In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map
η : S 3 → S 2 ,
and proved that η is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles
η − 1 ( x ) , η − 1 ( y ) ⊂ S 3 for any
x ≠ y ∈ S 2 .
It was later shown that the homotopy group π 3 ( S 2 ) is the infinite cyclic group generated by η . In 1951, Jean-Pierre Serre proved that the rational homotopy groups
π i ( S n ) ⊗ Q for an odd-dimensional sphere ( n odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree 2 n − 1 .
Let ϕ : S 2 n − 1 → S n be a continuous map (assume n > 1 ). Then we can form the cell complex
C ϕ = S n ∪ ϕ D 2 n , where D 2 n is a 2 n -dimensional disc attached to S n via ϕ . The cellular chain groups C c e l l ∗ ( C ϕ ) are just freely generated on the i -cells in degree i , so they are Z in degree 0, n and 2 n and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that n > 1 ), the cohomology is
H c e l l i ( C ϕ ) = { Z i = 0 , n , 2 n , 0 otherwise . Denote the generators of the cohomology groups by
H n ( C ϕ ) = ⟨ α ⟩ and
H 2 n ( C ϕ ) = ⟨ β ⟩ . For dimensional reasons, all cup-products between those classes must be trivial apart from α ⌣ α . Thus, as a ring, the cohomology is
H ∗ ( C ϕ ) = Z [ α , β ] / ⟨ β ⌣ β = α ⌣ β = 0 , α ⌣ α = h ( ϕ ) β ⟩ . The integer h ( ϕ ) is the Hopf invariant of the map ϕ .
Theorem: h : π 2 n − 1 ( S n ) → Z is a homomorphism. Moreover, if n is even, h maps onto 2 Z .
The Hopf invariant is 1 for the Hopf maps (where n = 1 , 2 , 4 , 8 , corresponding to the real division algebras A = R , C , H , O , respectively, and to the fibration S ( A 2 ) → P A 1 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 topological K-theory, that these are the only maps with Hopf invariant 1.
A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:
Let V denote a vector space and V ∞ its one-point compactification, i.e. V ≅ R k and
V ∞ ≅ S k for some
k .
If ( X , x 0 ) is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of V ∞ , then we can form the wedge products
V ∞ ∧ X .
Now let
F : V ∞ ∧ X → V ∞ ∧ Y be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of F is
h ( F ) ∈ { X , Y ∧ Y } Z 2 ,
an element of the stable Z 2 -equivariant homotopy group of maps from X to Y ∧ Y . Here "stable" means "stable under suspension", i.e. the direct limit over V (or k , if you will) of the ordinary, equivariant homotopy groups; and the Z 2 -action is the trivial action on X and the flipping of the two factors on Y ∧ Y . If we let
Δ X : X → X ∧ X denote the canonical diagonal map and I the identity, then the Hopf invariant is defined by the following:
h ( F ) := ( F ∧ F ) ( I ∧ Δ X ) − ( I ∧ Δ Y ) ( I ∧ F ) . This map is initially a map from
V ∞ ∧ V ∞ ∧ X to
V ∞ ∧ V ∞ ∧ Y ∧ Y ,
but under the direct limit it becomes the advertised element of the stable homotopy Z 2 -equivariant group of maps. There exists also an unstable version of the Hopf invariant h V ( F ) , for which one must keep track of the vector space V .
