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Hopf invariant one refers to homotopy classes of continuous functions between spheres of the form
hence to elements in the homotopy groups of spheres
whose Hopf invariant is equal to one:
The existence of an element of Hopf invariant one in $\pi_{2n-1}(S^n)$ is equivalent to the existence of an H-space structure on $S^n$.
A celebrated theorem due to (Adams 60, introducing and using the Adams spectral sequence) states that maps of Hopf invariant one correspond precisely to the the Hopf constructions on the four normed division algebras (see also at Hurwitz theorem): the real Hopf fibration, the complex Hopf fibration, the quaternionic Hopf fibration and the octonionic Hopf fibration.
(Adams 60)
Stable homotopy theory emerged as a distinct branch of algebraic topology with Adams‘ introduction of his eponymous spectral sequence and his spectacular conceptual use of the notion of stable phenomena in his solution to the Hopf invariant one problem.
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | $\phantom{AA}$spin group | normed division algebra | $\,\,$ brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\phantom{A}$ $\mathbb{R}$ the real numbers | super 1-brane in 3d |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\phantom{A}$ $\mathbb{C}$ the complex numbers | super 2-brane in 4d |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\phantom{A}$ $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) {\simeq}$ “SL(2,O)” | $\phantom{A}$ $\mathbb{O}$ the octonions | heterotic/type II string |
The original proof that the only maps of Hopf invariant one are the Hopf constructions on the four normed division algebras is due to
and made use of the classical Adams spectral sequence.
Review includes
Doug Ravenel, chapter 1, section 2 Methods of computing $\pi_\bullet(S^n)$ Complex cobordism and stable homotopy groups of spheres
John Rognes, around lemma 4.14, theorem 4.15 of The Adams spectral sequence, 2012 (pdf)
Joseph Victor, Section 2.4 of Stable Homotopy Groups of Spheres and The HopfInvariant One Problem, 2013 (pdf)
Further comments on the impact of this proof on the development of stable homotopy theory includes
Another proof that instead uses topological K-theory, Adams operations and the Atiyah-Hirzebruch spectral sequence was given in
This is reproduced for instance in
Dale Husemöller, chapter 15 of Fibre Bundles, Graduate Texts in Mathematics 20, Springer New York (1966)
Klaus Wirthmüller, section 12 of Vector bundles and K-theory, 2012 (pdf)
Allen Hatcher, section 2.3 of Vector bundles and K-theory (web)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 10.6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Review of this K-theoretic approach includes:
Ishan Banerjee, The Hopf invariant one problem, 2016 (pdf)
Review using the BP-Adams-Novikov spectral sequence includes
Last revised on May 1, 2019 at 01:17:48. See the history of this page for a list of all contributions to it.