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In this paper we consider the deformation theory of differential graded modules (DGM ’s) and differential graded algebras (DGA’s), where only the differential varies, the underlying module or algebra structure remaining fixed. At the outset we consider only individual modules or algebras and afterwards we examine deformations of sheaves. In most respects… (More)

The ring of invariant forms of the full linear group GL(V ) of a finite dimensional vector space over the finite field Fq was computed early in the 20th century by L. E. Dickson [5], and was found to be a graded polynomial algebra on certain generators {cn,i}. This ring of invariants, for q = p, has found use in algebraic topology in work of Milgram–Man… (More)

We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We conjecture that it is also the last one.

The purpose of this paper is to establish criteria for freeness and for the existence of a vanishing line for modules over sub-Hopf algebras of the modp Steenrod algebra. Here, a left module M over a connected K-algebra A is said to have a vanishing fine over A of slope d provided that there exists an intercept -c such that To&K, M) = 0 for all (s, t)… (More)

Remark. If G is a compact Lie group with maximal torus TG and Weyl Group WG, the finitely generated free Z-module LG = π1TG has a natural action of WG and is called the dual weight lattice of G. The group G is simple in the usual sense if and only if the center of G is trivial and the action of WG on Q ⊗ LG gives an irreducible representation of WG over Q.… (More)

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the… (More)

x1. Introduction J. Lannes has introduced and studied a remarkable functor T L1] which takes an unstable module (or algebra) over the Steenrod algebra to another object of the same type. This functor has played an important role in several proofs of the generalized Sullivan Conjecture L1] L2] DMN] and has led to homotopical rigidity theorems for classifying… (More)

One of the major problems in the homotopy theory of finite loop spaces is the classification problem for p-compact groups. It has been proposed to use the maximal torus normalizer (which at an odd prime essentially means the Weyl group) as the distinguishing invariant. We show here that the maximal torus normalizer does indeed classify many p-compact groups… (More)

The notion of p-compact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of p-compact groups, one for each prime number p. A key feature of the theory of compact Lie groups is the relationship between centers and… (More)

Suppose that G is a connected compact Lie group. We show that simple numerical information about the Weyl group of G can be used to obtain bounds, often sharp, on the size of the center of G. These bounds are obtained with the help of certain Coxeter elements in the Weyl group. Variants of the method use generalized Coxeter elements and apply to p-compact… (More)