Abstract
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 ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ 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 $12.$
Highlights
For n ≥ 1, let P(n) be the mod-2 cohomology group of the n-fold product of RP∞ with itself
A major problem in algebraic topology is that of determining A+P(n), the image problem of determining a basis for the of the action of the positively quotient vector space Q(n) =
The mod-2 Steenrod algebra A is the graded associative algebra generated over F2 by symbols S qi for i ≥ 0, called Steenrod squares subject to the Adem relations (Adem 1957) and S q0 = 1
Summary
For n ≥ 1, let P(n) be the mod-2 cohomology group of the n-fold product of RP∞ with itself. Let Pd(n) denote the homogeneous polynomials of degree d. The problem of determining A+P(n) is called the hit problem and has been studied by several authors, (Silverman, 1998; Singer, 1991) and (Wood, 1989). Let u ∈ P(n) be a monomial of degree d. When we write u ∈ Qd(n) we mean that u is an admissible monomial of degree d. We are in a position to define an order relation on monomials. Q(n) is known for , 1 ≤ n ≤ 4, and in some cases when n = 5 This gives rise to the following explicit formula for Qd(n) for d ≤ 5.
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