Types of Manifolds| 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 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 |
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
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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