Homotopy commutativity of 𝐻-spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod 3 3 H H -space with finitely generated mod 3 3 cohomology is homotopy equivalent to a finite product of K ( Z , 2 ) K({\mathbb Z},2) , S p ( 2 ) Sp(2) , the three-connected cover S p ( 2 ) ⟨ 3 ⟩ Sp(2)\langle 3\rangle and the homotopy fiber S p ( 2 ) ⟨ 3 ; 3 i ⟩ Sp(2)\langle 3;3^i\rangle of the map [ 3 i ] : S p ( 2 ) → K ( Z , 3 ) [3^i]:Sp(2)\to K({\mathbb Z},3) for i ≥ 1 i\ge 1 . Our result also shows that a connected C p C_p -space in the sense of Sugawara with finitely generated mod