Introduction. In [1], [2] and [7] the classification of rank 2 H-spaces was practically completed. The fact that there are only finitely many (homotopy classes of) such spaces is due primarily to the fact that the number of possible mod-2 cohomoloty rings of such spaces is very limited as the Steenrod algebra (as well as higher order operations) act on such rings in a very definite way.

In [4] it was shown that modulo odd primes non classical mod-p H-spaces of rank 2 exist: i. e. there exists an H-space X such that H*(X, ZP) is an exterior algebra on two generators. (Modulo odd primes we consider odd dimensional spheres as classical H-spaces.)

Consequently, the mod odd classification of mod odd rank 2 H-spaces is yet to be considered. Its implication for the classification problem of finite CW complexes is obvious, as the corollary to Theorem A illustrates.

Throughout this paper let n and m be fixed odd integers such that 3 ≤ n ≤ m. As Sn × Sm is a mod odd H-space there are mod odd H-spaces of type (n, m) for any given such pair, and the classification has to be carried out separately for each such pair.