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.
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 Rn or,
more generally, a locally convex topological
vector space (to cover the infinite-dimensional case).
* Classification: For
n = 1, the only connected manifolds are R1 (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.
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.
* 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.
$ Def: A topological
manifold which admits a locally finite cellular decomposition, M =
{Si |
i 
I 
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.
* Operations: Pachner moves.
@ References: Hudson 69; Rourke & Sanderson 72; Hirsch & Mazur
75; Kirby & Siebenmann
77; Barrett & Parker JAT(94)
[smooth limit].
Structured Space
$ Def: A non-empty tm 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 94; 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
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].
* Parallelizable manifold:
A manifold M is parallelizable if it
admits a continuous frame field, i.e., F(M) admits a cross-section; Examples:
Any Lie group; Sn or RPn, but only for n = 1,
3, 7.
@ Parallelizable: in Steenrod 51; Kervaire PNAS(58) [n-sphere for n > 7].
@ 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 Rn,
but otherwise quite similar): in Sikorsky 72; Gruszczak et al JMP(88),
FP(89); Multarzynski & Heller FP(90).
@ Non-associative: Wulkenhaar ht/96, ht/96,
PLB(97)ht/96 [standard
model], ht/96 [gut];
Nesterov & Sabinin CMUC(00)ht-in,
PRD(00)ht [and
spacetime]; > s.a. particle physics.
@ Einstein algebras: Geroch CMP(72); Heller & Sasin IJTP(95).
@ Other types: Parker JMP(79)
[distributional]; Liu & He RPMP(06)
[Dirac-Nijenhuis manifolds]; > s.a. distributions.
> Other types: see differential
geometry; Homology
Manifold; 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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Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jun 2008