Differentiable Manifolds and Differential Topology |

**In General** > s.a. diffeomorphisms;
differentiable maps; embedding;
manifolds.

* __Differentiable structure__:
An assignment of an equivalence class
of atlases, with charts related by differentiable transition functions.

* __Differentiable manifold__:
A space with a differentiable structure, or a topological manifold with a sheaf of
*k*-smooth functions (a ring space), or differentiable
relations between charts (some topological manifolds admit no such structures).

* __Motivation__: The differentiable
manifold structure is the most natural and general one in which to study differentiability;
Differential topology studies properties of differentiable manifolds without
additional structure (diffeos, forms, tensors and concomitants, etc);
They can be studied as ways of reducing a topological tangent bundle to a tangent
bundle [@ Milnor Top(64)].

* __History__: The first
to present a coordinate-independent account was Darboux.

* __Remark__: One atlas is
sufficient to specify the differentiable structure,
and any C^{1} atlas is C^{1}-equivalent
to a smooth one.

> __Online resources__: see Wikipedia page.

**Algebraic Description** > s.a. manifolds [Gel'fand-Naimark
theorem].

* __Idea__: The whole structure of a differentiable manifold can be recovered
just from that of an appropriate set of functions on the manifold, such as its ring of differentiable functions, by defining it to be the set
of maximal ideals of this ring.

@ __References__: in Milnor & Stasheff 74; Yodzis PRIA(75).

**Manifolds with Inequivalent Differentiable Structures** > s.a.
4D manifolds.

* __For R__

*

*n*:
7 8 9 10 11 12 13 14 15 16

*N*(*n*): 28 2 8 6 992 1 3 2 16256 2 ;

The exotic ones are also boundaries of manifolds, but not of contractible ones.

* __General results__: Any
closed, connected 2- or 3-manifold has a unique differentiable structure; In *n* ≥ 4
dimensions, some manifolds don't admit any, others many inequivalent ones;
For *n* ≥ 5, compact topological manifolds have a finite number of differentiable structures
[@ Kirby & Siebenmann 77], but the PL and Diff structures coincide for *n* up to 6.

@ __ Exotic spheres__: Castelvecchi SA(09)aug [classification, Kervaire problem]; Elwes plus(11)jan [the 4D case, I].

@ __ For Lorentzian manifolds__: Torres CQG(14)-a1407 [globally hyperbolic Lorentzian metrics pin down a smooth structure on the underlying manifold].

**Examples and Types** > s.a. 2D, 3D, 4D
manifolds; Contact
Manifold; spheres; Surfaces.

$ __Torus__: The *n*-dimensional
torus is the manifold T^{ n} ⊂ \(\mathbb C\)^{}^{n} given
by T^{ n}:=
{*z* ∈ \(\mathbb C\)^{}^{n} |
|*z*_{i}| = 1, *i* = 1, ..., *n*}.

$ __Stiefel manifold of k-frames__:
The manifold V

>

**References and Related Topics** > s.a. differential
forms; exterior calculus; integration;
operators [differential operators]; tensors.

@ __Texts__: Milnor 58, 65;
Wallace 66; Hirsch 76;
Chillingworth 77;
Bröcker & Jänich 82; Gauld 82;
Guillemin & Pollack 84; Pontrjagin 86;
Fomenko 87; Bredon 93;
Wang & Chen 99;
Lee 02;
Barden & Thomas 03;
Shastri 11; Wall 16 [including surgery and cobordism].

@ __For physicists__: Torres del Castillo 11; Baillon 13 [for experimental physicists].

@ __General references__: Kanakoglou a1204-ln [pedagogical].

@ __With boundaries__: Margalef-Roig & Outerelo Domínguez 92;
Scott & Szekeres
JGP(94)gq;
Fama & Clarke CQG(98)gq; Ciaglia et al IJGMP-a1705 [differential calculus on manifolds with boundaries and corners].

@ __Related topics__: Bott in(75) [invariants]; Dodson & Radivoiovici IJTP(82) [higher-order tangent structures];
> s.a. Jet Bundle.

> __Other topics__: see Isotopy;
morse theory; Stokes' Theorem;
tangent structures.

**Differentiable Structures and Physics**
> s.a. quantum spacetime / 4D manifolds;
singularities.

* __Remark__: Because spacetime is 4-dimensional,
at least at large scales, and taking into accont the result about the many inequivalent differentiable
structures on \(\mathbb R\)^{4}, it may be natural to consider the
differentiable structure as one of the variables to include in a theory of quantum gravity in which the
spacetime manifold is derived from a more basic structure; See however the results by Chernov &
Nemirovski on "smoothness censorship".

@ __General references__: Chamblin JGP(94)gq/95 [homotopy
type and causal structure]; Król FP(04)
[exotic \(\mathbb R\)^{4}
and non-commutative spaces]; Asselmeyer-Maluga & Brans 07; Brans GRG(08)
[rev].

@ __And phase transitions in field theory__: Goldin & Moschella JPA(95).

@ __Exotic structures and gravity__: Penrose in(84);
Witten CMP(85);
Rohm AP(89);
Brans & Randall GRG(93)gq/92;
Brans CQG(94)gq,
JMP(94)gq,
gq/96-proc;
Asselmeyer CQG(97)gq/96;
Schleich & Witt CQG(99)gq [7D
Euclidean quantum gravity]; Sładkowski gq/99 [consequences];
Aßelmeyer-Maluga & Brans GRG(02)gq/01 [and
cosmology]; Boyer et al EM(05)m.DG/03 [Einstein
metrics on exotic S^{7}, S^{11} and
S^{15}]; Król ht/05-conf
[model theory approach]; Sładkowski APPB-a0910-conf
[astrophysical consequences]; Duston IJGMP(11)-a0911 [semiclassical
euclidean quantum gravity]; Asselmeyer-Maluga CQG(10)-a1003; Asselmeyer-Maluga & Brans a1101 [and generation of spinor field]; Asselmeyer-Maluga & Król a1112; Chernov & Nemirovski CQG(13)-a1201 [smoothness censorship], Asselmeyer-Maluga & Brans a1401/GRG, GRG(15)-a1502 [and fermions]; Asselmeyer-Maluga a1601 [and quantization of geometry]; > s.a. cosmic censorship; inflationary scenarios.

@ __Matter and quantum theory__: Asselmeyer-Maluga & Rosé gq/05 [geometrization
of quantum mechanics]; Asselmeyer-Maluga & Król a1001 [exotic \(\mathbb R\)^{4} and
quantization];
Asselmeyer-Maluga & Rosé GRG(12)-a1006 [geometrization
of matter]; Asselmeyer-Maluga & Król a1107; Król JPCS(12)-a1111 [exotic \(\mathbb R\)^{4} and
matter, superstring theory]; Asselmeyer-Maluga & Mader JPCS(12)-a1112 [and operator algebras].

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