 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.

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$$^n$$: For $$n \ne 4$$, $$\mathbb R$$n admits a unique differentiable structure; $$\mathbb R$$4 has uncountably many inequivalent ones.
* For spheres: For S$$^n$$ with n ≤ 6 the differentiable structure is unique; For higher n, the number N(n) is

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$$^n \subset {\mathbb C}^n$$ given by T$$^n:= \{z \in {\mathbb C}^n \mid |z_i| = 1,\ i = 1, ..., n\}$$.$ Stiefel manifold of k-frames: The manifold Vk($$\mathbb R$$n) = Vn, n−k($$\mathbb R$$) = O(n)/O(nk) of all k-frames in n-dimensional space; Properties: πp Vn, n−k($$\mathbb R$$) = 0 for 0 ≤ pnk−1.
> Generalizations: see differential geometry; non-commutative geometry.

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]; Dundas 18 [II].
@ 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(17)-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-fs [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]; Asselmeyer-Maluga & Król a1801 [neutrino masses].