Math 124
Winter 2013
Current Problems in Topology
Lecture Plan
This lecture plan is tentative and will be updated irregularly. The homework page will be updated on the regular basis
Lectures 
Sections in Text 
Brief Description 
Monday January 7 
Chapter 1 
Topological manifolds and their properties. Examples. 
Wednesday January 9 
Chapter 1 
Smooth structures, atlases, Examples of smooth manifolds, manifolds with boundary 
Friday January 11 
Chapter 2 
Smooth functions and smooth maps, diffeomorphisms 
Monday January 14 
Chapter 2 
Partitions of Unity 
Wednesday January 16 
Chapter 2 
Partitions of Unity Continuation 
Friday January 18 
Chapter 3 
Tangent vectors and derivations 
Monday January 21 MLK day, no class 


Tuesday January 22 xhour instead of the class on January 21 
Chapter 3 
Pushforwards and computation in coordinates 
Wednesday January 23 
Chapter 3 
Tangent space to a manifold with boundary, tangent vectors to curves, alternative definitions of tangent vectors 
Friday January 25 
Chapter 3  Chapter 4 
Tangent bundle, Maps of constant rank, Inverse function theorem 
Monday January 28 
Chapter 4 
Proof of inverse function theorem 
Tuesday January 29 xhour possibly instead of one of the future lectures 
Chapter 4 
Rank Theorem, Implicit Function Theorem 
Wednesday January 30 
Chapter 4 
Immersions, submersions and constant rank maps between manifolds 
Friday February 1 
Chapter 5 
Embedded Submanifolds 
Monday February 4 Middle of the term presentation and discussion Monday February 4Friday February 8 
Chapter 5 
Immersed submanifolds 
Wednesday February 6 
Chapter 8 
Tangent bundle, Vector fields on manifolds 
Friday February 8 Winter Carnival No class 


Monday February 11 
Chapter 8 
Pushforwards of vector fields, Lie algebra of vector fields 
Tuesday February 12 xhour instead of the class on Friday February 8 
Chapter 10 
Vector bundles and examples, local and global sections of vector bundles 
Wednesday February 13 
Chapter 10 
Bundle maps and constructions with bundles 
Friday February 15 
Chapter 11 
Covectors and tangent convectors on manifolds, cotangent bundle 
Monday February 13 
Chapter 11 
Differential of a function, pullbacks 
Wednesday February 15 
Chapter 12 
Algebra of tensors and tensor fields on manifolds 
Monday February 18 
Chapter 14 
Algebra of alternating tensors, differential forms 
Wednesday February 20 
Chapter 14 

Friday February 22 
Chapter 14 
Exterior Derivative, cohomology 
Monday February 25 
Chapter 15 
Orientation, orientation of the boundary of a manifold 
Wednesday February 27 
Chapter 16 
Fubini Theorem without proof, Integration of differential forms on manifolds 
Friday March 1 
Chapter 16 
Stokes Theorem 
Monday March 4 
Chapter 16 
Stokes Theorem continuation 
Wednesday March 6 
Chapter 16 
Vector calculus theorems and their relation to the stokes Theorem. Bordism groups and the pairing between cohomology and bordism groups given by the Stokes Theorem. 
Friday March 8 
Wrap up 
Wrap up 
End of the term presentation and discussion Sunday March 10 – Wednesday March 13 

