Mathematics / advanced
Differential topology and transversality
Understand smooth manifolds through regular values, transversality, degree, embeddings, and Morse-theoretic decompositions.
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Inside this module
Module 01 advanced
Differential topology and transversality
Understand smooth manifolds through regular values, transversality, degree, embeddings, and Morse-theoretic decompositions.
- 01
Smooth atlases and compatible structures
Separate a manifold's topology from its chosen smooth coordinates and test whether overlapping descriptions define the same calculus.
28 min - 02
Inverse and implicit function theorems
Turn invertible derivative information into local coordinates, while keeping global injectivity and graph choices separate.
28 min - 03
Submersions, immersions, and embedded submanifolds
Use derivative rank to identify local models and explain why a parametrized immersed image need not be an embedded manifold.
29 min - 04
Sard's theorem and regular values
Distinguish critical points from critical values and use the correct dimension-dependent differentiability threshold.
29 min - 05
Transversality and preimage manifolds
Check normal directions instead of demanding full surjectivity, and include the boundary restriction when forming neat preimages.
30 min - 06
Degree theory and oriented counts
Count regular preimages with signs and track the compactness or properness that makes the count invariant.
30 min - 07
Whitney embedding and smooth approximation
Distinguish guaranteed embedding dimensions, general-position arguments, and approximation of a particular map.
30 min - 08
Vector fields and the Poincaré–Hopf theorem
Measure the local index of an isolated zero and relate the global sum to Euler characteristic under the correct boundary condition.
30 min - 09
Morse functions and handle attachments
Read a nondegenerate Hessian as a local topological transition and state the compact-band hypotheses behind a global handle attachment.
31 min - 10
Cobordisms and boundary orientations
Track incoming and outgoing signs, use collars when gluing smooth boundaries, and connect Stokes' theorem to geometric interpolation.
31 min