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Hopf invariant - Wikipedia

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Homotopy invariant of maps between n-spheres

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

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

η : S 3 → S 2 , {\displaystyle \eta \colon S^{3}\to S^{2},}

and proved that η {\displaystyle \eta } is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles

η − 1 ( x ) , η − 1 ( y ) ⊂ S 3 {\displaystyle \eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}}

is equal to 1, for any x ≠ y ∈ S 2 {\displaystyle x\neq y\in S^{2}} .

It was later shown that the homotopy group π 3 ( S 2 ) {\displaystyle \pi _{3}(S^{2})} is the infinite cyclic group generated by η {\displaystyle \eta } . In 1951, Jean-Pierre Serre proved that the rational homotopy groups [1]

π i ( S n ) ⊗ Q {\displaystyle \pi _{i}(S^{n})\otimes \mathbb {Q} }

for an odd-dimensional sphere ( n {\displaystyle n} odd) are zero unless i {\displaystyle i} is equal to 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 {\displaystyle 2n-1} .

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

C φ = S n ∪ φ D 2 n , {\displaystyle C_{\varphi }=S^{n}\cup _{\varphi }D^{2n},}

where D 2 n {\displaystyle D^{2n}} is a 2 n {\displaystyle 2n} -dimensional disc attached to S n {\displaystyle S^{n}} via φ {\displaystyle \varphi } . The cellular chain groups C c e l l ∗ ( C φ ) {\displaystyle C_{\mathrm {cell} }^{*}(C_{\varphi })} are just freely generated on the i {\displaystyle i} -cells in degree i {\displaystyle i} , so they are Z {\displaystyle \mathbb {Z} } in degree 0, n {\displaystyle n} and 2 n {\displaystyle 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 {\displaystyle n>1} ), the cohomology is

H c e l l i ( C φ ) = { Z i = 0 , n , 2 n , 0 otherwise . {\displaystyle H_{\mathrm {cell} }^{i}(C_{\varphi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\text{otherwise}}.\end{cases}}}

Denote the generators of the cohomology groups by

H n ( C φ ) = ⟨ α ⟩ {\displaystyle H^{n}(C_{\varphi })=\langle \alpha \rangle } and H 2 n ( C φ ) = ⟨ β ⟩ . {\displaystyle H^{2n}(C_{\varphi })=\langle \beta \rangle .}

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

H ∗ ( C φ ) = Z [ α , β ] / ⟨ β ⌣ β = α ⌣ β = 0 , α ⌣ α = h ( φ ) β ⟩ . {\displaystyle H^{*}(C_{\varphi })=\mathbb {Z} [\alpha ,\beta ]/\langle \beta \smile \beta =\alpha \smile \beta =0,\alpha \smile \alpha =h(\varphi )\beta \rangle .}

The integer h ( φ ) {\displaystyle h(\varphi )} is the Hopf invariant of the map φ {\displaystyle \varphi } .

Theorem: The map h : π 2 n − 1 ( S n ) → Z {\displaystyle h\colon \pi _{2n-1}(S^{n})\to \mathbb {Z} } is a homomorphism. If n {\displaystyle n} is odd, h {\displaystyle h} is trivial (since π 2 n − 1 ( S n ) {\displaystyle \pi _{2n-1}(S^{n})} is torsion). If n {\displaystyle n} is even, the image of h {\displaystyle h} contains 2 Z {\displaystyle 2\mathbb {Z} } . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. h ( [ i n , i n ] ) = 2 {\displaystyle h([i_{n},i_{n}])=2} , where i n : S n → S n {\displaystyle i_{n}\colon S^{n}\to S^{n}} is the identity map and [ ⋅ , ⋅ ] {\displaystyle [\,\cdot \,,\,\cdot \,]} is the Whitehead product.

The Hopf invariant is 1 {\displaystyle 1} for the Hopf maps, where n = 1 , 2 , 4 , 8 {\displaystyle n=1,2,4,8} , corresponding to the real division algebras A = R , C , H , O {\displaystyle \mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O} } , respectively, and to the fibration S ( A 2 ) → P A 1 {\displaystyle 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 Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.

Whitehead integral formula[edit]

J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.[2][3]: prop. 17.22  Given a map φ : S 2 n − 1 → S n {\displaystyle \varphi \colon S^{2n-1}\to S^{n}} , one considers a volume form ω n {\displaystyle \omega _{n}} on S n {\displaystyle S^{n}} such that ∫ S n ω n = 1 {\displaystyle \int _{S^{n}}\omega _{n}=1} . Since d ω n = 0 {\displaystyle d\omega _{n}=0} , the pullback φ ∗ ω n {\displaystyle \varphi ^{*}\omega _{n}} is a closed differential form: d ( φ ∗ ω n ) = φ ∗ ( d ω n ) = φ ∗ 0 = 0 {\displaystyle d(\varphi ^{*}\omega _{n})=\varphi ^{*}(d\omega _{n})=\varphi ^{*}0=0} . By Poincaré's lemma it is an exact differential form: there exists an ( n − 1 ) {\displaystyle (n-1)} -form η {\displaystyle \eta } on S 2 n − 1 {\displaystyle S^{2n-1}} such that d η = φ ∗ ω n {\displaystyle d\eta =\varphi ^{*}\omega _{n}} . The Hopf invariant is then given by

∫ S 2 n − 1 η ∧ d η . {\displaystyle \int _{S^{2n-1}}\eta \wedge d\eta .}
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 {\displaystyle V} denote a vector space and V ∞ {\displaystyle V^{\infty }} its one-point compactification, i.e. V ≅ R k {\displaystyle V\cong \mathbb {R} ^{k}} and

V ∞ ≅ S k {\displaystyle V^{\infty }\cong S^{k}} for some k {\displaystyle k} .

If ( X , x 0 ) {\displaystyle (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 ∞ {\displaystyle V^{\infty }} , then we can form the wedge products

V ∞ ∧ X . {\displaystyle V^{\infty }\wedge X.}

Now let

F : V ∞ ∧ X → V ∞ ∧ Y {\displaystyle 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 {\displaystyle F} is

h ( F ) ∈ { X , Y ∧ Y } Z 2 , {\displaystyle h(F)\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}},}

an element of the stable Z 2 {\displaystyle \mathbb {Z} _{2}} -equivariant homotopy group of maps from X {\displaystyle X} to Y ∧ Y {\displaystyle Y\wedge Y} . Here "stable" means "stable under suspension", i.e. the direct limit over V {\displaystyle V} (or k {\displaystyle k} , if you will) of the ordinary, equivariant homotopy groups; and the Z 2 {\displaystyle \mathbb {Z} _{2}} -action is the trivial action on X {\displaystyle X} and the flipping of the two factors on Y ∧ Y {\displaystyle Y\wedge Y} . If we let

Δ X : X → X ∧ X {\displaystyle \Delta _{X}\colon X\to X\wedge X}

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

h ( F ) := ( F ∧ F ) ( I ∧ Δ X ) − ( I ∧ Δ Y ) ( I ∧ F ) . {\displaystyle h(F):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).}

This map is initially a map from

V ∞ ∧ V ∞ ∧ X {\displaystyle V^{\infty }\wedge V^{\infty }\wedge X} to V ∞ ∧ V ∞ ∧ Y ∧ Y , {\displaystyle V^{\infty }\wedge V^{\infty }\wedge Y\wedge Y,}

but under the direct limit it becomes the advertised element of the stable homotopy Z 2 {\displaystyle \mathbb {Z} _{2}} -equivariant group of maps. There exists also an unstable version of the Hopf invariant h V ( F ) {\displaystyle h_{V}(F)} , for which one must keep track of the vector space V {\displaystyle V} .


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