nLab
exotic smooth structure

Idea

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 $X$.

Properties

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 $S^n\hookrightarrow {ℝ}^{n+1}$.

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 $n\ge 5$) and the number of isotopy classes ${\pi }_0(\mathrm{Diff}(S^{n-1}))$ 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 ${ℝ}^4$ (Gompf 1985, Freedman/Taylor 1986, Taubes 1987), but a unique smooth structure on ${ℝ}^n$ for $n\ne 4$ (Stallings, Zeeman 1962). There is a unique maximal exotic ${ℝ}^4$ into which all other ‘versions’ of ${ℝ}^4$ smoothly embed as open subsets (Freedman/Taylor 1986, DeMichelis/Freedman 1992).

There are two classes of exotic ${ℝ}^4$’s: large and small. A large exotic ${ℝ}^4$ cannot be embedded in the 4-sphere $S^4$ (Gompf 1985, Taubes 1987) whereas a small exotic ${ℝ}^4$ admits such an embedding (DeMichelis/Freedman 1992). A large exotic ${ℝ}^4$ 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 ${ℝ}^4$ (or ribbon ${ℝ}^4$) 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 ${ℝ}^4$ 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 ${ℝ}^4$ 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).

References

For the mathematical theory

The first construction of exotic smooth structures was on the 7-sphere in

• John Milnor (1956), “On manifolds homeomorphic to the 7-sphere”, Annals of Mathematics (Annals of Mathematics) 64 (2): 399–405

(…)

• Smale, Stephen (1962), “On the structure of manifolds” , Amer. J. of Math. 84 : 387-399 (#Smale)

• John Milnor (1965), Lectures on the h-cobordism theorem (Princeton Univ. Press, Princeton)

• Kervaire, Michel; Milnor, John W. (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).

• Stallings, John R.; Zeeman, E. C. (1962), “The piecewise-linear structure of Euclidean space”, Proceedings of the Cambridge Philosophical Society 58: 481–488

• 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 ${ℝ}^4$’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.

• Donaldson, Simon (1990) “Polynomial invariants for smooth four manifolds”, Topology 29, pp. 257-315.

• Gompf, Robert (1985) “An infinite set of exotic ${ℝ}^4$’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 ${ℝ}^4$’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)

(see Wikipedia (spheres, $R^4$) for more for now)

For applications to physics

A survey of applications of the theory of exotic smooth structures in physics is in

• Torsten Asselmeyer-Maluga, Carl Brans, Exotic smoothness and physics. Differential topology and spacetime models Singapore: World Scientific (2007) (ZMATH)

below we list more references sorted by the kind of physical application they deal with.

General relativity

The original argument that exotic spheres are to be regarded as instantons of gravity is on p. 12 of

• Edward Witten, Global gravitational anomalies Comm. Math. Phys. Volume 100, Number 2 (1985), 197–229. (EUCLID)

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 ${ℝ}^4$:

• Carl Brans Localized exotic smoothness Class. Quant. Grav., 11, (1994), 1785–1792.

A more philosophical discussion can be found in:

• Carl Brans Absolute spacetime: the twentieth century ether Gen. Rel. Grav. 31, (1999), 597–609

Generation of source terms (fields)

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):

• Torsten Asselmeyer Generation of source terms in general relativity by differential structures Class. Quant. Grav. 14, (1996), 749–758

Using the invariant of L. Taylor arXiv, Sladkowski confirmed the conjecture for the exotic ${ℝ}^4$ in:

• Jan Sładkowski Gravity on exotic R4 with few symmetries Int.J. Mod. Phys. D, 10, (2001) 311–313

Via a detailed analysis of the Casson handle (as the main object to construct exotic smoothness for $4$-manifolds via the Akbulut cork), it is possible to get the action of Dirac field and of the gauge bosons from the Einstein-Hilbert action (with a boundary term) in:

• Torsten Asselmeyer-Maluga, Helge Rose, On the geometrization of matter by exotic smoothness, arXiv:1006.2230

Gerbes, WZW models, foliations

The first paper describes a construction of an exotic ${ℝ}^4$ via a codimension-$1$ foliation of a homology $3$-sphere in the interior (classified by the Godbillon–Vey invariant). One of the results is a charge quantization without a monopole:

• Torsten Asselmeyer-Maluga, Jerzy Król Gerbes, SU(2) WZW models and exotic smooth ${ℝ}^4$ arXiv

Then one gets various relations to orbifolds and the WZW model (level of the conformal field theory)

• Torsten Asselmeyer-Maluga, Jerzy Król Gerbes on orbifolds and exotic smooth ${ℝ}^4$ arXiv

Quantum (field) theory

The first real connection between exotic smoothness and quantum field theory is Witten’s TQFT:

• Edward Witten, Topological quantum field theory Comm. Math. Phys., 117, (1988), 353–386.

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

The first conjectured (direct) relation (identifying submanifolds with operators) to quantum mechanics (using singular connections) was established here:

• Torsten Asselmeyer-Maluga, Helge Rose, Differential Structures – the Geometrization of Quantum Mechanics arXiv

Using the relation to foliations and noncommutative geometry for the leave space, one can reproduce the vacuum sector of a QFT (the factor ${\mathrm{III}}_1$) via exotic smoothness as explained here:

• Torsten Asselmeyer-Maluga, Jerzy Król Exotic smooth ${ℝ}^4$, noncommutative algebras and quantization arXiv

String theory

An argument for interpreting exotic smooth spheres as gravitational instanton?s and to cancel the gravitational anomalies of string theory is in

• Edward Witten, Global gravitational anomalies Comm. Math. Phys. Volume 100, Number 2 (1985), 197–229. (web)

The influence of exotic smoothness for Kaluza-Klein models was discussed here:

• Matthias Kreck, Stefan Stolz, A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with $\mathrm{𝔰𝔲}(3)\times \mathrm{𝔰𝔲}(2)\times 𝔲(1)$-symmetry Ann. Math. 127, (1988), 373–388.

A discussion of topological effects (also of string theory) in relation to exotic smoothness is in

• Ryan Rohm, Topological Defects and Differential Structures Annals Of Physics, 189, (1989), 223–239.

There are also some relations between string backgrounds, quantum D-branes and exotic ${ℝ}^4$

• Torsten Asselmeyer-Maluga, Jerzy Król Exotic smooth ${ℝ}^4$, geometry of string backgrounds and quantum D-branes, arXiv

Cosmology

The first idea to use exotic smoothness to explain some cosmological anomalies (dark energy etc.) can be found here:

• Carl Brans, Torsten Asselmeyer-Maluga Cosmological anomalies and exotic smoothness structures, Gen. Rel. Grav. 34, (2002), 597–607.

Then this idea was used to discuss a model of the cosmos including the cosmological constant (only the present value of the constant can be reproduced, no explanation of the scaling behavior!):

• Torsten Asselmeyer-Maluga, Helge Rose, Dark energy and $3$-manifold topology. Act. Phys. Pol., 38 (2007) 3633–3639

An overview can be also found in

• Jan Sładkowski, Exotic smoothness and astrophysics Act. Phys. Polon. B, 40, (2009), 3157–3163

Quantum gravity

A first calculation of the state sum in quantum gravity by inclusion of exotic smoothness

• Kristin Schleich, Donald Witt, Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions Class.Quant.Grav., 16, (1999), 2447–2469

A semi-classical approach to the functional integral is discussed here:

• Christofer Duston, Exotic smoothness in 4 dimensions and semiclassical Euclidean quantum gravity arxiv

A complete expression for the state sum in quantum gravity for compact $4$-manifolds by inclusion of exotic smoothness using the knot surgery a la Fintushel and Stern

• Torsten Asselmeyer-Maluga, Exotic smoothness and quantum gravity , Class. Quantum Grav. 27 (2010) 165002. (web) or (arXiv)

The inclusion of singularities for asymptotically flat spacetimes is discussed here (with an example of a singularity coming from exotic smoothness):

• Kristin Schleich, Donald Witt, Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes, arxiv