In our earlier paper [7], we defined and studied a certain class of spaces which in many ways behave like p ‐completed classifying spaces of finite groups. These spaces occur as “classifying spaces” of certain algebraic objects called p ‐local finite groups. The purpose of this paper is to generalize the concept of p ‐local finite groups to what we call p ‐local compact groups. The motivation for introducing this family comes from the observation that p ‐completed classifying spaces of finite and compact Lie groups, as well as classifying spaces of p ‐compact groups (see Dwyer and Wilkerson [11]), share many similar homotopy theoretic properties, but earlier studies of these properties usually required different techniques for each case. Moreover, while p ‐completed classifying spaces of finite and, more generally, compact Lie groups arise from the algebraic and geometric structure of the groups in question, p ‐compact groups are purely homotopy theoretic objects. Unfortunately, many of the techniques used in the study of p ‐compact groups fail for p ‐completed classifying spaces of general compact Lie groups. With the approach presented here, we propose a framework general enough to include p ‐completed classifying spaces of arbitrary compact Lie groups as well as p ‐compact groups. The new idea here is to replace fusion systems over finite p ‐groups, as handled in [7], by fusion systems over discrete p ‐toral groups. A discrete p ‐toral group is a group which contains a discrete p ‐torus (a group of the form .Z=p 1 / r for finite r 0) as a normal subgroup of p ‐power index. A p ‐local compact group consists of a triple

/pdf/discrete-models-for-the-p-local-homotopy-theory-of-compact-2ns6r8wews.pdf

Infinite Abelian groups

Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups

In this paper we construct symmetric powers in the motivic homotopy categories of morphisms and finite correspondences associated with f-admissible subcategories in the categories of schemes of finite type over a field. Using this construction we provide a description of the motivic Eilenberg-MacLane spaces representing motivic cohomology on some f-admissible categories including the category of semi-normal quasi-projective schemes and, over fields which admit resolution of singularities, on some admissible subcategories including the category of smooth schemes. This description is then used to give a complete computation of the algebra of bistable motivic cohomological operations on smooth schemes over fields of characteristic zero and to obtain partial results on unstable operations which are required for the proof of the Bloch-Kato conjecture.

/pdf/motivic-eilenberg-maclane-spaces-3hkfdcto2g.pdf

Motivic Eilenberg-MacLane spaces

Every cotorsion-free algebra is an endomorphism algebra

A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3-manifolds to the more general class of compact 3-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a 3-manifold is a subgroup of the additive group of rationals. (1) Introduction. This paper is primarily concerned with the classification of those solvable groups which can occur as the fundamental group of a compact 3-manifold. We also consider the problem of determining the structure of a compact 3-manifold whose fundamental group is known to be solvable. Our results are complete except in the category of almost sufficiently large 3-manifolds and the category of 3-manifolds whose nontrivial second homotopy group is generated by projective plane boundary components. If M is a compact, sufficiently large 3-manifold with trivial second homotopy group, and if ir^M) is solvable, then ^(M) appears in the following list of groups: (1) Z,Z©Z, or Jf, the fundamental group of the Klein bottle, (2) an extension 1 -> A -> iti(M) -> Z -> 1 where A is either Z © Z or ^T, (3) a free product of two copies of JT amalgamated along certain subgroups isomorphic with Z © Z. These groups may be presented by (a, b, x, y \\ bab~1 = a~1, yxy~1 = x~1, a = xpy2q, b2 = xTy2s) where//, q, r, s are integers such that//s — rq= ± 1. Further, the above list is complete. That is, for each group G listed above, there is a compact sufficiently large 3-manifold M with n2(M) = 0, and tt1(M)~G. If the restrictions that M be sufficiently large and that ir2(M) = 0 are dropped, then further groups must be added to the list. Such groups are discussed in detail in §§3, 4, 5, and 6. C. Thomas [16] has listed those nilpotent groups which can act as the fundamental group of a closed 3-manifold. Making use of the information we gain Received by the editors February 2, 1971. AMS 1970 subject classifications. Primary 57A10, 55A05; Secondary 20E15, 20K45.

/pdf/solvable-fundamental-groups-of-compact-3-manifolds-1yoofmwe56.pdf

Solvable fundamental groups of compact 3-manifolds

The purpose of this paper is to address several problems posed by V. I. Kuzminov [Ku] regarding cohomological dimension of noncompact spaces. In particular, we prove the following results: Theorem A. Suppose X is metrizable and G is the direct limit of the direct system {G s , h s',s , S} of abelian groups. Then, dim G X ≤ max {dim Gs X|s ∈ S}. Theorem B. Let X be a metrizable space and let G be an abelian group. Let l = {p\p.(G/Tor G) ¬= G/TorG}. (a) If G = Tor G, then dim G X = max {dim H X\H ∈ σ(G)}, (b) dim G X = max {dim TorG X, dim G/Tor G X}, (c) dim G X ≥ dim Q X if G ¬= Tor G, (d) dim G X ≥ dim Zl X, where Z l is the group of l-adic integers, (e) max(dim G X, dim Q X + 1) ≥ max {dim H X\H ∈ σ (G)}, (f) dim G X ≤ dim Zl X ≤ dim G X + 1 if G ¬= 0 is torsion-free

/pdf/cohomological-dimension-and-metrizable-spaces-dvpx9ufdi0.pdf

Infinite Abelian groups

Citations

Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups

Motivic Eilenberg-MacLane spaces

Every cotorsion-free algebra is an endomorphism algebra

Solvable fundamental groups of compact 3-manifolds

Cohomological dimension and metrizable spaces

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