Some Relations between Admissible Monomials for the Polynomial Algebra

Abstract

Let P(n)=F2[x1,…,xn] be the polynomial algebra in n variables xi, of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(n) according to well known rules. A major problem in algebraic topology is of determining A+P(n), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q(n)=P(n)/A+P(n). Q(n) has been explicitly calculated for n=1,2,3,4 but problems remain for n≥5. Both P(n)=⨁d≥0Pd(n) and Q(n) are graded, where Pd(n) denotes the set of homogeneous polynomials of degree d. In this paper, we show that if u=x1m1⋯xn-1mn-1∈Pd′(n-1) is an admissible monomial (i.e., u meets a criterion to be in a certain basis for Q(n-1)), then, for any pair of integers (j,λ), 1≤j≤n, and λ≥0, the monomial hjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1)(n) is admissible. As an application we consider a few cases when n=5.