Summary: Relatives of the quotient of the complex projective
plane by the complex cojugation
V.I.Arnold \Lambda
Abstract
It is proved, that the quotient space of the four­dimensional quaternionic pro­
jective space by the automorphism group of the quaternionic algebra becomes the
13­dimensional sphere while quotioned the the quaternionic conjugation.
This fact and its various generalisations are proved using the results of the
theory of the hyperbolic partial differential equations, providing also the proof
of the theorem (which was, it seems, known to L.S.Pontriagin already in the
thirties) claiming that the quotient of the complex projective plane by the complex
conjugation is the 4­sphere.
1 Introduction
In the paper [1] on the topology of the real algebraic curves, written in 1971; I have used
the fact, that the quotient space of the complex projective plane by complex conjugation
is the four­dimensional sphere. The attempts to find a reference for this fact in the
literature were not successful at this time 1 . However V.A.Rokhlin told me that this
result had been known to L.S.Pontriagin already in the thirties.
I do not know how had Pontriagin proved this theorem. My proof (published later
in[2] ) was based on the theory of the hyperbolic partial differential equations. Thinking

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

Collections: Mathematics