synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
An exotic smooth structure is, roughly speaking, a smooth structure on a topological manifold $X$ which makes the resulting smooth manifold be non-diffeomorphic to the smooth manifold given by some evident ‘standard’ smooth structure on $X$.
Mostly the term is used for smooth structures on Euclidean space $\mathbb{R}^n$ and on the n-spheres $S^n$, for $n \in \mathbb{N}$. The standard smooth structure on $\mathbb{R}^n$ is exhibited by the identity atlas, and the standard smooth structure on $S^n$ is that given by the atlas by the two hemisphere as given by stereographic projection.
For special values of $n$ there may exist smooth structure not equivalent to these. They are the exotic smooth structures.
A classification of smooth, PL? and topological structures on manifolds in dimension 5 and higher, in terms of various groups from algebraic topology (many not known) was established by Kirby and Siebenmann (1977) using obstruction theory.
Rado (1925) proved that in dimension 2 there are no exotic differentiable structures (or the uniqueness of the standard structure). The classification of 1-dimensional manifolds and the uniqueness of the smooth structure can be found in the Appendix of Milnor (1965b).
Moise (1952) proved that in dimension 3 there are no exotic differentiable structures, or to put in another way, 3-dimensional differentiable manifolds which are homeomorphic are diffeomorphic. In this way the 3-sphere $S^3$ inherits a unique differentiable structure, no matter which $\mathbb{R}^4$ it is considered to be embedded in.
There exists a unique smooth structure on the Euclidean space $\mathbb{R}^n$ for $n\neq 4$ (Stallings 1962).
There exists uncountably many exotic smooth structures on the Euclidean space $\mathbb{R}^4$ of dimension 4 (Gompf 1985, Freedman/Taylor 1986, Taubes 1987). See also at exotic R^4.
There is a unique maximal exotic $\mathbb{R}^4$ into which all other ‘versions’ of $\mathbb{R}^4$ smoothly embed as open subsets (Freedman/Taylor 1986, DeMichelis/Freedman 1992).
There are two classes of exotic $\mathbb{R}^4$‘s: large and small. A large exotic $\mathbb{R}^4$ cannot be embedded in the 4-sphere $S^4$ (Gompf 1985, Taubes 1987) whereas a small exotic $\mathbb{R}^4$ admits such an embedding (DeMichelis/Freedman 1992):
A large exotic $\mathbb{R}^4$ is constructed by using the failure to smoothly split a smooth 4-manifold (the K3 surface for instance) as a connected sum of some factors (where a topological splitting exits).
The small exotic $\mathbb{R}^4$ (or ribbon $\mathbb{R}^4$) is constructed by using the failure of the smooth h-cobordism theorem in dimension 4 (Donaldson 1987, 1990). Bizaca and Gompf (1996) are able to present an infinite handle body of a small exotic $\mathbb{R}^4$ which serve as a coordinate representation.
It is open whether the 4-sphere admits an exotic smooth structure. See (Freedman-Gompf-Morrison-Walker 09 for review).
Milnor (1956) gave the first examples of exotic smooth structures on the 7-sphere, finding at least seven.
The 7-spheres constructed in Milnor 1956 are all examples of fibre bundles over the 4-sphere $S^4$ with fibre the 3-sphere $S^3$, with structure group the special orthogonal group $SO(4)$.
By the classification of bundles on spheres via the clutching construction, these correspond to homotopy classes of maps $S^3 \to SO(4)$, i.e. elements of $\pi_3(SO(4))$. From the table at orthogonal group – Homotopy groups, this latter group is $\mathbb{Z}\oplus\mathbb{Z}$. Thus any such bundle can be described, up to isomorphism, by a pair of integers $(n,m)$. When $n+m=1$, then one can show there is a Morse function with exactly two critical points on the total space of the bundle, and hence this 7-manifold is homeomorphic to a sphere.
The fractional first Pontryagin class $\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}$ of the bundle is given by $n-m$. Milnor constructs, using cobordism theory and Hirzebruch's signature theorem for 8-manifolds, a mod-7 diffeomorphism invariant of the manifold, so that it is standard 7-sphere precisely when $\frac{p_1}{2}^2 -1 = 0 (mod 7)$.
By using the connected sum operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an abelian group. For the 7-sphere, it is the cyclic group $Z/{28}$ and Brieskorn (1966) found the generator $\Sigma$ so that $\underbrace{\Sigma\#\cdots\#\Sigma}_28$ is the standard sphere.
For review see for instance (Kreck 10, chapter 19, McEnroe 15).
Via the celebrated h cobordism theorem of Smale (Smale 1962, Milnor 1965) one gets a relation between the number of smooth structures on the $n$-sphere $S^n$ (for $n \geq 5$) and the number of isotopy classes $\pi_0 (Diff(S^{n-1}))$ of the equator $S^{n-1}$.
Then Kervaire and Milnor (1963) proved that for each $n \geq 5$ there are only finitely many exotic smooth structures on the n-sphere $S^n$ (possibly none).
By using the connected sum operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an abelian group.
The only odd-dimensional spheres with no exotic smooth structure are the circle $S^1$, the 3-sphere $S^3$, as well as $S^5$ and $S^{61}$ (Wang-Xu 16, corollary 1.13)
In the range $5 \leq n \leq 61$ the only $n$-spheres with no exotic smooth structures are $S^5$, $S^6$, $S^{12}$, $S^{56}$ and $S^{61}$ (Wang-Xu 16, corollary 1.15).
It is conjectured that this exhausts in fact all examples of $n$-spheres without exotic smooth structure for $n \geq 5$ (Wang-Xu 16, conjecture 1.17).
See also
The first construction of exotic smooth structures was on the 7-sphere in
(…)
Stephen Smale, On the structure of manifolds, Amer. J. of Math. 84 : 387-399 (1962)
John Milnor (1965), Lectures on the h-cobordism theorem (Princeton Univ. Press, Princeton)
Michel Kervaire, ; John Milnor, (1963) “Groups of homotopy spheres: I”, Ann. Math. 77, pp. 504 - 537.
Kirby, R.; Siebenmann, L. (1977) Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Studies (Princeton University Press, Princeton).
John R. Stallings, The piecewise-linear structure of Euclidean space, Proceedings of the Cambridge Philosophical Society 58: 481–488 (1962) (pdf)
Moise, Edwin E. (1952) “Affine structures on 3-manifolds”, Ann. Math. 56, pp. 96-114
Freedman, Michael H.; Taylor, Laurence (1986) “A universal smoothing of four-space”, J. Diff. Geom. 24, pp. 69-78
De Michelis, Stefano; Freedman, Michael H. (1992) “Uncountably many exotic $\mathbb{R}^4$‘s in standard 4-space”, J. Diff. Geom. 35, pp. 219-254.
Donaldson, Simon (1987) “Irrationality and the h-cobordism conjecture”, J. Diff. Geom. 26, pp. 141-168.
Simon Donaldson, (1990) “Polynomial invariants for smooth four manifolds”, Topology 29, pp. 257-315.
Gompf, Robert (1985) “An infinite set of exotic $\mathbb{R}^4$‘s”, J. Diff. Geom. 21, pp. 283-300.
Taubes, Clifford H. (1987) “Gauge theory on asymptotically periodic 4-manifolds”, J. Diff. Geom. 25, pp. 363-430
Bizaca, Z.; Gompf, Robert (1996) “Elliptic surfaces and some simple exotic $\mathbb{R}^4$‘s”, J. Diff. Geom. 43, pp. 458-504.
Rado, T. (1925) “Über den Begriff der Riemannschen Fläche” , Acta Litt. Scient. Univ. Szegd 2, pp. 101-121
Milnor, John W. (1965b) Topology from the Differentiable Viewpoint (University Press of Virginia)
Guozhen Wang, Zhouli Xu, The triviality of the 61-stem in the stable homotopy groups of spheres (arXiv:1601.02184)
On the open issue of exotic 4-spheres:
Review includes
Matthias Kreck, chapter 19 “Exotic 7-spheres” of Differential Algebraic Topology – From Stratifolds to Exotic Spheres, AMS 2010
Rachel McEnroe, Milnor’ construction of exotic 7-spheres, 2015 (pdf)
See also
The relevance of exotic smooth structure to physics is tantalizing but remains by and large unclear. Some of the following references probably ought to be handled with care.
The original argument that exotic spheres are to be regarded as instantons of gravity is on p. 12 of
Further discussion of exotic $4$-manifolds from the general relativity point of view is in
Carl Brans, Duane Randall, Exotic differentiable structures and general relativity Gen. Rel. Grav., 25 (1993) 205–220
Carl Brans Exotic smoothness and physics J. Math. Phys. 35, (1994), 5494–5506.
The following paper contained a first proof to localize exotic smoothness in an exotic $\mathbb{R}^4$:
A more philosophical discussion can be found in:
Brans conjectured in the papers above, that exotic smoothness should be a source of an additional gravitational field (Brans conjecture). This conjecture was confirmed for compact $4$-manifolds (using implicitly a mapping of basic classes):
Using the invariant of L. Taylor arXiv, Sladkowski confirmed the conjecture for the exotic $\mathbb{R}^4$ in:
The first real connection between exotic smoothness and quantum field theory is Witten’s TQFT:
and the whole work of Seiberg and Witten leading to the celebrated invariants.
The relation to particle physics by using the algebra of smooth functions can be found in
Jan Sładkowski, Exotic smoothness, noncommutative geometry and particle physics Int. J. Theor. Phys., 35, (1996), 2075–2083
Jan Sładkowski, Exotic smoothness and particle physics Acta Phys. Polon., B 27, (1996), 1649–1652
Jan Sładkowski, Exotic smoothness, fundamental interactions and noncommutative geometry arXiv
The relation between TQFT and differential-topological invariants of smooth manifolds was clarified in:
Hendryk Pfeiffer Quantum general relativity and the classification of smooth manifolds arXiv
Hendryk Pfeiffer Diffeomorphisms from finite triangulations and absence of ‘local’ degrees of freedom Phys.Lett. B, 591, (2004), 197-201
An argument for interpreting exotic smooth spheres as gravitational instanton?s and to cancel the gravitational anomalies of string theory is in
The influence of exotic smoothness for Kaluza-Klein models was discussed here:
A discussion of topological effects (also of string theory) in relation to exotic smoothness is in
An overview can be also found in
A first calculation of the state sum in quantum gravity by inclusion of exotic smoothness
A semi-classical approach to the functional integral is discussed here:
The inclusion of singularities for asymptotically flat spacetimes is discussed here (with an example of a singularity coming from exotic smoothness):