ALGEBRAIC π₂(S²)

Abstract

For the algebraic sphere S² defined as the zero set of the equation x² ₁ + x² ₂ + x² ₃ = 1 in C³, we see that there are polynomial maps f_n: S² → S² of each degree. For the algebraic sphere S² defined as the suspension of S¹ = C-(0) in the category AFF_C of based affine schemes of countable type over C, the set Hom_AFFC(S², S²) modulo homotopy is seen, using standard definitions adapted to our algebraic situation, to be (ZxZ)/ψ where ψ collapses (Zx0) ∪ (0xZ) to a point.