Some Relations between Admissible Monomials for the Polynomial Algebra

Mbakiso Fix Mothebe,Lafras Uys

doi:10.1155/2015/235806

Abstract

LetP(n)=F2[x1,…,xn]be the polynomial algebra innvariablesxi, of degree one, over the fieldF2of two elements. The mod-2 Steenrod algebraAacts onP(n)according to well known rules. A major problem in algebraic topology is of determiningA+P(n), the image of the action of the positively graded part ofA. We are interested in the related problem of determining a basis for the quotient vector spaceQ(n)=P(n)/A+P(n).Q(n)has been explicitly calculated forn=1,2,3,4but problems remain forn≥5. BothP(n)=⨁d≥0Pd(n)andQ(n)are graded, wherePd(n)denotes the set of homogeneous polynomials of degreed. In this paper, we show that ifu=x1m1⋯xn-1mn-1∈Pd′(n-1)is an admissible monomial (i.e.,umeets a criterion to be in a certain basis forQ(n-1)), then, for any pair of integers (j,λ),1≤j≤n, andλ≥0, the monomialhjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1)(n)is admissible. As an application we consider a few cases whenn=5.

Highlights

For n ≥ 1 let P(n) be the mod-2 cohomology group of the nfold product of RP∞ with itself
The mod-2 Steenrod algebra A is the graded associative algebra generated over F2 by symbols Sqi for Steenrod squares subject to the Adem relations i ≥ 0, [1] and called Sq0 =
While in general it is true that LB(5, d) ≤ dim(Qd(5)), there are cases, even nontrivial, where equality holds

Summary

Introduction

For n ≥ 1 let P(n) be the mod-2 cohomology group of the nfold product of RP∞ with itself. Let u ∈ P(n − 1) be a monomial of degree d󸀠, where α(d󸀠 + n − 1) ≤ n − 1. Given any explicit admissible monomial basis for Q(n − 1) one may compute LB(n, d), the dimension of the subspace of Qd(n) generated by all monomials of the form hjλ(u) ∈ Pd(n), λ ∈ C(n, d) and u ∈ Qd−(2λ−1)(n − 1). While in general it is true that LB(5, d) ≤ dim(Qd(5)), there are cases, even nontrivial, where equality holds This is demonstrated with the aid of known results for dim(Qd(5)) (cited in Table 1) but we will show later that the same conclusions can be reached independently.

Preliminaries

Proof of Theorem 4

Application of Theorem 4