nLab Arnold-Kuiper-Massey theorem

Statements

Proposition

(Arnold-Kuiper-Massey theorem)

The 4-sphere is the quotient space of the complex projective plane by the O(1)-action by complex conjugation (on homogeneous coordinates):

$$\mathbb{C}P^2 / \mathrm{O}(1) \simeq S^4$$

(Arnold 71, Massey 73, Kuiper 74, Arnold 88)

In fact, this is is the beginning of a small pattern indexed by the real normed division algebras:

Proposition

The 7-sphere is the quotient space of the (right-)quaternionic projective plane by the left multiplication action by U(1) $\subset$ Sp(1):

$$\mathbb{H}P^2 / \mathrm{U}(1) \simeq S^7$$

(Arnold 99, Atiyah-Witten 01, Sec. 5.5)

Proposition

The 13-sphere is the quotient space of the (right-)octonionic projective plane by the left multiplication action by Sp(1):

$$\mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13}$$

(Atiyah-Berndt 02)

References

AKM-theorem for the complex projective plane

The original proof that the 4-sphere is a quotient of the complex projective plane by an action of Z/2:

• Vladimir Arnold, On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.

• William Massey, The quotient space of the complex projective space under conjugation is a 4-sphere, Geometriae Didactica 1973 (pdf)

• Nicolaas Kuiper, The quotient space of $\mathbb{C}P(2)$ by complex conjugation is the 4-sphere, Mathematische Annalen, 1974 (doi:10.1007/BF01432386)

• Vladimir Arnold, Ramified covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology, Siberian Math. Journal 1988, V. 29, N 5, P.36-47

See also

• José Seade, Section V.5 in: On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, Birkhauser 2006 (ISBN:978-3-7643-7395-5)

• J. A. Hillman, An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$ (arXiv:1705.05038)

The SO(3)-equivariant enhancement:

• Le, José Seade, Alberto Verjovsky, Quadrics, orthogonal actions and involutions in complex projective space, L’Enseignement Mathématique, t. 49 (2003) (e-periodica:001:2003:49::488)

Generalization to the quaternionic projective plane

The generalization to the 7-sphere being a U(1)-quotient of the quaternionic projective plane is due to

• Vladimir Arnold, Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 (mathnet:tm691)

and independently due to

• Michael Atiyah, Edward Witten, Section 5.5 of: $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177, doi:10.4310/ATMP.2002.v6.n1.a1)

  (in the context of M-theory on G₂-manifolds)

Generalization to the octonionic projective plane

Another proof of these cases and further generalization to the 13-sphere being an Sp(1)-quotient of the octonionic projective plane:

• Michael Atiyah, Jürgen Berndt, Projective planes, Severi varieties and spheres, in: Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 (arXiv:math/0206135, doi:10.4310/SDG.2003.v8.n1.a1)