Types of Manifolds |

**Pseudomanifold**

$ __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 S^{1} (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 S^{n–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*
= {*S*_{i} |
*i* ∈ *I* ⊂ \(\mathbb N\)}.

* __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* :
*M* → *N* and *g* : *N* → *M* such
that *f* \(\circ\) *g* is homotopic to the identity map
id_{M} 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; S^{n} or
\(\mathbb R\)P^{n}, 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];
> 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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