Manifolds, Stacks & Classifying Spaces

← Back to Domain-Specific Families

Abstractions about geometric-topology spaces built from local models — manifold variants (Fréchet manifold, solvmanifold, quaternion-Kähler manifold), stack and groupoid presentations (gerbe, differentiable stack, Haefliger structure), and classification constructions such as classifying space, the Kirwan map and topological quantum field theory.

17 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.

• Cantor tree surface — Form the unique orientable boundaryless surface of genus zero whose end space is a Cantor set and whose every end is planar, equivalently a sphere with a Cantor set removed.
• Classifying space — A space BG representing principal G-bundles up to homotopy, obtained from a contractible free G-space EG and characterized by pullback classification.
• Cousin problems — Ask whether compatible local meromorphic data on a complex manifold glue to a global meromorphic function, with additive and multiplicative versions carrying distinct cohomological obstructions.
• 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.
• Differential Structure — A maximal compatible atlas that determines which coordinate descriptions on a topological manifold count as differentiable.
• Fréchet manifold — Build an infinite-dimensional smooth manifold from charts valued in Fréchet spaces, while making the chosen smooth calculus and the failure of general Banach inverse-function machinery explicit.
• Gerbe — A stack locally equivalent to the classifying stack of a group, serving as a degree-two geometric analogue of a principal bundle and encoding obstruction and twisting data.
• 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.
• 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.
• Locally simply connected space — Require every point to have a neighborhood basis of simply connected open sets, separating local loop triviality from global simple connectedness and weaker semilocal conditions.
• Morse–Smale System — A smooth complete flow with finitely many hyperbolic recurrent pieces whose stable and unstable manifolds meet transversely.
• 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.
• Polychromatic Symmetry — Symmetry of a pattern with at least three color classes under compatible spatial operations and nontrivial global color permutations.
• Quaternion-Kähler Manifold — A Riemannian manifold of dimension divisible by four whose Levi-Civita holonomy lies in Sp(n)Sp(1), carrying a parallel rank-three quaternionic structure rather than a globally selected complex structure.
• Solvmanifold — A homogeneous manifold of a connected solvable Lie group, with compact lattice quotients forming the special convention central to topology and Lie theory.
• Topological quantum field theory — A quantum field theory whose observables depend only on topological structure, mathematically formalized as a symmetric monoidal functor from a cobordism category to vector spaces or related algebraic categories.
• Topological Surface — A connected two-dimensional topological manifold whose points have planar neighborhoods, with half-plane neighborhoods allowed at a boundary.