Nilpotence in the Steenrod Algebra

While all of the relations in the Steenrod algebra, A, can be deduced in principle from the Adem relations, in practice, it is extremely difficult to determine whether a given polynomial of elements in A is zero for all but the most elementary cases. In his original paper [Mi] Milnor states “It would be interesting to discover a complete set of relations between the given generators of A”. In particular Milnor shows that every positive dimensional element of A is nilpotent. Thus it would be desirable to find a simple closed form for nilpotence relations in A. Let x ∈ A. We say that x has nilpotence k, if x = 0 and xk−1 6= 0 (take x = 1). In this case we write Nil(x) = k. In this paper we investigate Nil(x) for several infinite families of Milnor basis elements of A at the prime 2. The paper is organized as follows. First, an infinite family of subalgebras and isomorphisms between them are constructed. The isomorphisms are used to produce infinite families of elements having the same nilpotence. Next, we compute strong upper and lower bounds for the nilpotence of Milnor basis elements in these subalgebras. Comparing these bounds and extending to the families produced via the isomorphisms shows that Sq(2(2 − 1) + 1) has nilpotence k + 2. Finally a strong lower bound for the nilpotence of P s t is computed for all s, t ∈ N. The main results are stated and discussed in Sections II and III. Detailed proofs are presented in Section IV.