First-year graduateGeometry & TopologyLesson available

Differential topology

Smooth manifolds up to diffeomorphism: regular values and Sard's theorem, transversality, the degree of a map (mod 2 and oriented), the Brouwer fixed-point theorem, vector fields and the Poincaré–Hopf theorem, the hairy ball theorem and a glimpse of Morse theory.

math.geom.differential-topology

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Builds on

Smooth manifolds

Required prerequisites, all the way down: 42

You will learn to

State Sard's theorem and use regular values to define the degree of a smooth map between compact manifolds.
Prove the Brouwer fixed-point theorem and the hairy ball theorem using degree.
State the Poincaré–Hopf theorem and compute indices of zeros of vector fields.

Definition

A point y is a regular value of a smooth map f : M → N if the derivative of f is surjective at every point of f⁻¹(y).
Two submanifolds are transverse if at each intersection point their tangent spaces together span the tangent space of the ambient manifold.
The degree of a smooth map f : M → N between closed connected oriented manifolds of the same dimension is the signed count of preimages of a regular value.

Theorem

Sard: the critical values of a smooth map have measure zero.
The degree is independent of the regular value and is a homotopy invariant.
Hairy ball: every continuous tangent vector field on S²ⁿ vanishes somewhere.
Poincaré–Hopf: the sum of the indices of the isolated zeros of a vector field on a closed manifold equals its Euler characteristic.

Common misconceptions

Thinking 'almost every value is regular' means every value is regular; critical values exist, but they form a set of measure zero.
Believing homeomorphic manifolds are always diffeomorphic; Milnor's exotic 7-spheres are homeomorphic but not diffeomorphic to the standard S⁷.

Leads to

Low-dimensional topology: 3- and 4-manifolds

Optional discoveries

Low-dimensional topology: 3- and 4-manifolds In dimension 4 the smooth and topological worlds part company: ℝ⁴ admits uncountably many pairwise non-diffeomorphic smooth structures, and no other ℝⁿ admits more than one.
Dynamical systems and chaos Morse theory and the Poincaré–Hopf theorem show how the topology of a manifold forces critical points of functions and zeros of flows.
← Dynamical systems and chaos

Learning pathways

Geometry and topology