nLab Arnold-Kuiper-Massey theorem




topology (point-set topology, point-free topology)

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(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):

P 2/O(1)S 4 \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:


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

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

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


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

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

(Atiyah-Berndt 02)


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 P(2)\mathbb{C}P(2) by complex conjugation is the 4-sphere, Mathematische Annalen, 1974 (doi:10.1007/BF01432386)

  • Vladimir Arnold, Ramified covering P 2S 4\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 P 2\mathbb{C}P^2 to S 4S^4 (arXiv:1705.05038)

The SO(3)-equivariant enhancement:

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

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:

Last revised on September 20, 2021 at 13:58:19. See the history of this page for a list of all contributions to it.