In projective geometry in mathematics, the Arnold–Kuiper–Massey theorem (or AKM theorem) includes three connected results about projective planes and their connection to spheres. The Arnold–Kuiper–Massey theorem is named after Vladimir Arnold, Nicolaas Kuiper und William Massey, who described its complex version in 1971, 1974 and 1973, respectively. The quaternionic generalization was described by Vladimir Arnold in 1999 as well as by Michael Atiyah and Edward Witten in 2001. The octonionic generalization was described by Michael Atiyah and Jürgen Berndt in 2002.
Complex AKM theorem The first orthogonal group of unit real numbers acts on the complex projective space by complex conjugation. For the complex projective plane, the quotient space is homeomorphic to a sphere:Massey 1973, p. 371Atiyah & Berndt 2003, Eq. (1.1)
Quaternionic AKM theorem The first unitary group of unit complex numbers acts on the imaginary components of the quaternionic projective space by multiplication. For the quaternionic projective plane, the quotient space is homeomorphic to a sphere:Atiyah & Witten 2001, Eq. (5.50)
Octonionic AKM theorem The first symplectic group of unit quaternions acts on the second octonionic projective space or Cayley plane (there is no for due to the octonions not being associative) with:
Literature
External links
Arnold-Kuiper-Massey theorem on nLab
References