Differential Topology

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7 domain-specific abstractions whose origin domain is Differential Topology.

• Collar neighbourhood — A neighborhood of a manifold's boundary that is identified with the product of that boundary and a half-open interval while fixing the boundary at interval coordinate zero.
• Gelfand–Fuks cohomology — Continuous Lie-algebra cohomology for topological Lie algebras of smooth vector fields.
• Haefliger structure — Encode generalized codimension-q foliation data by local maps to transverse Euclidean space whose transition germs form a cocycle, permitting pullback and classification beyond regular foliations.
• Morse homology — A homology theory whose chain groups are generated by critical points of a Morse function and whose boundary counts gradient-flow trajectories between adjacent indices.
• Smooth functor — A functor on finite-dimensional real vector spaces whose action varies smoothly in smoothly parameterized families.
• Stratifold — A stratified topological space equipped with a sheaf of smooth functions and manifold strata satisfying controlled local conditions, used as a geometric model for homology theories.
• Submersion (mathematics) — A smooth map between manifolds whose differential is surjective at every point, making each target direction locally attainable.