MAT 531 Topology/Geometry II

Spring 2005

(Last Updated: May 6, 2005)

__Room:__ Mathematics P131 __Time:__ TuTh 9:50-11:10 am
__Instructor:__ Vladlen
Timorin

e-mail: mailto:timorin@math.sunysb.edu

Office: 4-102 Phone: 2-8265

Office hours: Wed 2-4, Th 5-6

__Grader:__ Yakov Savelyev

e-mail: mailto:yasha@math.sunysb.edu

__Textbook:__ Michael Spivak, *A Comprehensive introduction to
differential geometry*

Homeworks will be assigned approximately weekly. They are due the following
week.

There will be a midterm and a final exam.

Grading policy: 30%
homeworks, 20% midterm, 50% final exam.

__Syllabus __(from the Graduate Student Handbook):

**Differentiable manifolds and maps**- Inverse and implicit function theorems
- Submanifolds, immersions and submersions

**The tangent bundle**- Vector bundles, transition functions
- Reconstruction of a vector bundle from transition functions
- Equivalence classes of curves and derivations; tangent vectors
- The tangent bundle of a manifold as a vector bundle, examples
- Vector fields, differential equations and flows
- Lie derivatives and bracket

**Differential forms**- Exterior differential, closed and exact forms
- Distributions, foliations and Frobenius integrability theorem
- Poincare Lemma

**Integration**- Stokes' Theorem
- Integration and volume on manifolds
- De Rham cohomology
- Chain and cochain complexes
- Homotopy theorem
- The degree of a map
- The Mayer-Vietoris theorem

**Homeworks:** **1**
[ps] [pdf]
due on Tuesday, February 8 **2**
[ps] [pdf]
due on Thursday, February 17 **3**
[ps] [pdf]
due on Thursday, March 3 **4**
[ps] [pdf]
due on Thursday, March 17 **5**
[ps] [pdf]
due on Thursday, April 14 **6**
[ps] [pdf]
due on Tuesday, April 26 **7**
[ps] [pdf]
due on Tuesday, May 3