Math 104
Winter 2015
Topics 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 5 
Chapter 1 
Topological manifolds and their properties. Examples. 
Wednesday January 7 
Chapter 1 
Smooth structures, atlases, Examples of smooth manifolds,
manifolds with boundary 
Friday January 9 
Chapter 2 
Smooth functions and smooth maps, diffeomorphisms 
Monday January 12 
Chapter 2 
Partitions of Unity 
Wednesday January 14 
Chapter 2 
Partitions of Unity Continuation 
Friday January 16 
Chapter 3 
Tangent vectors and derivations 
Monday January 19 MLK day classes moved to xhour 


Tuesday January 20 xhour instead of the class on January 19 
Chapter 3 
Pushforwards and
computation in coordinates 
Wednesday January 21 
Chapter 3 
Tangent space to a manifold with boundary, tangent vectors
to curves, alternative definitions of tangent vectors 
Friday January 23 
Chapter 4 
Tangent bundle, Vector fields on manifolds 
Monday January 26 
Chapter 4 
Pushforwards of vector
fields, Lie algebra of vector fields 
Wednesday January 28 
Chapter 5 
Vector bundles and examples, local and global sections of
vector bundles 
Friday January 30 
Chapter 5 
Bundle maps and constructions with bundles 
Monday February 2 
Chapter 6 
Covectors and
tangent convectors on manifolds, cotangent bundle 
Tuesday February 3 xhour instead of the class on February 6 
Chapter 6 
Differential of a function, pullbacks 
Wednesday February 4 
Chapter 7 
Maps of constant rank, Inverse function theorem 
Friday February 6 Carnival Holiday classes moved to xperiods 
Chapter 7 
Proof of inverse function theorem 
Monday February 9 Middle of the term presentation and discussion Monday
February 9Friday February 13 
Chapter 7 
Rank Theorem, Implicit Function Theorem 
Wednesday February 11 
Chapter 7 
Immersions, submersions and constant rank maps between
manifolds 
Friday February 13 
Chapter 8 
Embedded Submanifolds 
Monday February 16 
Chapter 8 
Immersed submanifolds 
Wednesday February 18 
Chapter 11 
Algebra of tensors and tensor fields on manifolds 
Friday February 20 
Chapter 12 
Algebra of alternating tensors, differential forms 
Monday February 23 
Chapter 12 

Wednesday February 25 
Chapter 12 
Exterior Derivative, cohomology 
Friday February 27 
Chapter 13 
Orientation, orientation of the boundary of a manifold 
Monday March 2 
Chapter 14 
Fubini Theorem without
proof, Integration of differential forms on manifolds 
Wednesday March 4 
Chapter 14 
Stokes Theorem 
Friday March 6 
Chapter 14 
Stokes Theorem continuation 
Monday March 9 
Chapter 14 
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. 
End of the term presentation and discussion Thursday March
12 – Saturday March 14 

