Realization and nonrealization of Poincaré duality quotients of ${\Bbb F}_2[x, y]$ as topological spaces
Volume 177 / 2003
Abstract
Let ${\bf d}_{2,0} = x^2y + xy^2,$ ${\bf d}_{2, 1} = x^2 + xy + y^2 \in {\mathbb F}_2[x, y]$ be the two Dickson polynomials. If $a$ and $b$ are positive integers, the ideal $( {\bf d}_{2,0}^a, {\bf d}_{2,1}^b) \subset {\mathbb F}_2[x, y]$ is invariant under the action of the mod $2$ Steenrod algebra ${\scr A}^{\kern 2pt*}$ if and only if when we write $b = 2^t\cdot k$ with $k$ odd, then $a \leq 2^t$. The quotient algebra ${\mathbb F}_2[x, y]/ ( {\bf d}_{2,0}^a, {\bf d}_{2,1}^b)$ is a Poincaré duality algebra and for such $a$ and $b$ admits an unstable action of ${\scr A}^{\kern 2pt*}$. It has trivial Wu classes if and only if $a=2^t$ for some $t \geq 0$ and $b = 2^t(2^s - 1)$ for some $s > 0$. We ask under what conditions on $a$ and $b$, ${\mathbb F}_2[x, y]/( {\bf d}_{2,0}^a, {\bf d}_{2,1}^b)$ appears as the mod $2$ cohomology of a manifold. In this note we show that for $a = 2^t = b$ there is a topological space whose cohomology is ${\mathbb F}_2[x, y]/( {\bf d}_{2,0}^{2^t}, {\bf d}_{2,1}^{2^t})$ if and only if $t = 0, 1, 2,$ or $3$, and in these cases the space may be taken to be a smooth manifold.
