The existence and uniqueness of simply connected P-compact groups with Weyl groups W such that the order of W is not divisible by the square of P

Changsheng Xu, Purdue University

Abstract

This thesis is centered around the following problem which was posed by N. Steenrod in the 1960's. Steenrod's problem: Determine the polynomial rings $F\sb{p}\lbrack x\sb1,\...,x\sb{n}$) regarded as graded algebras over the Steenrod algebra ${\cal A}\sb{p}$ which can be realized as the mod p cohomology of a space. Assume that p is an odd prime and W is a finite irreducible p-adic reflection group such that $p\sp2 \vert\ \vert W\vert.$ We prove that (1) The ring of invariants $F\sb{p}\lbrack t\sb1,\...,t\sb{n}\rbrack\sp{W}$ is a polynomial algebra. (2) There is a p-compact group $\chi$ such that $H\sp*(B\chi, F\sb{p})\ \cong F\sb{p}\lbrack t\sb1,\...,t\sb{n}\rbrack\sp{W},$ i.e., the Weyl group $W(\chi$) of $\chi$ is isomorphic to W. (3) Zabrodsky's space which realizes the ring of invariants of the general linear group $Gl(2,F\sb3)$ is unique up to homotopy. (It is reasonable to guess that all such spaces in (2) are unique up to homotopy.) Shephard-Todd and Clark-Ewing's classifications of irreducible pseudoreflection groups are used to study the rings of invariants. Theorems from computer algebra are also used to show that the rings of invariants of certain reflection groups are polynomial algebras. For a polynomial algebra of invariants of a modular irreducible p-adic reflection group W with $p\sp2 \vert\ \vert W\vert,$ a special subcategory of Rector's category associated with the algebra are studied and the homotopy colimit structure is then used to construct a p-compact group to realize this polynomial algebra. Dwyer-Wilkerson's theorem about the set of equivalent classes of "centric" topology diagrams enables us to prove that the classifying space of p-compact group which realizes the ring of invariants of $W = Gl(2, F\sb3)$ is unique.

Degree

Ph.D.

Advisors

Wilkerson, Purdue University.

Subject Area

Mathematics

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