Topological realization of a family of pseudoreflection groups

Dietrich Notbohm¹

¹

Fundamenta Mathematicae, Tome 155 (1998) no. 1, pp. 1-31

Résumé

We are interested in a topological realization of a family of pseudoreflection groups $G ⊂ GL(n,{\sym F}_p )$; i.e. we are looking for topological spaces whose mod-p cohomology is isomorphic to the ring of invariants ${\sym F}_p [x_1,..., x_n]^G$. Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over ${\sym F}_p $ can appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product $ℤ/q ≀Σ_n$ where q divides p - 1 and where p is odd. Let G be such a subgroup acting on the polynomial algebra $A:= {\sym F}_p [x_1,..., x_n]$. We show that there exists a space X such that $H*(X;{\sym F}_p )≅ A^G$ which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod-p cohomology of the classifying spaces of special orthogonal groups or of symplectic groups.  The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group G. Using this construction we also show that the homotopy type of the p-adic completion of these spaces is completely determined by the mod-p cohomology considered as an algebra over the Steenrod algebra. Moreover, we calculate the set of homotopy classes of self maps of the completed spaces.

Détail

DOI : 10.4064/fm-155-1-1-31

Keywords: pseudoreflection groups, p-compact group, classifying spaces, compact Lie group, polynomial algebra