differential topology

mathematics

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July 14, 2026

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Differential topology is a branch of mathematics that studies smooth, or differentiable, manifolds. A manifold is a topological space that locally resembles Euclidean space but can have complex global properties. These spaces are studied using methods from analysis and algebraic topology.1

Differential topology imposes a structure of "smoothness" on manifolds, allowing for the use of calculus and differentiation. This field has found applications in modern physics. A key figure in differential topology is John Willard Milnor, who received the Fields Medal for his work in this area, including the discovery of "exotic spheres".23

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topology
Differential topology
Many tools of algebraic topology are well-suited to the study of manifolds. In the field of differential topology an additional structure involving “smoothness,” in the sense of differentiability (see analysis: Formal definition of the derivative), is imposed on manifolds. Since early investigation in…
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Many tools of algebraic topology are well-suited to the study of manifolds. In the field of differential topology an additional structure involving “smoothness,” in the sense of differentiability (see analysis: Formal definition of the derivative), is imposed on manifolds. Since early investigation in…
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John Willard Milnor
…1962 for his work in differential topology and the Abel Prize in 2011 for his work in topology, geometry, and algebra.
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…1962 for his work in differential topology and the Abel Prize in 2011 for his work in topology, geometry, and algebra.
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