Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres. In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map 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 is equal to 1, for any x ≠ y ∈ S 2 {displaystyle x eq yin 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 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 phi colon S^{2n-1} o S^{n}} be a continuous map (assume n > 1 {displaystyle n>1} ). Then we can form the cell complex 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 phi } .The cellular chain groups C c e l l ∗ ( C ϕ ) {displaystyle C_{mathrm {cell} }^{*}(C_{phi })} 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