Resumen de The structure of p-local compact groups
The importance of Lie groups in all areas of mathematics is indisputable, They have been studied from several points of view, and the richness of their structure makes them to be present not only in all areas of mathematics, but also in other areas of science.
The theory of Lie groups was completely developed with the classification of all simple connected compact Lie groups, which cannot be attributed to a single mathematician but was achieved thanks to the contributions of several people, like \'E. Cartan, W. Killing, A. Borel, E. B. Dynkin and H. S. M. Coxeter among others.
As algebraic topologists, the natural step to take from that point was then to develop models isolating some of the topological properties of compact Lie groups. There is not a single way to proceed, and several models could be defined, depending on the properties we want to study.
Our particular interest then lies on the topological properties of p-completions of classifying spaces of compact Lie groups. Roughly speaking, by p-completing a space, we isolate the information that we may obtain from the cohomology of the space at the prime p from other information that may make the picture more difficult to understand.
From this point of view, a first generalization was introduced in [Dwyer-Wilkerson, "Homotopy fixed-point methods for Lie groups and finite loop spaces"]: the well-known p-compact groups. In fact, this model does not generalizes all compact Lie groups, but only those compact Lie groups whose group of components is a p-group. This new class of spaces was completely classified in [Andersen-Grodal-Møller-Viruel, "The classification of p-compact groups for p odd"] at the prime p odd, and in [Andersen-Grodal, "The classification of 2-compact groups"] and [Møller, "N-determined 2-compact groups"] at the prime p = 2.
More recently, p-local compact groups were introduced in [Broto-Levi-Oliver, "Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups"] as such a more general model. It has been proved in that paper that indeed p-local compact groups include all (p-completions of classifying spaces of) compact Lie groups, as well as all p-compact groups. However, the more general the model is, the more difficult its study becomes. In this sense, we are far from describing all p-local compact groups in terms of a (smaller) well-understood list of p-local compact groups, and several basic properties of them have to be ''explored'' before this can be done.
In this work, then, we try to cover some gaps in the newborn theory of p-local compact groups, such as the definition and (some) properties of connected p-local compact groups, and the mod p cohomology rings of p-local compact groups. While the hopes are high that most of the constructions and results in this work will sooner or later be proved to hold in the general case, the theory of p-local compact groups is rather evasive, and it will not be an easy task.
Below we summarize briefly the work done in this memory, and we refer the reader to each chapter for further details on a specific subject.
The first chapter introduces the notion of a p-local compact group as a triple G = (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S (roughly speaking, a category whose objects are the subgroups of S and such that the morphisms among objects are actual group monomorphisms satisfying some set of axioms(I), (II) and (III)), and where L is a centric linking system associated to F (roughly speaking, a category whose objects are the F-centric subgroups of S, and such that the morphism sets are extensions of the corresponding morphism sets in F, again satisfying a set of axioms (A), (B) and (C)). We also summarize all the results from [Broto-Levi-Oliver, "Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups"] that we will use at some point in this memory. We complement the definitions and results from this source with some useful properties of p-local compact groups which do not appear in [Broto-Levi-Oliver, "Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups"], like the fact that the normalizer fusion subsystem of a fully normalized subgroup is always saturated (a feature which was proved to hold for p-local finite groups in [Broto-Levi-Oliver, "The homotopy theory of fusion systems"]), and an equivalent set of axioms for the saturation of a fusion system, inspired in the paper [Kessar-Stancu, "A reduction theorem for fusion systems of block"].
Each of the following chapters introduces original work on p-local compact groups and saturated fusion systems.
The second chapter studies the realization of saturated fusion systems by infinite groups, in the same way as this was done for finite fusion systems in [Robinson, "Amalgams, blocks, weights, fusion systems and finite simple groups"], that is, using amalgams of locally finite artinian groups. Solving this problem in particular requires extending the results on constrained saturated fusion systems done in section 4 of [Broto-Castellana-Grodal-Levi-Oliver, "Subgroups families controlling p-local finite groups"], which turns out not to be difficult in the exercise in the case of saturated fusion systems over discrete p-toral groups, thanks to the results on higher limits developed in [Broto-Levi-Oliver, "Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups"]. While we were originally planning to apply the results on realization of saturated fusion systems to avoid assuming the existence of a centric linking system, in general the groups one obtains using the results in chapter 2 are rather difficult to work with (this is not at all surprising, in view of the results in [Robinson, "Amalgams, blocks, weights, fusion systems and finite simple groups"], where it is already shown that in the case of finite fusion systems one already will find the resulting groups difficult to work with).
The third chapter introduces a notion of connectivity of saturated fusion systems and p-local compact groups, and studies the existence of connected components in the case the group S has rank 1. In this sense, we first give a list of all connected p-local compact groups of rank 1, and also show that each connected fusion system over a rank 1 p-local compact group has a unique associated linking system. This list just confirms what was initially expected, i.e., that rank 1 connected p-local compact groups correspond to connected compact Lie groups of rank 1 (that is, S^1, SO(3) and S^3). Note that the same was proved to happen when classifying p-compact groups.
In addition to this list, we also prove that each saturated fusion system over a rank 1 discrete p-toral group determined a unique connected saturated fusion subsystem over a discrete p-toral subgroup of rank 1, which can be then considered as the connected component of the original fusion system. The corresponding result on rank 1 p-local compact groups is also proved, although needs a better explaining. In this case, such a p-local compact group G determines a unique connected rank 1 p-local compact group G_0, which we think of as the connected component of G. However, to properly consider G_0 as a p-local compact subgroup of G, some kind of inclusion, at least at the level of linking systems, had to be constructed. Such an inclusion functor has indeed been constructed, but in a rather \textit{ad hoc} way. This inclusion, in fact, provides an example of a morphism (i.e., functor) between linking systems which does not send centric subgroups to centric subgroups. More examples of such functors are provided in chapter 5. It remains to be solved the problem of a proper classification of all rank 1 p-local compact groups in terms of the list of connected rank 1 p-local compact groups that we provide.
The fourth chapter introduces unstable Adams operations for p-local compact groups. The first two sections in this chapter contain the construction of such operations on p-local compact groups originally done in [Junod, "Unstable Adams operations on p-local compact groups"], and the third section uses this ideas to construct unstable Adams operations on the groups realizing saturated fusion systems which we previously studied in chapter 2. There is in fact not much difficulty in doing so, since both the construction of such groups and the construction of unstable Adams operations in [Junod, "Unstable Adams operations on p-local compact groups"] are somehow similar.
The fifth chapter studies the action of unstable Adams operations on a given p-local compact group, and is probably one of the most important chapters in this memory, together with chapter 3. More concretely, we study the fixed points of a given p-local compact group G under the action of families of unstable Adams operations {\Psi_i\}.
This problem can be approached from different points of view, and we have chosen an algebraic way of treating the problem. Thus, we propose a definition of the invariants of G under the action of each \Psi_i which is not the obvious one (since it seems not to work in general) and we prove that there always exists some M such that, for all i > M, we obtain triples G_i = (S_i, F_i, L_i) which are almost p-local finite groups. However, the last condition to show in order to prove that the G_i are p-local finite groups is rather technical and quite difficult to prove in general.
Nevertheless, we prove that indeed the G_i are p-local finite groups for some specific situations, the most important being when G is a rank 1 p-local compact group.
This study allows us to conclude, for instance, that the classifying space of G is the (p-completion of the) homotopy colimit of the classifying spaces of the G_i, which in fact come equipped with inclusions G_i ----> G_{i+1}. In particular, when the G_i are proved to be p-local finite groups, this again provides examples of functors between linking systems which do not send centric subgroups to centric subgroups. Furthermore, if the G_i are p-local finite groups, then one can prove a version of the Stable Elements theorem for p-local compact groups, using the same result on p-local finite groups (proved in [Broto-Levi-Oliver, "The homotopy theory of fusion systems"]).
The two appendices at the end contain technical results needed all along this memory. We have chosen to place them at the end and in different chapters due to the extension of the contents in each of them.
The first appendix deals with extensions of p-local compact groups by discrete p-toral groups, using the more general setting of transporter systems introduced for p-local finite groups in [Oliver-Ventura, "Extensions of linking systems with p-groups kernel"]. In fact, in this chapter we just prove that the results in the former paper extend as expected to the compact case.
The second chapter deals with saturated fusion subsystems of p-power index and index prime to p of a given saturated fusion system. Again, we extend the known results from [Broto-Castellana-Grodal-Levi-Oliver, Extensions of p-local finite groups"] for p-local finite groups to the compact case, while in this case we cannot extend the whole of the contents in the former paper since a better understanding of quasicentric subgroups would be needed first. In particular, we prove the existence of a unique minimal saturated subsystem of index prime to p, a result which can be extended to a result on p-local compact groups, and which is fundamental in the definition of connectivity for p-local compact groups.
