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Arnold-Kuiper-Massey theorem

Contents

Context

Topology

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

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Complex geometry

Contents

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):

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:

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):

P 2/U(1)S 7 \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):

𝕆P 2/Sp(1)S 13 \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 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 09:58:19. See the history of this page for a list of all contributions to it.