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

Source abstract

Let G > GL ( n ; F p ) G > {\text {GL}}(n;{{\mathbf {F}}_p}) be a p p -group, p p an odd prime, and R ∗ := F p [ x 1 , … , x n ] G {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 ∗ {R^*} is a graded polynomial algebra, where deg ⁡ x 1 = ⋯ = deg ⁡ x n = 2 \deg {x_1} = \cdots = \deg {x_n} = 2 , then there is no space X X such that H ∗ ( X : F p ) ≃ R ∗ {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 ∤ [ G ; 1 ] p\nmid [G;1] .