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