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Register a new userCellular approximations of p-local compact groups
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Let A be the classifying space of an abelian p-torsion group. We compute A-cellular approximations (in the sense of Chacholski and Farjoun) of classifying spaces of p-local compact groups, with special emphasis in the cases which arise from honest compact Lie groups.
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A $p$-local compact group is an algebraic object modelled on the homotopy theory associated with $p$-completed classifying spaces of compact Lie groups and p-compact groups. In particular $p$-local compact groups give a unified framework in which one may study $p$-completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p-compact groups, $p$-local compact groups admit unstable Adams operations - self equivalences that are characterised by their cohomological effect. Unstable Adams operations on $p$-local compact groups were constructed in a previous paper by F. Junod and the authors. In the current paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of operations.
A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this paper we define unstable Adams operations within the theory of p-local compact groups, and show that such operations exist under rather mild conditions. More precisely, we prove that for a given p-local compact group G and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of p-adic units which are congruent to 1 modulo p^m to the group of unstable Adams operations on G
A p-compact group is a mod p homotopy theoretical analogue of a compact Lie group. It is determined the homotopy nilpotency class of a p-compact group having the homotopy type of the $p$-completion of the direct product of spheres.
For a compact Lie group $G$ with maximal torus $T$, Pittie and Smith showed that the flag variety $G/T$ is always a stably framed boundary. We generalize this to the category of $p$-compact groups, where the geometric argument is replaced by a homotopy theoretic argument showing that the class in the stable homotopy groups of spheres represented by $G/T$ is trivial, even $G$-equivariantly. As an application, we consider an unstable construction of a $G$-space mimicking the adjoint representation sphere of $G$ inspired by work of the second author and Kitchloo. This construction stably and $G$-equivariantly splits off its top cell, which is then shown to be a dualizing spectrum for $G$.
The primary goal of this paper is to study Spanier-Whitehead duality in the $K(n)$-local category. One of the key players in the $K(n)$-local category is the Lubin-Tate spectrum $E_n$, whose homotopy groups classify deformations of a formal group law of height $n$, in the implicit characteristic $p$. It is known that $E_n$ is self-dual up to a shift; however, that does not fully take into account the action of the Morava stabilizer group $mathbb{G}_n$, or even its subgroup of automorphisms of the formal group in question. In this paper we find that the $mathbb{G}_n$-equivariant dual of $E_n$ is in fact $E_n$ twisted by a sphere with a non-trivial (when $n>1$) action by $mathbb{G}_n$. This sphere is a dualizing module for the group $mathbb{G}_n$, and we construct and study such an object $I_{mathcal{G}}$ for any compact $p$-adic analytic group $mathcal{G}$. If we restrict the action of $mathcal{G}$ on $I_{mathcal{G}}$ to certain type of small subgroups, we identify $I_{mathcal{G}}$ with a specific representation sphere coming from the Lie algebra of $mathcal{G}$. This is done by a classification of $p$-complete sphere spectra with an action by an elementary abelian $p$-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the $K(n)$-local Spanier-Whitehead duals of $E_n^{hH}$ for select choices of $p$ and $n$ and finite subgroups $H$ of $mathbb{G}_n$.
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