Lecture summaries (2025)

1/28   The syllabus.  Introduction to the course.  Intuition behind manifolds: some manifolds, some manifolds-with-boundary, and some objects that are not manifolds.  Open sets and smooth functions.  Reference: Textbook 1.1, supplementary notes on topology of Euclidean space on the course webpage

1/30 Smooth functions on arbitrary subsets.  Diffeomorphisms.  The definition of a manifold.  Parametrizations.  Products of manifolds are manifolds.  Reference: Textbook 1.1

2/4  The tangent space.  Derivatives of maps.  The chain rule.  Reference: Textbook 1.2

2/6 More about the chain rule.  Invariance of dimension under diffeomorphism.  The inverse function theorem.  Writing a function in co-ordinates. Immersions and submersions.  Reference: Textbook, 1.3

2/11 Proof of the local immersion theorem.  Immersions at a point are immersions in a neighborhood of that point.  When is the image of an immersion a submanifold?  Some non-examples.  Reference: Textbook, 1.3

2/13 Closed and compact sets.  Proper maps.  Proof that the image of a proper injective immersion is a manifold. An injective immersion whose image is not a manifold; more examples and non-examples.    Reference: Textbook, 1.3  

2/18 Regular values.  The preimage of a regular value is a manifold.  Textbook, 1.4

2/20 Transversality.  Examples.  Reference: Textbook, 1.5

2/25 Ideas behind the proofs of transversality.  Homotopy and stability.  Statement of the stability theorem.  Reference: Textbook, 1.6  

2/27 Proof of the stability theorem.  Dense sets and measure zero sets.  Statement of Sard's theorem.  Reference: Textbook, 1.7

3/4 Measure zero subsets of manifolds.  Examples and non-examples.   Applications of Sard's Theorem: non-existence of surjective maps from a lower dimensional manifold to a higher dimensional one; embedding k-dimensional manifolds into R^{2k+1} (statement).  Reference: Textbook, 1.7, 1.8

END OF MATERIAL FOR MIDTERM 1