Dimension Result for the Polynomial Algebra as a Module over the Steenrod Algebra

Mbakiso Fix Mothebe

doi:10.1155/2013/150704

Abstract

For let be the polynomial algebra in variables of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well-known rules. Let denote the image of the action of the positively graded part of A major problem is that of determining a basis for the quotient vector space Both and are graded where denotes the set of homogeneous polynomials of degree A spike of degree is a monomial of the form where for each In this paper we show that if and can be expressed in the form with then where is the number of spikes of degree

Highlights

For n > 1 let P(n) be the mod-2 cohomology group of the nfold product of RP∞ with itself
In this paper we show that if n ≥ 2 and d ≥ 1 can be expressed in the form d
P(n) is a module over the mod-2 Steenrod algebra A according to well-known rules

Summary

Introduction

For n > 1 let P(n) be the mod-2 cohomology group of the nfold product of RP∞ with itself. A polynomial u is said to be hit if it belongs to the set. The problem of determining A+P(n) is called the hit problem and has been studied by several authors [1,2,3]. Some of the motivation for studying these problems is mentioned in [6]. It stems from the Peterson conjecture proved in [3] and various other sources [10, 11]. Let u ∈ P(n) be a monomial of degree d. We note that a spike can never appear as a term in a hit polynomial.

Preliminaries

Main Result

Proof of Lemma 9

Proof of Theorem 8