A new proof of the Frobenious conjecture on the dimensions of real algebras without zero dividers

A new way to prove the Frobenious conjecture on the dimensions of real algebras without zero dividers is given in the present paper. Firstly, the proof of nonexistence of real algebras without zero dividers in all dimensions except 1,2,4 and 8 was given in [1]. It was based on the simplicial cohomology operation technique. Later on methods of K-theory cohomology operations gave one a possibility to obtain a more simple proof of the Frobenious conjecture (see [2]). For proving the Frobenious conjecture we suggest a new approach different from [1, 2]. We demonstrate that the restriction on the dimensions of real algebras without zero dividers 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 Frobenious 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 Frobenious conjecture to the question of parallelizibility of real projective spaces. Finally, we obtain exact dimensions of real algebras without zero dividers. Let ξ n be one-dimensional real Hopf’s vector bundle over RP , and let ξ n be its orthogonal complement. Denote one-dimensional Hopf’s complex vector bundle over CP by η n. We give the simplest calculation of K (RP). Our method to calculate K-functor of RP is based on the following geometric observation. Complex Hopf’s bundle π : S → CP with fiber S can be passed through real Hopf’s bundle π1 : S → RP 2n+1 with fiber Z2. Under these conditions we obtain the bundle π2 : RP 2n+1 → CP whose fiber is also a circle. More over, the following theorem holds. Theorem 1. The bundle π2 : RP 2n+1 → CP 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 ×ρ S, where ρ : S × S → S is defined by the formula ρ(u, v) = uv. Construct the map g : RP 2n+1 → S(η) supposing g(x1 : x2 : ... : x2n+1 : x2n+2) = ((w̄z1, ..., w̄zn+1), w )