nLab exotic smooth structure

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

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 $X$.

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.

Existence and Examples

No exotic smooth structure in dimensions $\leq 3$

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.

No exotic Euclidean space away from dimension 4

There exists a unique smooth structure on the Euclidean space $\mathbb{R}^n$ for $n\neq 4$ (Stallings 1962).

Exotic Euclidean 4-space

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.

Exotic 4-spheres?

It is open whether the 4-sphere admits an exotic smooth structure. See (Freedman-Gompf-Morrison-Walker 09 for review).

Exotic 7-sphere

Milnor (1956) gave the first examples of exotic smooth structures on the 7-sphere, finding at least seven.

The exotic 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) (see also at 8-manifold the section With exotic boundary 7-spheres):

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). For more see at exotic 7-sphere.

From the point of view of M-theory on 8-manifolds, these 8-manifolds $X$ with (exotic) 7-sphere boundaries correspond to near horizon limits of black M2 brane spacetimes $\mathbb{R}^{2,1} \times X$, where the M2-branes themselves would be sitting at the center of the 7-spheres (if that were included in the spacetime, see also Dirac charge quantization).

Exotic $n$-spheres for $n \geq 5$

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).

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

(…)

• 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.

• Simon Donaldson (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

• John Milnor, (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)

• Llohann D. Sperança, Pulling back the Gromoll-Meyer construction and models of exotic spheres, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 (arXiv:1010.6039)

• Llohann D. Sperança, Explicit Constructions over the Exotic 8-sphere (pdf, pdf)

• C. Duran, A. Rigas, Llohann D. Sperança, Bootstrapping Ad-equivariant maps, diffeomorphisms and involutions, Matematica Contemporanea, 35:27–39, 2010 (pdf)

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)

For applications to physics

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.

General relativity

The original argument that exotic spheres are to be regarded as instantons of gravity is in

• Edward Witten, p. 12 of 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 BransExotic smoothness and physics J. Math. Phys. 35, (1994), 5494–5506.

The following paper contained a first proof to localize exotic smoothness in an exotic $\mathbb{R}^4$:

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

A more philosophical discussion can be found in:

• Carl BransAbsolute 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):

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

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

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 PfeifferQuantum general relativity and the classification of smooth manifolds arXiv

• Hendryk PfeifferDiffeomorphisms from finite triangulations and absence of ‘local’ degrees of freedom Phys.Lett. B, 591, (2004), 197-201

String theory

An argument for interpreting exotic smooth spheres as instantons of gravity and to cancel the gravitational anomalies of string theory is in (Witten 85).

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 $\mathfrak{su}(3) \times \mathfrak{su}(2) \times \mathfrak{u}(1)$-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.

Cosmology

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

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