nLab
exotic 7-sphere

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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

A topological 7-sphere equipped with an exotic smooth structure is called an exotic 7-sphere.

Milnor’s construction

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 4S^4 with fibre the 3-sphere S 3S^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 3SO(4)S^3 \to SO(4), i.e. elements of π 3(SO(4))\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)(n,m). When n+m=1n+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 p 12H 4(S 4)\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z} of the bundle is given by nmn-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 the standard 7-sphere precisely when p 12 21=0(mod7)\frac{p_1}{2}^2 -1 = 0 (mod 7).

By using the connected sum operation, the set of smooth, non-diffeomorphic structures on the nn-sphere has the structure of an abelian group. For the 7-sphere, it is the cyclic group Z/28Z/{28} and Brieskorn (1966) found the generator Σ\Sigma so that Σ##Σ 28\underbrace{\Sigma\#\cdots\#\Sigma}_28 is the standard sphere.

Review includes (Kreck 10, chapter 19, McEnroe 15, Joachim-Wraith).

Examples

Properties

As near-horizon geometries of black M2-branes

From the point of view of M-theory on 8-manifolds, these 8-manifolds XX with (exotic) 7-sphere boundaries in Milnor’s construction correspond to near horizon limits of black M2 brane spacetimes 2,1×X\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).

(Morrison-Plesser 99, section 3.2, FSS 19, 4.6))

References

General

  • Matthias Kreck, chapter 19 of Exotic 7-spheres of Differential Algebraic Topology – From Stratifolds to Exotic Spheres, AMS 2010

  • Rachel McEnroe, Milnor’ construction of exotic 7-spheres, 2015 (pdf)

  • Michael Joachim, D. J. Wraith, Exotic spheres and curvature (pdf)

See also

In M-theory

Last revised on April 26, 2019 at 12:11:49. See the history of this page for a list of all contributions to it.