.stanford.edu/~galatius/215B09/

Classes TTh 12:50 - 2:05 in 381-T

Instructor Soren Galatius

Office 383-DD

e-mail galatius@stanford.edu

Office hours TTh 9:30 - 11

Course Assistant Robin Koytcheff

Office 381-N

e-mail robink@math.stanford.edu

Office hours MW 1-2:30

Textbooks
The following two textbooks will be used for 215B and also next quarter in 215C.
Topology and Geometry, by Glen E. Bredon. Main textbook for 215B. The course will be about chapters 3, 4 and 5.
Algebraic Topology, by Allen Hatcher. Supplementary textbook. Chapters 1, 2 and 3. This book can be downloaded for free. It is also on reserve in the library.

Prerequisites Apart from formal prerequisites, here are some of the mathematical concepts and notions that I will use over and over. I will assume you are intimately familiar with these. Topological prerequisites: lots of point set topology. You need to know the various constructions on spaces, in particular constructing the quotient topology (B.I.13). You also need to know about compactness and how to use it (B.I.7-8). Algebraic prerequisites: Groups, rings, homomorphisms of such. Equivalence relations, quotient sets (B.appendix). Quotient of a group by a normal subgroup.

Exams

Midterm 1 Take home exam, due 2/10. Get the problem set here. Here's correction to problem 1. Sketch of solutions to problem 1, problem 3, problems 2,5,6 with figures, and problem 7.

Midterm 2 In class exam. Open books, open notes. March 9 , 6PM - 9PM in 380D. Solutions.

Final There will be no final exam.

Exams
Midterm 1	Take home exam, due 2/10. Get the problem set here. Here's correction to problem 1. Sketch of solutions to problem 1, problem 3, problems 2,5,6 with figures, and problem 7.
Midterm 2	In class exam. Open books, open notes. March 9 , 6PM - 9PM in 380D. Solutions.
Final	There will be no final exam.

Lecture plan. See below for a tentative list of topics covered in the class. Will be updated during the quarter.

Homework problems will be posted below, about a week before they are due. You are encouraged to discuss the problems with each other, but you must work on your own when you write them down. Late homework not be accepted, but your lowest homework score will be thrown away before calculating the average. The letter B denotes problems in Bredon's book.

Due date Problems

1/13 Prove that S¹ is homotopy equivalent to R² - {0}. If you need to, first find and read the definition of homotopy equivalence. B.III.2#1,2,3. If you need to, find and read the definition of "topological group". The Klein bottle is defined by a picture in figure I-3 in Bredon. B.III.3#1.

1/20 These exercises.

1/27 These exercises.

2/3 These exercises.

2/10 1st midterm due. Get the problem set here. (problem 1 is wrong, see next week).

2/17 Corrected version of the first problem on the midterm. Homework set.

2/24 These exercises. Here's a solution to the first problem.

3/5 These exercises

Due date	Problems
1/13	Prove that S¹ is homotopy equivalent to R² - {0}. If you need to, first find and read the definition of homotopy equivalence. B.III.2#1,2,3. If you need to, find and read the definition of "topological group". The Klein bottle is defined by a picture in figure I-3 in Bredon. B.III.3#1.
1/20	These exercises.
1/27	These exercises.
2/3	These exercises.
2/10	1st midterm due. Get the problem set here. (problem 1 is wrong, see next week).
2/17	Corrected version of the first problem on the midterm. Homework set.
2/24	These exercises. Here's a solution to the first problem.
3/5	These exercises

Classes.

Date Topic

1/6 Introduction: spaces, maps and homotopies. Fundamental group: group structure and functoriality. Dependence on the base point. (B.III.2)

1/8 Covering spaces: definition, examples, lifting properties. (B.III.3,4). Note: The definition in Bredon is a little more restrictive than what's necessary. We (and most other people) will work with a simpler definition, omitting assumptions about Hausdorff, path connected, and locally path connected. This is also the definition found in Hatcher's book.

1/13 More lifting properties (Theorem 4.1). Action of the fundamental group on the fiber (acts on the right). The action is transitive if X is path connected and free if it's simply connected (B.III.5). This "calculates" the fundamental group. Maps between path connected covering spaces (with basepoint): Exist when inclusion hold between subgroups of fundamental group. Isomorphism between covering spaces (with basepoint): when subgroups of fundamental group are equal. Existence of universal covering space. (B.III.8)

1/15 Deck transformations. Classification of covering spaces. Graphs and free groups. (B.III.6,7,8).

1/20 Graphs and free groups. van Kampen's theorem. Examples. Presentations of groups. Wirtinger presentation of the fundamental group of a knot complement (B.III.9,7, Hatcher exercise 1.2.22).

1/22 Pushout diagram. Attaching cells. CW complexes (B.IV.8). I wrote the definition slightly differently, here's a short note about that. Fundamental groups of CW complexes (Hatcher, page 50ff).

1/27 Fundamental groups of CW complexes (Hatcher, page 50ff). Compact subsets of CW complexes (Bredon, IV.8). Homology: Definition of the singular chain complex. (B.IV.1-5).

1/29 Homology, homological algebra. (B.IV.2-5)

2/3 More homological algebra: split exact sequences, 5-lemma. Axioms for homology. Basic calculations of homology of spheres and disks, from axioms. Degree of a map (assuming existence of a homology theory). (B.IV.6)

2/5 Some classical applications of homology (assuming it exists): Brouwer's fixed point theorem, vector fields on spheres. Cellular homology: Definition of cellular chain complex. (B IV.10)

2/10 The differential in cellular homology. Example calculations using cellular homology (real projective space). Functoriality of cellular homology. B(IV.7,10,14)

2/12 Singular homology: Proof of homotopy invariance. Cross product. (B IV.15-16). I also talked a bit about natural transformations and Yoneda's lemma. A brief introduction to Categories, Functors, and Natural transformations is in Hatcher, p162-165. Here's 1-page introduction to Yoneda's lemma.

2/17 Singular homology: Excision. The subdivision operator. (B.IV.17)

2/19 Proof of excision. Mayer-Vietoris. (B.IV.17-18). Cellular maps (B.IV.11). Euler characteristics (B.IV.13).

2/24 Classical applications. (B.IV.18-20).

2/26 Definition of cohomology. Review of tensor product. Tensor product is right exact, but not exact. Projective modules. Projective resolutions: existence and uniqueness up to chain homotopy equivalence. We'll only need this theory for modules over Z (a.k.a. abelian groups), but I'll discuss the theory in more generality: For modules over an arbitrary commutative ring Lambda.

3/3 Homological algebra continued: Uniqueness and functoriality (up to chain homotopy equivalence) of projective resolutions. Short exact sequence of modules induce long exact sequence of Tor groups. Proof that tensor product is right exact. An example: Lambda = Z[t]/(tⁿ-1). (B.V.6)

3/5 Ext. Injective modules. Universal coefficient theorems. (B.V.6-7).

3/10 Products (B.VI). Cross product and Kunneth theorem. Cup product.

3/12

Date	Topic
1/6	Introduction: spaces, maps and homotopies. Fundamental group: group structure and functoriality. Dependence on the base point. (B.III.2)
1/8	Covering spaces: definition, examples, lifting properties. (B.III.3,4). Note: The definition in Bredon is a little more restrictive than what's necessary. We (and most other people) will work with a simpler definition, omitting assumptions about Hausdorff, path connected, and locally path connected. This is also the definition found in Hatcher's book.
1/13	More lifting properties (Theorem 4.1). Action of the fundamental group on the fiber (acts on the right). The action is transitive if X is path connected and free if it's simply connected (B.III.5). This "calculates" the fundamental group. Maps between path connected covering spaces (with basepoint): Exist when inclusion hold between subgroups of fundamental group. Isomorphism between covering spaces (with basepoint): when subgroups of fundamental group are equal. Existence of universal covering space. (B.III.8)
1/15	Deck transformations. Classification of covering spaces. Graphs and free groups. (B.III.6,7,8).
1/20	Graphs and free groups. van Kampen's theorem. Examples. Presentations of groups. Wirtinger presentation of the fundamental group of a knot complement (B.III.9,7, Hatcher exercise 1.2.22).
1/22	Pushout diagram. Attaching cells. CW complexes (B.IV.8). I wrote the definition slightly differently, here's a short note about that. Fundamental groups of CW complexes (Hatcher, page 50ff).
1/27	Fundamental groups of CW complexes (Hatcher, page 50ff). Compact subsets of CW complexes (Bredon, IV.8). Homology: Definition of the singular chain complex. (B.IV.1-5).
1/29	Homology, homological algebra. (B.IV.2-5)
2/3	More homological algebra: split exact sequences, 5-lemma. Axioms for homology. Basic calculations of homology of spheres and disks, from axioms. Degree of a map (assuming existence of a homology theory). (B.IV.6)
2/5	Some classical applications of homology (assuming it exists): Brouwer's fixed point theorem, vector fields on spheres. Cellular homology: Definition of cellular chain complex. (B IV.10)
2/10	The differential in cellular homology. Example calculations using cellular homology (real projective space). Functoriality of cellular homology. B(IV.7,10,14)
2/12	Singular homology: Proof of homotopy invariance. Cross product. (B IV.15-16). I also talked a bit about natural transformations and Yoneda's lemma. A brief introduction to Categories, Functors, and Natural transformations is in Hatcher, p162-165. Here's 1-page introduction to Yoneda's lemma.
2/17	Singular homology: Excision. The subdivision operator. (B.IV.17)
2/19	Proof of excision. Mayer-Vietoris. (B.IV.17-18). Cellular maps (B.IV.11). Euler characteristics (B.IV.13).
2/24	Classical applications. (B.IV.18-20).
2/26	Definition of cohomology. Review of tensor product. Tensor product is right exact, but not exact. Projective modules. Projective resolutions: existence and uniqueness up to chain homotopy equivalence. We'll only need this theory for modules over Z (a.k.a. abelian groups), but I'll discuss the theory in more generality: For modules over an arbitrary commutative ring Lambda.
3/3	Homological algebra continued: Uniqueness and functoriality (up to chain homotopy equivalence) of projective resolutions. Short exact sequence of modules induce long exact sequence of Tor groups. Proof that tensor product is right exact. An example: Lambda = Z[t]/(tⁿ-1). (B.V.6)
3/5	Ext. Injective modules. Universal coefficient theorems. (B.V.6-7).
3/10	Products (B.VI). Cross product and Kunneth theorem. Cup product.
3/12