Hybrid spaces with interesting cohomology

Lesh, Kathryn

doi:10.1090/S0002-9947-1995-1316857-9

Hybrid spaces with interesting cohomology
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by Kathryn Lesh

Trans. Amer. Math. Soc. 347 (1995), 3247-3262

DOI: https://doi.org/10.1090/S0002-9947-1995-1316857-9

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Abstract:

Let $p$ be an odd prime, and let $R$ be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to $p$. To such an algebra is associated a $p$-adic pseudoreflection group $W$, and we assume that $W$ is of order prime to $p$ and irreducible. Adjoin to $R$ a one-dimensional element $z$, and give $R[z]$ an action of the Steenrod algebra by $\beta z = 0$ and $\beta x = (\left | x \right |/2)zx$ for an even dimensional element $x$. We show that the subalgebra of elements of $R[z]$ consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a $p$-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson.

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