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Realization and nonrealization of Poincaré duality quotients of 𝔽₂[x,y] as topological spaces

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Meyer D., Smith L.

Fundamenta Mathematicae

2003

tom 177

nr 3

241-250

Let $d_{2,0} = x²y + xy², d_{2,1} = x² + xy + y² ∈ 𝔽₂[x, y]$ be the two Dickson polynomials. If a and b are positive integers, the ideal $(d_{2,0}^a,d_{2,1}^b) ⊂ 𝔽₂[x,y]$ is invariant under the action of the mod 2 Steenrod algebra 𝒜* if and only if when we write $b = 2^t · k$ with k odd, then $a ≤ 2^t$. The quotient algebra $𝔽₂[x,y]/(d_{2,0}^a,d_{2,1}^b)$ is a Poincaré duality algebra and for such a and b admits an unstable action of 𝒜*. It has trivial Wu classes if and only if $a = 2^t$ for some t ≥ 0 and $b = 2^t(2^s - 1)$ for some s > 0. We ask under what conditions on a and b, 𝔽₂[x,y]/(d_{2,0}^a,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 $𝔽₂[x,y]/(d_{2,0}^{2^t},d_{2,1}^{2^t})$ if and only if t = 0, 1, 2, or 3, and in these cases the space may be taken to be a smooth manifold.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Meyer D.

1 Smith L.

1 2003

Wyniki wyszukiwania

Realization and nonrealization of Poincaré duality quotients of 𝔽₂[x,y] as topological spaces