Clarence W. Wilkerson

Homotopy fixed-point methods for Lie groups and finite loop spaces

A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We will say that a loop space X is finite if the integral homology H*(X, Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem.

Homotopical uniqueness of classifying spaces

If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24] asked in general for a determination of all spaces X such that H∗(X,Fp) is a finitely generated polynomial algebra (i.e., such that X has a polynomial cohomology ring); at that time, the only examples known were spaces of the form X = BG. There has been a lot of subsequent progress on this problem. On one hand, the topological constructions of Sullivan [25, p. 4.28], as exploited by Clark-Ewing [8] and Wilkerson [27], have led to the discovery of exotic spaces X with polynomial cohomology rings. On the other hand, the algebraic arguments of Wilkerson [26] and Adams-Wilkerson [1] have shown in some generality that if X is any space with a polynomial cohomology ring then H∗(X,Fp) must be one of the polynomial algebras listed in [8]. However, there is still a gap here between topology and algebra; in this paper we act to narrow the gap and in some cases to close it. Suppose that p is an odd prime. Let K be the category of unstable algebras [18] over the mod p Steenrod algebra Ap and Kpoly the full subcategory of K consisting of objects which as rings are finitely generated polynomial algebras. If X is a p-complete space with H(X,Fp) ∈ Kpoly we extend the ideas of [1] by associating to X a finite p-adic linear group WX generated by pseudoreflections (see 1.1); given any such finite group W such that p does not divide the order of W, we then show (1.2) that there is up to homotopy exactly one p-complete space X with H∗(X,Fp) ∈ Kpoly and WX = W. In a variety of particular situations (1.3, 1.4) this gives a bijective correspondence between finite group data and homotopy types of p-complete spaces with polynomial cohomology rings.

A new finite loop space at the prime two

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.

A cohomology decomposition theorem

In [9] Jackowski and McClure gave a homotopy decomposition theorem for the classifying space of a compact Lie group G; their theorem states that for any prime p the space BG can be constructed at p as the homotopy direct limit of a specific diagram involving the classifying spaces of centralizers of elementary abelian p-subgroups of G. In this paper we will prove a parallel algebraic decomposition theorem for certain kinds of unstable algebras over the mod p Steenrod algebra. This algebraic result gives a new proof of the theorem of Jackowski and McClure and has the potential to lead to homotopy decompositon theorems for many spaces which are not of the form BG (see §6). Before stating our results we will recall some material from [9]. Choose a prime p. Let G be a compact Lie group, and let AG be the category whose objects are the non-trivial elementary abelian p-subgroups of G; a morphism A → A′ in AG is a homomorphism f : A → A′ of abelian groups with the property that there exists an element g ∈ G such that f(x) = gxg−1 for all x ∈ A. There is a functor from A G to the category of topological spaces which sends A to the Borel construction EG ×G (G/C(A)), where C(A) denotes the centralizer of A in G. (Notice that this Borel construction has the homotopy type of the classifying space BC(A).) Jackowski and McClure prove that the natural map from the homotopy direct limit of this functor to EG×G ∗ = BG is an isomorphism on mod p cohomology. They derive this from a spectral sequence argument [2, XII, 5.8] and the following calculation with the inverse limit functor lim ← and its right derived functors lim ← . Let H∗ denote mod p cohomology and αG the functor on AG which sends A to H∗(EG×G (G/C(A)). Theorem 1.1 [9, Prop. 3–4]. The natural map H∗BG → lim ← αG is an isomorphism and the groups lim ← αG vanish for i > 0. The proof of Theorem 1.1 in [9] uses the Feshbach double coset formula and so depends heavily on the presence of a genuine compact Lie group.

Smith theory and the functorT

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 spaces [DMW1] [DMW2]. In this paper we will use techniques of Smith theory [DW] to calculate the functor T explicitly in certain key special situations (see 1.1 and 1.3). On the one hand, our calculation gives general structural information (1.4) about T itself. On the other hand, up to a convergence question which we will not discuss here our calculation produces a direct analogue of Smith theory (1.2) for actions of elementary abelian p-groups on certain infinite-dimensional complexes; this analogue differs from Smith theory only in that “homotopy fixed point set” is substituted for “fixed point set”. We will now state the main results, which are completely algebraic in nature although they have a geometric motivation. Fix a prime p; the field Fp with p elements will be the coefficient ring for all cohomology. Let Ap denote the mod p Steenrod algebra, and U (resp. K) the category of unstable modules (resp. unstable algebras) over Ap (see [L1]). If R is an object of K, an unstable Ap R module M is by definition an object of U which is also an R module in such a way that the multiplication map R⊗M →M obeys the Cartan formula; we will denote the category of Ap R modules by U(R). An object of U(R) typically arises from a map q : E → B of spaces; in this case the induced cohomology map q∗ makes H∗E an object of U(R) for R = H∗B. Let V be an elementary abelian p-group, ie., a finite-dimensional vector space over Fp, and H V the classifying space cohomology H∗BV . Lannes [L1] has constructed a functor T V : U → U which is left adjoint to the functor given by tensor product (over Fp) with H V and has shown that T V lifts to a functor K → K which is also left adjoint to tensoring

The Elementary Geometric Structure of Compact Lie Groups

We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p -compact groups.

Kahler Differentials, the

Let W be a finite group acting on a finite dimensional vector space V over the field k, U a subset of V , and WU the subgroup of W which fixes U pointwise. Suppose that the algebra Sym(V #)W is a polynomial algebra. Steinberg has shown that if k has characteristic zero the algebra Sym(V #)WU is also a polynomial algebra. We extend this result to the case k = Fp (see also [14]) by proving that Lannes’s functor “T” preserves polynomial algebras. We give examples to show that if k = Fp and W is generated by reflections, it is not necessarily the case that WU is generated by reflections.

T

In the late 1930’s, P. A. Smith began the investigation of the cohomological properties of a group G of prime order p acting by homeomorphisms on a topological space X. This thread has continued now for almost fifty years. Smith was successful in calculating the cohomology of the the fixed point sets for involutions on spheres [Smith 1] and projective spaces [Smith 2]. In the 1950’s, Smith theory was reformulated by the introduction of the Borel construction XG = EG×GX and equivariant cohomology H(XG) = H ∗ G(X). Borel made the key observation [Borel 1] that the cohomology of the fixed point set was closely related to the torsion-free (with respect to H∗ G(pt)) quotient of H ∗ G(X) . In the 1960’s, this was formalized as the “localization theorem” of Borel-Atiyah-Segal-Quillen [Atiyah-Segal],[Quillen]. (The localization theorem is described below.) The localization theorem has previously been used to deduce the actual cohomology of the fixed point set for particular examples in an ad hoc fashion, but a general algorithmic computation of the cohomology of the fixed point set has not been provided in the literature.

-functor, and a Theorem of Steinberg

We determine the groups which can appear as the normalizer of a maximal torus in a connected 2–compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie group, and then showing that this description can be extended to the 2–compact case.

Smith Theory Revisited

We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the pcompletion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.