Hopf invariant

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In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.

Motivation[edit]

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

\eta\colon S^3 \to S^2,

and proved that \eta is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles

\eta^{-1}(x),\eta^{-1}(y) \subset S^3 for any x \neq y \in S^2.

It was later shown that the homotopy group \pi_3(S^2) is the infinite cyclic group generated by \eta. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

\pi_i(S^n) \otimes \mathbb{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 2n-1. There is an interesting way of seeing this:

Definition[edit]

Let \phi \colon S^{2n-1} \to S^n be a continuous map (assume n>1). Then we can form the cell complex

C_\phi = S^n \cup_\phi D^{2n},

where D^{2n} is a 2n-dimensional disc attached to S^n via \phi. The cellular chain groups C^*_\mathrm{cell}(C_\phi) are just freely generated on the n-cells in degree n, so they are \mathbb{Z} in degree 0, n and 2n 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^i_\mathrm{cell}(C_\phi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \\ 0 & \mbox{otherwise}. \end{cases}

Denote the generators of the cohomology groups by

H^n(C_\phi) = \langle\alpha\rangle and H^{2n}(C_\phi) = \langle\beta\rangle.

For dimensional reasons, all cup-products between those classes must be trivial apart from \alpha \smile \alpha. Thus, as a ring, the cohomology is

H^*(C_\phi) = \mathbb{Z}[\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0, \alpha\smile\alpha=h(\phi)\beta\rangle.

The integer h(\phi) is the Hopf invariant of the map \phi.

Properties[edit]

Theorem: h\colon\pi_{2n-1}(S^n)\to\mathbb{Z} is a homomorphism. Moreover, if n is even, h maps onto 2\mathbb{Z}.

The Hopf invariant is 1 for the Hopf maps (where n=1,2,4,8, corresponding to the real division algebras \mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}, respectively, and to the double cover S(\mathbb{A}^2)\to\mathbb{PA}^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.

Generalisations for stable maps[edit]

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^\infty its one-point compactification, i.e. V \cong \mathbb{R}^k and

V^\infty \cong 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^\infty, then we can form the wedge products

V^\infty \wedge X.

Now let

F \colon V^\infty \wedge X \to V^\infty \wedge Y

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of F is

h(F) \in \{X, Y \wedge Y\}_{\mathbb{Z}_2},

an element of the stable \mathbb{Z}_2-equivariant homotopy group of maps from X to Y \wedge 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 \mathbb{Z}_2-action is the trivial action on X and the flipping of the two factors on Y \wedge Y. If we let

\Delta_X \colon X \to X \wedge X

denote the canonical diagonal map and I the identity, then the Hopf invariant is defined by the following:

h(F) := (F \wedge F) (I \wedge \Delta_X) - (I \wedge \Delta_Y) (I \wedge F).

This map is initially a map from

V^\infty \wedge V^\infty \wedge X to V^\infty \wedge V^\infty \wedge Y \wedge Y,

but under the direct limit it becomes the advertised element of the stable homotopy \mathbb{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.

References[edit]

  • Adams, J.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104, doi:10.2307/1970147, JSTOR 1970147 
  • Adams, J.F.; Atiyah, M.F. (1966), "K-Theory and the Hopf Invariant", The Quarterly Journal of Mathematics 17 (1): 31–38, doi:10.1093/qmath/17.1.31