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The nonrealizability of modular rings of polynomial invariants by the cohomology of a topological space


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 86 (1982), 339-340
MSC: Primary 55R35; Secondary 55S10
DOI: https://doi.org/10.1090/S0002-9939-1982-0667302-4
MathSciNet review: 667302
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Abstract: Let $ G < {\text{GL}}(n;{{\mathbf{F}}_p})$ be a $ p$-group, $ p$ an odd prime, and $ {R^*}: = {{\mathbf{F}}_p}{[{x_1}, \ldots ,{x_n}]^G}$ the ring of invariants. The purpose of this note is to prove that in the case where $ {R^*}$ is a graded polynomial algebra, where $ \deg {x_1} = \cdots = \deg {x_n} = 2$, then there is no space $ X$ such that $ {H^*}(X:{{\mathbf{F}}_p}) \simeq {R^*}$. This complements the work of Clark and Ewing [3] and Adams and Wilkerson [1] on the case $ p\nmid [G;1]$.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0667302-4
Article copyright: © Copyright 1982 American Mathematical Society

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