Types of Manifolds  

$ Recursive def: An n-dimensional pseudomanifold is a set of points, each having a neighborhood homeomorphic to a cone over an (n−1)-dimensional pseudomanifold; A 0D pseudomanifold is just a set of disjoint points.
* Examples: Any manifold is a pseudomanifold; A non-trivial example is a graph with intersections.
@ References: Altshuler a1004 [spacetime as a pseudomanifold]; Benedetti NPB(17)-a1608 [Mogami pseudomanifolds, and 3-spheres].

Topological Manifold > s.a. 2D, 3D, 4D manifolds; Whitehead Continua.
$ Def: A Hausdorff topological space, such that every point has a neighborhood homeomorphic to an open set in \(\mathbb R\)n or, more generally, a locally convex topological vector space (to cover the infinite-dimensional case).
* Classification: For n = 1, the only connected manifolds are \(\mathbb R\)1 (non-compact) and S1 (compact); For n = 2, they are classifiable; For n = 3 it is not known; For n ≥ 4 they are not classifiable (for n = 5, not even a list with repetitions is possible!).
* Decidability: Closed n-manifolds with n ≤ 3 are algorithmically decidable; With n = 4 it is not known; With n ≥ 5 they are not.
@ References: Kirby & Siebenmann 77; Chapman 81; Daverman 86; Ranicki 92; Lee 00.

Combinatorial Manifold > s.a. discrete geometries.
$ Def: An n-dimensional combinatorial manifold is a simplicial complex in which the link of every vertex is a combinatorial Sn–1.
* Result: Any two closed combinatorial manifolds are PL-isomorphic if and only if they are related by a finite sequence of Pachner moves.
* Relationships: For n < 7, all combinatorial manifolds have a smooth counterpart; A description of a differentiable manifold in terms of combinatorial manifolds incorporates both topology and differentiable structure.
@ References: Schleich in(94); Anderson Top(99) [and PL manifolds].

Piecewise Linear Manifold (PL) > s.a. 3D manifold; cell complex; euler classes; topological field theories; Whitehead Theorem.
$ Def: A topological manifold which admits a locally finite cellular decomposition, M = {Si | iI ⊂ \(\mathbb N\)}.
* Results: Every 1D, 2D, and 3D topological manifold admits an essentially unique PL (and differentiable) structure; In 4 dimensions, every PL manifold admits a unique induced differentiable structure, but the transition from topological to PL is still open; In 5 or more dimensions, although locally a PL structure always exists, there may be global obstructions, that can be characterized by cohomology classes.
* Results: In up to six dimensions, each PL-manifold admits a smoothing, and the resulting smooth manifold is unique up to diffeomorphism.
* Smoothing: A differentiable structure on a manifold is a smoothing of a PL structure there if it satisfies a compatibility condition.
* Operations: see Pachner Moves.
@ General references: Hudson 69; Rourke & Sanderson 72; Hirsch & Mazur 75; Kirby & Siebenmann 77; Barrett & Parker JAT(94) [smooth limit]; Rudyak 16 [classification, readable].
@ Physics-related topics: Schrader JPA(16)-a1508 [Einstein metrics and Ricci flows]; Korepanov AACA(17)-a1605 [free fermions].

Structured Space
$ Def: A non-empty topological manifold with a sheaf of functions satisfying a closure axiom.
@ And general relativity: Heller & Sasin JMP(95).

With Mild Singularities > s.a. Homogeneous and Symmetric Spaces; Orbifold.
@ With singularities: Botvinnik 92; Lesch Top(93).
@ Conifold: Fursaev & Solodukhin PRD(95); Schleich & Witt NPB(93)gq, NPB(93)gq.
@ Stratified: Weinberger 95; Rudolph et al JPA(02) [gauge orbits]; Hübsch & Rahman JGP(05)m.AG/02 [from supersymmetric theories]; Vilela Mendes JPA(04)mp/02 [gauge orbits]; > s.a. geometrodynamics; quantum field theory on general backgrounds; quantum gauge theory; Quasifold; symplectic manifolds.
@ Orientifold: Dabholkar ht/98-ln [and duality, intro].

Other Types and Generalizations > s.a. fiber bundles; differentiable manifolds; manifolds [including supermanifolds].
* In general: There are two influential ways of defining a generalized notion of space; One, inspired by Gel'fand duality, states that the category of 'non-commutative spaces' is the opposite of the category of C*-algebras; The other, loosely generalizing Stone duality, maintains that the category of 'pointfree spaces' is the opposite of the category of frames.
* Homotopy equivalence: Two topological manifolds M and N are called homotopy equivalent if there exists a pair of continuous maps f : MN and g : NM such that f \(\circ\) g is homotopic to the identity map idM of M; Homeomorphism implies homotopy equivalence; The converse holds in 1D and 2D.
* Parallelizable manifold: A manifold M is parallelizable if it admits a continuous frame field, i.e., F(M) admits a cross-section defining an absolute parallelism structure; Examples: Any Lie group; Sn or \(\mathbb R\)Pn, but only for n = 1, 3, 7.
@ General references: Heunen et al JAMS(11)-a1010 [Gel'fand spectrum of a non-commutative C*-algebra]; Lin T&A(12) [infinite-dimensional, using the language of categories and functors].
@ Parallelizable: in Steenrod 51; Kervaire PNAS(58) [n-sphere for n > 7]; Youssef & Elsayed RPMP(13)-a1209 [geometry, global approach].
@ Non-metrizable: Balogh & Gruenhage T&A(05) [perfectly normal].
@ Families converging to graphs: Exner & Post JGP(05) [and Laplace-Beltrami spectrum].
@ d-spaces (not locally diffeomorphic to \(\mathbb R^n\), but otherwise quite similar): in Sikorsky 72; Gruszczak et al JMP(88), FP(89); Multarzyński & Heller FP(90).
@ Other types: Parker JMP(79) [distributional]; Liu & He RPMP(06) [Dirac-Nijenhuis manifolds]; Bellettini et al a1106 [Lorentzian varifolds]; Delphenich a1809 [geometry of non-parallelizable manifolds]; Vysoky a2105 [graded manifolds]; > s.a. distributions.
> Other types: see Deformations; differential geometry; Einstein Algebras; Homology Manifold; Non-Associative Geometry; non-commutative geometry; quantum group; Topos.
> Related topics: see connection; laplace equation; partial differential equations; path-integral quantum gravity; regge calculus [polymerized / random].

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