A few examples of $R-$good and $R-$bad classifying spaces

TL;DR

This paper explores the properties of classifying spaces related to a commutative ring R, highlighting the limitations of their 'good' and 'bad' classifications beyond prime-based completions.

Contribution

It provides examples illustrating the constraints on classifying spaces being simultaneously 'good' and 'bad' for arbitrary rings R.

Findings

Spaces cannot be arbitrarily good or bad for a given ring R.

The contrast with Bousfield-Kan completion at a prime is emphasized.

Examples demonstrate the specific behavior of classifying spaces in this context.

Abstract

For a commutative ring $R$, in contrast to the completion in the sense of Bousfield and Kan at just a prime integer, there cannot exist spaces which are good and bad in an arbitrary way.

Full text

1 Introduction

In general, it is a very hard question to decide whether a space is good or bad in the sense of Bousfield and Kan [Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1972] and there are not many explicit examples in the literature. In this note we give results about $R-$completed classifying spaces for various solid rings $R$. In particular, we show that spaces cannot be good and bad at different solid rings in any arbitrary combination. The special case for primes only was a question raised during the talk of Robert Oliver on $p-$local homotopy theory at the MSRI Berkeley at the beginning of the algebraic topology program in 2014 concerning the possible compatibilty of completions at two different primes at the same time and was answered positively by the author in [17] who was supported by a postdoctoral fellowship at the University of Haifa under the direction of Professor David Blanc. I would like to thank him and Emmanuel Dror Farjoun for discussions on the topic.

2 Preliminaries

The $p$-completion $X^{\wedge}_{p}$ of a topological space $X$ in the sense of Bousfield-Kan [4] is a special case of the $R-$completion functor of Bousfield and Kan which is defined for solid rings $R$. This is a functor from the category of simplicial sets to itself. The category of simplicial sets will be denoted by spaces thoughout this article. A ring $R$ is solid if the map $R\otimes_{\mathbb{Z}}R\rightarrow R$ is an isomorphism. The solid rings are $\mathbb{Z}/n\mathbb{Z}$, subrings of the rationals $\mathbb{Z}[J^{-1}]$ for any set $J$ of primes, the product rings $\mathbb{Z}[J^{-1}]\times\mathbb{Z}/n\mathbb{Z}$ where each prime factor of $n$ is in $J$, and direct limits of these three types of rings [5]. The Bousfield-Kan completion is related to completions and localizations in the sense of Malcev [13], Sullivan [18], and Quillen [16]. In $p-$local homotopy theory the classifying space $|\mathcal{L}|^{\wedge}_{p}$ of a $p-$local finite group $(S,\mathcal{F},\mathcal{L})$ is one of the main objects of study where $(-)^{\wedge}_{p}$ denotes the $p-$completion functor with respect to the ring $\mathbb{Z}/p\mathbb{Z}$. A space X is called $R-$complete if the map $X\rightarrow X^{\wedge}_{R}$ is a mod $R-$equivalence and $R-$good if the natural map $(H_{*};R)\rightarrow H_{*}(X^{\wedge}_{R};R)$ is an isomorphism. Otherwise the space is called $R$-bad. Examples of spaces which are good for all primes and the integers are classifying spaces of finite groups. Spaces with finite homotopy groups are good for all solid rings $R$. A space $X$ is called $p-$good for a prime $p$ if it is good for the ring $\mathbb{Z}/p\mathbb{Z}$. For spaces with finite homotopy groups or finite homology groups the $\mathbb{Z}-$completion is up to homotopy, the product of the $\mathbb{Z}/p\mathbb{Z}-$completions. The space $\mathbb{R}P^{2}$ is good for all primes and bad for $\mathbb{Z}_{(J)}$ as long as $2\in J$, and bad for the integers [4]. Recall that the space $S^{1}\vee S^{1}$ is bad for all primes and the integers [3]. A group $\pi$ acts on a group $G$ if there is given a homomorphism $\alpha:\pi\rightarrow Aut(G)$. Such an action is called nilpotent if there exists a finite sequence of subgroups of $G$: $G=G_{1}\supset\cdots\supset G_{j}\supset\cdots\supset G_{n}=1$ such that for each $j$ we have $G_{j}$ is closed under the action of $\pi$, $G_{j+1}$ is normal in $G_{j}$, and $G_{j}/G_{j+1}$ is abelian, and the induced action on $G_{j}/G_{j+1}$ is trivial.

A (possibly infinite) group $G$ has a Sylow $p-$subgroup $S$ if it has a $p-$subgroup isomorphic to $S$ and all other subgroups of $G$ whose order is a power of $p$ are subconjugate to $S$. Examples of infinite groups with finite Sylow $p-$subgroups are group models for fusion systems in the sense of Robinson and Leary-Stancu and ourselves together with Leip and ourselves together with Libman. Subgroups of amalgamated products are described by Kurosh’s subgroup Theorem. The most general version states that [15] if $H$ is a subgroup of the free amalgamated product of groups $\underset{i\in I}{*}G_{i}=G$, then $H=F(X)*(*_{j\in J}g_{j}H_{j}g_{j}^{-1})$, where $X$ is a subset of $G$ and $J$ is some index set and $g_{j}\in G$ and each $H_{j}$ is a subgroup of some $G_{i}$. Recall that for any pair of groups $G,H$ we have a weak equivalence of classifying spaces $B(G*H)\simeq BG\vee BH$, [15]. The mod $R-$Fibre Lemma [4, Lemma 5.1] states that the $R$-completion preserves fibrations of connected spaces up to homotopy $F\rightarrow E\rightarrow B$ for which the fundamental group of the base space $\pi_{1}(B)$ acts nilpotently on every reduced homology group of the fibre with coefficients in the ring of the completion $\overline{H}_{i}(F;R)$. It is used to show that spaces are good or bad without constructing the completion tower. A group $G$ is virtually finite if it has a free subgroup $H<G$ of finite index. A group $G$ is a virtual finite $p$-group if it has a free subgroup of index a power of $p$. A space is nilpotent if the fundamental group acts nilpotently on all higher homotopy groups [4]. For a ring $R$ a group $G$ is $R-$nilpotent if it has a finite central series $G=G_{1}\subset\cdots G_{j}\cdots G_{n}=1$ such that each quotient $G_{j}/G_{j+1}$ admit an $R$-module structure.A map $f:\{G_{s}\}\rightarrow\{H_{s}\}$ between two towers of groups is a pro-isomorphism if, for every group $B$, it induces an isomorphism $\underset{\leftarrow}{lim}\text{ }Hom_{(groups)}(H_{s},B)\cong\underset{\leftarrow}{lim}\text{ }Hom_{(groups)}(G_{s},B)$. The completion tower $R_{n}X$defined by Bousfield and Kan preserves the homology with $R$-coefficients [9]. This property of preserving $R$-homology characterizes the tower completely [4]. For a ring $R$ and a space $X$ denote $E^{R}X$ the $HR-$localization functor defined in [2]. While it is possible to prove that a space is bad for prime numbers using the Fibre Lemma, this is not possible for arbitrary solid rings as we showed in [17].

The following proves that in contrast to the case of primes in the general case a space cannot be both $R$-good and $R^{\prime}$-bad for arbitrary solid rings $R$ and $R^{\prime}$ at the same time.

Theorem 2.1

Let either

• •

$R=\mathbb{Z}/m\mathbb{Z}$* and $R^{\prime}=\mathbb{Z}/n\mathbb{Z}$, where $m$ and $n$ are integers such that $m$ and $n$ are coprime, or*

• •

$R=\mathbb{Z}[J^{-1}]$* and $R^{\prime}=\mathbb{Z}/n\mathbb{Z}$, where $n$ is an integer all of whose prime factors are in $J$.*

Let $X$ be a space which is good for the ring $R$ and bad for the ring $R^{\prime}$. Then $X$ is bad for $R\times R^{\prime}$.

Proof: We have from [4, Proposition 9.5] that $X\widehat{{}_{R\times R^{\prime}}}\simeq X^{\wedge}_{R}\times X^{\wedge}_{R^{\prime}}$. The result then follows from the Künneth Theorem.$\Box$

Remark 2.2

Let $p$ be a prime and $X$ a $p-$good space. Then $X$ can be both good or bad for $\mathbb{Z}_{(p)}$.

Theorem 2.3** ([11], Corollary 9.2.)**

For a finitely presented metabelian group $X$, the natural map $E^{R}X\rightarrow\hat{X}_{R}$ is an isomorphism for $R=\mathbb{Q}$ or $\mathbb{Z}/n\mathbb{Z}$.

Remark 2.4** ([11], Theorem 9.1.)**

The above result cannot be generalized to the case $R=\mathbb{Z}$.

Remark 2.5

Recall that $\mathbb{R}P^{\infty}\simeq BC_{2}$ is good for the integers and all primes but bad for $\mathbb{Z}_{(2)}$.

Remark 2.6

The space $\mathbb{R}P^{2}$ is bad for the prime $2$ and good for all other primes because the reduced homology $\overline{H}(\mathbb{R}P^{2};\mathbb{F}_{p})$ is trivial for $p$ odd.

Theorem 2.7

Let $R$ be a nonzero solid ring. The classifying space $S^{1}\vee S^{1}\simeq B(\mathbb{Z}*\mathbb{Z})$ is $R$-bad.

Proof: Assume the contrary that $S^{1}\vee S^{1}$ is $R-$good. Then any finite wedge is $R-$good as well. Recall that $(S^{1}\vee S^{1})^{\wedge}_{R}\simeq(B(\mathbb{Z}*\mathbb{Z}))^{\wedge}_{R}\simeq B((\mathbb{Z}*\mathbb{Z})^{\wedge}_{R})$ in this situation because $\mathbb{Z}*\mathbb{Z}$ has finite virtual cohomological dimension. This is a contradiction in low dimensions because through the group completion we have that $(\mathbb{Z}*\mathbb{Z})^{\wedge}_{R}$ is not finitely generated in low degrees as proved in [2].$\Box$