Differential topology

Differential topology studies smooth manifolds and smooth maps, focusing on properties preserved by smooth deformation and equivalence.

Smooth manifold: A topological space locally modeled on Euclidean space, with smoothly compatible coordinate charts. Differential topology studies the global properties of these spaces using their smooth structure.
Sard's theorem: A theorem stating that the critical values of a sufficiently smooth map have measure zero. It ensures regular values are plentiful enough to apply the regular value theorem.
Hairy ball theorem: A theorem stating that every continuous tangent vector field on an even-dimensional sphere has a zero. It links the existence of smooth vector fields to a sphere's topological structure.
Henri Poincaré: A French mathematician whose work helped establish topology and the study of manifolds. His foundational questions about manifolds helped shape the subject's topological ambitions.
Algebraic topology: The study of topological spaces using algebraic invariants such as homology and homotopy groups. It often forgets smooth structure, while differential topology uses smooth maps and derivatives.
Smooth map: A map between smooth manifolds whose coordinate expressions have derivatives of every order. Smooth maps are the transformations whose topological behavior the field investigates.
Morse theory: A method that studies a manifold through the critical points of a smooth real-valued function. It converts the derivative data of functions into information about manifold topology.
Brouwer fixed-point theorem: A theorem stating that every continuous map from a closed ball to itself has a fixed point. Its degree-theoretic proof illustrates how smooth methods establish topological conclusions.
Hassler Whitney: An American mathematician who developed foundational results in differential topology and singularity theory. His embedding and approximation theorems established central tools for studying smooth manifolds.
Differential geometry: The study of geometric structures involving derivatives, such as metrics, curvature, and connections. Differential geometry adds metric or connection data that differential topology does not require.

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