Differential Topology & Geometric Structure

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Abstractions about differentiable stacks and manifolds, decompositions, root systems, orientations, stratified spaces, and major geometric-topological conjectures.

11 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Chow–Rashevsky theorem — A sub-Riemannian accessibility theorem stating that any two points of a connected manifold can be joined by a horizontal path when the allowed distribution is bracket generating.
• Differentiable stack — Represent quotient-like smooth geometry as a stack on manifolds admitting a smooth atlas, equivalently through a Lie-groupoid presentation considered up to Morita equivalence.
• Eells–Kuiper Manifold — Recognize the exceptional closed manifolds in dimensions 2, 4, 8, or 16 that admit a three-critical-point Morse function and have projective-plane-like compactification and cohomology structure.
• Exotic R4 — A smooth four-manifold homeomorphic but not diffeomorphic to ordinary Euclidean four-space, exposing the unique dimension-four split between topological and smooth equivalence.
• Iwasawa decomposition — A factorization G=KAN of a connected real semisimple Lie group into maximal compact, abelian and nilpotent subgroups, generalizing matrix QR decomposition.
• Kirwan map — Restrict equivariant cohomology classes of a Hamiltonian group space to a regular moment-map level set and identify them with ordinary cohomology classes on the resulting symplectic quotient.
• Novikov conjecture — The conjecture that higher signatures of closed oriented manifolds are invariant under oriented homotopy equivalence.
• Orientation character — Classify loops in a manifold by whether their transport preserves or reverses local orientation, encoding the result as a homomorphism from the fundamental group to the two-element sign group.
• Root System — Encode reflection symmetry by a finite spanning set of Euclidean vectors closed under root reflections, with integral Cartan pairings in the crystallographic setting.
• 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.
• Thurston Elliptization Conjecture — A proved three-manifold classification theorem: a closed three-manifold has finite fundamental group exactly when it admits a spherical metric of constant positive sectional curvature.