A new proof of the Frobenius conjecture on the dimensions of real algebras without zero divisors

A new way to prove the Frobenius conjecture on the dimensions of real algebras without zero divisors is given in the present paper. Firstly, the proof of nonexistence of real algebras without zero divisors in all dimensions except 1,2,4 and 8 was given in [1]. It was based on the simplicial cohomology operation technique. Later on the methods of K-theory cohomology operations gave one a possibility to obtain a more simple proof of the Frobenius conjecture (see [2]). For proving the Frobenius conjecture we suggest a new approach different from [1, 2]. We demonstrate that the restriction on the dimensions of real algebras without zero divisors follows elementary from the structure of K-functors of real projective spaces. The general idea is to use K-theory characteristic classes for investigation the question of parallelizibility of real projective spaces that is equivalent to the Frobenius conjecture(see, e.g. [3]). Simplification of this scheme lies in the basis of our proof. The structure of this paper is as follows. We begin with the calculation of K(RP , ∅). Then we give without proof the reduction of the Frobenius conjecture to the question of parallelizibility of real projective spaces. Finally, we obtain exact dimensions of real algebras without zero divisors. Let ξ n be the one-dimensional real Hopf vector bundle over RP , and let ξ n be its orthogonal complement. Denote the one-dimensional Hopf complex vector bundle over CP n by η n. We give the simplest calculation of K (RP ). Our method to calculate K-functor of RP n is based on the following geometric observation. The complex Hopf bundle π : S → CP n with fiber S can be passed through the real Hopf bundle π1 : S 2n+1 → RP 2n+1 with fiber Z2. Under these conditions we obtain the bundle π2 : RP 2n+1 → CP n whose fiber is also a circle. More over, the following theorem holds. Theorem 1. The bundle π2 : RP 2n+1 → CP n is isomorphic to the spherical bundle of the tensor square of the bundle η n and also π ∗ 2η 1 n ∼= C ⊗ ξ 2n+1. Proof. Denote the tensor square of the bundle η n by η . Let us construct an equivariant with respect to S-action homeomorphism g of the spaces RP 2n+1 and S(η). Observe that S(η) ∼= S×ρS , where ρ : S×S → S is defined by the formula ρ(u, v) = uv. Construct the map g : RP 2n+1 → S(η) supposing