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 $X$.
Milnor (1956) gave the first examples of exotic smooth structures on the 7-sphere constructed via a $S^3$-bundle over the 4-sphere $S^4$, finding at least seven. Note that spheres inherit a canonical smooth structure from their canonical (topological) embedding into a Cartesian space $S^n \hookrightarrow \mathbb{R}^{n+1}$.
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 there are only finitely many exotic smooth structures on all spheres in dimension 5 or higher. 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.
A complete classification of smooth, PL and topological structures on manifolds in dimension 5 and higher was established by Kirby and Siebenmann (1977) using obstruction theory.
Note that there exist uncountably many exotic smooth structures on $\mathbb{R}^4$ (Gompf 1985, Freedman/Taylor 1986, Taubes 1987), but a unique smooth structure on $\mathbb{R}^n$ for $n\neq 4$ (Stallings, Zeeman 1962). 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 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.
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.
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).
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, 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 – Homtopy 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 $\frac{p_1}{2}^2 -1 = 0 (mod 7)$.
It is open whether the 4-sphere admits an exotic smooth structure. See (Freedman-Gompf-Morrison-Walker 09 for review).
The first construction of exotic smooth structures was on the 7-sphere in
(…)
Stephen Smale?, (1962), “On the structure of manifolds” , Amer. J. of Math. 84 : 387-399
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, E. C. Zeeman, The piecewise-linear structure of Euclidean space, Proceedings of the Cambridge Philosophical Society 58: 481–488 (1962)
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)
On the open issue of exotic 4-spheres:
See also
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):