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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\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)

Spheres

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 modulo-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 /28\mathbb{Z}/{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, 3.8))

References

General

See also

In M-theory

Last revised on June 8, 2022 at 17:58:46. See the history of this page for a list of all contributions to it.