Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

摘要

Let ${P}(n) ={F}[x_1,ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $F$ 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 that 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 ${Q}(n) = {P}(n)/A^{+}P(n)$. Both ${P }(n) =igoplus_{d geq 0} {P}^{d}(n)$ and ${Q}(n)$ are graded, where ${P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${Q}(n)$ in degrees less than or equal? to $12.$