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Abstract.Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors). The mod-2 Steenrod algebra A acts on Ps according to well-known rules. If A C A denotes the augmentation ideal, then we are interested in determining the image of the action A ® Ps -* Ps: the space of elements in Ps that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [PI] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials.

Communicated by E.M. Friedlander MSC: 55Q10; 55Q45; 55S05; 55S10; 55T15 a b s t r a c tWe examine the dual of the so-called ''hit problem'', the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as a module over the Steenrod Algebra A at the prime 2. The dual problem is to determine the set of A-annihilated elements in homology. The set of A-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each k ≥ 0, the set of k partially A-annihilateds, the set of elements that are annihilated by Sq i for each i ≤ 2 k , itself forms a free associative algebra.

We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard action of † s , the symmetric group on s letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring .P˝s/ † s as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces Z=2˝A .P˝s/ † s . Our main result is stated in terms of a "bigradedSteenrod algebra" H. The generators of this algebra H, like the generators of the classical Steenrod algebra A, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that Sq 0 is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces Z=2˝A .P˝s/ † s , for all s 0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over H.

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