Properties
Theorem: is a homomorphism. Moreover, if is even, maps onto .
The Hopf invariant is for the Hopf maps (where, corresponding to the real division algebras, respectively, and to the double cover sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of K-theory, that these are the only maps with Hopf invariant 1.
Read more about this topic: Hopf Invariant
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)