SYSTEMATIC MATHEMATICS

Differential Topology, Vector Bundles & Morse Theory

When does a system of smooth equations cut out a manifold, how can a tiny perturbation make two geometric objects meet cleanly, why can signed preimages determine a global degree, and how can the critical points of one function reveal the topology of an entire space? This course answers those questions in sixty-three visible bilingual chapters across eleven content-sized units of lengths 6, 7, 6, 7, 5, 6, 5, 6, 7, 4, and 4. It develops smooth rank and regular-value methods, embedded and immersed submanifolds, normal and tubular neighborhoods, partitions of unity, smooth approximation and isotopy, Sard’s theorem and transversality, oriented and mod-two intersections, degree, vector bundles and sections, characteristic classes and obstruction evidence, vector-field indices and Poincaré–Hopf, Morse functions, handles, inequalities, cobordism, Pontryagin–Thom boundaries, surgery boundaries, and a complete embedded-torus dossier. Every manifold, dimension, chart, tangent map, compactness condition, orientation, coefficient system, perturbation class, local/global distinction, and theorem boundary is named. The course proves and calculates the finite-dimensional smooth results it uses; it does not claim to replace full courses in geometric measure theory, singularity theory, symplectic topology, gauge theory, stable homotopy theory, high-dimensional surgery, or research-level h-cobordism.

Before this course: Completed Differential Geometry & Manifolds, Topology, Algebraic Topology, Multivariable Calculus, Vector Calculus, Linear Algebra, and Real Analysis. Students should know smooth manifolds and maps, tangent and cotangent spaces, differential forms and Stokes, compactness and quotient spaces, fundamental groups, homology and cohomology, determinants and orientations, multivariable derivatives, inverse and implicit function theorems, and proof by local coordinates followed by invariant comparison. No prior transversality, vector-bundle characteristic classes, Morse theory, cobordism, or surgery theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Smoothness is a chart-independent statement about one map

Objective: Why is the single formula θ↦2θ on (−π,π) not by itself a complete global proof?

A manifold has many valid coordinate systems. A formula may look different after changing coordinates, so differential topology cannot define smoothness by favoring one chart. Instead, every compatible source and target chart must turn the map into an ordinary smooth Euclidean function wherever their domains meet. Before calculating, name the source manifold, target manifold, dimensions, charts, map, point, tangent spaces, and every regularity or compactness hypothesis. The exact conclusion to understand is: It is enough to verify smoothness in one smooth atlas on M and one smooth atlas on N; every other compatible chart gives a smooth coordinate expression automatically.

The new object means: Let M and N be smooth manifolds. A map f:M→N is smooth when for every chart (U,φ) of M and every chart (V,ψ) of N, the coordinate expression ψ∘f∘φ^{-1} is a smooth map between open subsets of Euclidean spaces on φ(U∩f^{-1}(V)). The symbol ∘ means composition: apply the rightmost map first. Read every derivative as a linear map between named tangent spaces and every geometric picture as evidence for a statement that still needs a coordinate-independent proof.

Rebuild the worked problem “Prove that f:S¹→S¹ given in complex notation by f(z)=z² is smooth, using the angle charts away from one cut point and checking what happens on chart overlaps.” from the definitions. Verify dimensions and hypotheses before invoking a theorem, and keep this failure boundary visible: A continuous map need not be smooth, and a formula smooth in one arbitrary coordinate parameter is not enough unless that parameter belongs to the declared smooth atlas. At a chart boundary, one must move to an overlapping chart rather than differentiate outside a chart domain.

It is enough to verify smoothness in one smooth atlas on M and one smooth atlas on N; every other compatible chart gives a smooth coordinate expression automatically.

Take a verified coordinate expression g=ψ∘f∘φ^{-1}. In new charts φ̃ and ψ̃, insert identity maps to write ψ̃∘f∘φ̃^{-1}=(ψ̃∘ψ^{-1})∘g∘(φ∘φ̃^{-1}).

The two outside maps are transition maps between compatible smooth charts, so both are smooth on their named overlap domains.

A composition of smooth Euclidean maps is smooth. Therefore the new coordinate expression is smooth, proving the definition does not depend on the chosen atlas charts.

Prove that f:S¹→S¹ given in complex notation by f(z)=z² is smooth, using the angle charts away from one cut point and checking what happens on chart overlaps.

  1. On an angle chart write z=e^{iθ}. Wherever the target angle branch is fixed, f(e^{iθ})=e^{i2θ}, so the coordinate expression is θ↦2θ followed, if needed, by adding an integer multiple of 2π to remain in the chosen target interval.
  2. The map θ↦2θ+2πm is an ordinary smooth real function on every connected overlap piece, because m is constant on that piece.
  3. Angle-chart transition maps also add integer multiples of 2π. Composing with them preserves smoothness, so all chart expressions are smooth and f is a smooth circle map.

Result: Every local angle expression is affine θ↦2θ+2πm on its overlap component; hence z↦z² is smooth on all of S¹.