A few examples of $R-$good and $R-$bad classifying spaces
Nora Seeliger
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. %In contrast to the case for completions at primes in the general case spaces cannot be good and bad for arbitrary solid rings in the sense of Bousfield and Kan \cite{BK in arbitrary combination.
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\title{A few examples of $R-$good and $R-$bad classifying spaces}
\author{Nora Seeliger}
\date{}
\begin{document}
\maketitle
\begin{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.
%In contrast to the case for completions at primes in the general case spaces cannot be good and bad for arbitrary solid rings in the sense of Bousfield and Kan \cite{BK} in arbitrary combination.
\end{abstract}
\section{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 [\textit{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 \cite{pgoodpbad} % We continue to illustrate the impact of the existence of a Sylow $p$-subgroup on the homotopy type of the classifying space of a group. The universal property of $R-$badness allows to prove certain spaces not to be retracts of others. This will be shown to work in the case of certain projective spaces. We give an argument that illustrates the limitation of the mod-$R$ fibre Lemma to show that a space is bad for subrings of the rationals and some results relating the $R-$completion and $R$-localisation of a space to various other $R$-homological structures including the structure of its $\mathbb{F}_p-$
%homology structure as an unstable coalgebra over the dual Steenrod algebra and a collection of examples. %We finish with a proof due to Farjoun that the space $S^1\vee S^1$  is bad for all nonzero solid rings.
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.% I would like to thank David Benson, Bill Dwyer, Emmanuel Farjoun, Achim Krause, Robert Oliver.
\section{Preliminaries}
The $p$-completion $X\pcom$ of a topological space $X$ in the sense of Bousfield-Kan \cite{BK} 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 \textbf{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 \cite{BK2}. The Bousfield-Kan completion is related to completions and localizations in the sense of Malcev \cite{Malcev}, Sullivan \cite{Sullivan}, and Quillen \cite{Quillen}.
 In $p-$local homotopy theory the
classifying space  $|\mathcal{L}|\pcom$ of a $p-$local finite group $(S,\mathcal{F},\mathcal{L})$ is one of the main 
objects of study where $(-)\pcom$
denotes the $p-$completion functor with respect to the ring $\mathbb{Z}/p\mathbb{Z}$.  A space X is called \textbf{$R-$complete} if the map $X\rightarrow X^{\wedge}_R$ is a mod $R-$equivalence and  \textbf{$R-$good} if the natural map $(H_*;R)\rightarrow H_*(X^{\wedge}_R ;R)$ is an isomorphism. Otherwise the space is called \textbf{$R$-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 \cite{BK}. Recall that the space $S^1\vee S^1$ is bad for all primes and the integers \cite{Bousfield}.
 A group $\pi$ acts on a group $G$
if there is given a homomorphism
$\alpha :\pi\rightarrow Aut(G)$. Such an action is called \textbf{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 \textbf{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 \cite{Massey} if $H$ is a subgroup of the free amalgamated product of groups $ \underset{i\in I}{*}G_i = G$, then $H=F(X)*(*_{j\in J} g_jH_jg_j^{-1})$, 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 $B(G*H)\simeq BG\vee BH$, \cite{Massey}. The mod $R-$Fibre Lemma \cite[Lemma 5.1]{BK} 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 \textbf{virtually finite} if it has a free subgroup $H<G$ of finite index. A group $G$ is \textbf{a virtual finite $p$-group} if it has a free subgroup of index a power of $p$. A space is \textbf{nilpotent} if the fundamental group acts nilpotently on all higher homotopy groups \cite{BK}. 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 \textbf{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 $R_nX$defined by Bousfield and Kan preserves the homology with $R$-coefficients \cite{Farjoun2}. This property of preserving $R$-homology characterizes the tower completely \cite{BK}. 
For a ring $R$ and a space $X$ denote $E^RX$  the $HR-$localization functor defined in \cite{Bousfield1}.
%There exists a tower version of the mod-$R$ fibre lemma \cite{BK}. 
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 \cite{pgoodpbad}.\\[0.3cm] The following proves that in contrast to the case of primes in the general case a space cannot be both $R$-good and $R'$-bad for arbitrary solid rings $R$ and $R'$ at the same time. 
\begin{Theorem}
Let either
\begin{itemize}
\item $R=\mathbb{Z}/m\mathbb{Z}$ and $R'=\mathbb{Z}/n\mathbb{Z}$, where $m$ and $n$ are integers such that $m$ and $n$ are coprime, or
\item $R=\mathbb{Z}[J^{-1}]$ and $R'=\mathbb{Z}/n\mathbb{Z}$, where $n$ is an integer all of whose  prime factors are in $J$.
\end{itemize}

Let $X$ be a space which is good for the ring $R$ and bad for the ring $R'$. Then $X$ is bad for $R\times R'$.
\end{Theorem}
\underline{Proof:} We have from \cite[Proposition 9.5]{BK} that $X\widehat{_{R\times R'}}\simeq X^{\wedge}_R\times X^{\wedge}_{R'}$. The result then follows from the K\"unneth Theorem.$\Box$
\begin{Remark}
Let $p$ be a prime and $X$ a $p-$good space. Then $X$ can be both good or bad for $\mathbb{Z}_{(p)}$.
\end{Remark}
\begin{Theorem}[\cite{IvanovMikhailov1}{, Corollary 9.2.}]
For a finitely presented metabelian group $X$, the natural map $E^RX\rightarrow \hat{X}_R$ is an isomorphism for $R=\mathbb{Q}$ or $\mathbb{Z}/n\mathbb{Z}$.
\end{Theorem}
\begin{Remark}[\cite{IvanovMikhailov1}{, Theorem 9.1.}]
The above result cannot be generalized to the case $R=\mathbb{Z}$.
\end{Remark}
\begin{Remark}
Recall that $\mathbb{R}P^{\infty}\simeq BC_2$ is good for the integers and all primes but bad for $\mathbb{Z}_{(2)}$. 
\end{Remark}
\begin{Remark}
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.
\end{Remark}

\begin{Theorem}
Let $R$ be a nonzero solid ring. The classifying space $S^1\vee S^1\simeq B(\mathbb{Z}*\mathbb{Z})$ is $R$-bad.
\end{Theorem}
\underline{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 \cite{Bousfield1}.$\Box$

\begin{thebibliography}
{}
\bibitem{AKO} M.\ Aschbacher, R.\ Kessar, B.\ Oliver, \textit{Fusion systems in algebra and topology}, London Mathematical Society Lecture Note Series: 31, Cambridge University Press, 2011.
%\bibitem{BGH} C.\ Benson, J.\ Grodal, E.\ Henke, \textit{Group cohomology and control of $p$-fusion}, arXiv:1210.1564.
%\bibitem{BW} D.\ Benson, C.\ Wilkerson, \textit{Finite simple groups and Dickson invariants}, Homotopy Theory and its Applications, ed. Adem, Milgram and Ravenel, Contemp. Math. 188, 31--42, American Math. Society, 1995.
\bibitem{Bousfield1} A.\ K.\ Bousfield, \textit{Homological localization towers for groups and $\pi -$modules}, Mem.\ Amer.\ Math.\ Soc, no. 186, 1977. 
\bibitem{Bousfield} A.\ K.\ Bousfield, \textit{On the $p-$adic completions of nonnilpotent spaces}, Transactions of the American Mathematical Society, Volume 331, Number 1, May 1992, 335--359.
\bibitem{BK} A.\ K.\ Bousfield, D.\ M.\ Kan, \textit{Homotopy Limits, Completions and Localizations}, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
\bibitem{BK2} A.\ K.\ Bousfield, D.\ M.\ Kan, \textit{The core of a ring}, J.\ Pure Applied Algebra 2 (1972), 73--81.
%\bibitem{BCGLO1} C.\ Broto, N.\ Castellana, J.\  Grodal, B.\ Oliver, \textit{Subgroup families controlling $p-$local finite groups}, Proc., London Math. Soc. 91 (2005), 325--354.
%\bibitem{BCGLO2} C. Broto, N. Castellana, J. Grodal, B. Oliver, \textit{Extensions of $p-$local finite groups}, Transactions of the American Mathematical Society, 359 (2007), 3791-3858.
%\bibitem{BLO1} C.\ Broto, R.\ Levi B.\ Oliver, \textit{Homotopy Equivalences of $p-$Completed Classifying Spaces of Finite Groups}, Invent. Math. 151 (2003), 611--664.
%\bibitem{BLO2} C.\ Broto, R.\ Levi, B.\ Oliver, \textit{The Homotopy Theory of Fusion Systems}, J. Amer. Math. Soc. \textbf{16} (2003), no. 4, 779--856.
%\bibitem{BLO3} C.\ Broto, R.\ Levi, B.\ Oliver, \textit{Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups}, Geometry and Topology 11 (2007) 315--427.
%\bibitem{BLO4} C.\ Broto, R.\ Levi, B.\ Oliver, \textit{A Geometric Construction of Saturated Fusion Systems}, Contemp. Math. 399, 2006, 11--39.
% \bibitem{CE} H.\ Cartan, S.\ Eilenberg, \textit{Homological Algebra}, Princeton University Press, Princeton, New Jersey, 1956.
%\bibitem{Chermak} A.\ Chermak, \textit{Fusion systems and localities}, preprint 2011.
\bibitem{CurtisReiner} C.\ W.\ Curtis, I.\ Reiner, \textit{Representation Theory of Finite Groups and Associative Algebras}, Interscience Publishers, New York, 1962.
%\bibitem{McCleary} J.\ McCleary, \textit{A user's guide to spectral sequences}, Cambridge studies in Advanced Mathematics 58, second edition, Cambridge University Press, Cambridge, UK, 2001.
%\bibitem{Dwyer} W. G. Dwyer, \textit{Classifying Spaces and Homology Decompositions}, Homotopy Theoretic Methods inn Group Cohomology, Advanced Courses in Mathematics CRM Barcelona, Birkh?user Verlag, Basel-Boston-Berlin, 2001.
\bibitem{DwyerFarjounRavenel} W.\ G.\ Dwyer, E.\ D.\ Farjoun, and D.\ C. Ravenel, \textit{Bousfield localizations of classifying spaces of nilpotent groups}, Proceedings of the American Mathematical Society, Volume 127, Pages 1855--1861.
%\bibitem{DK} W. G. Dwyer and D. M. Kan, \textit{Centric Maps and Realization of Diagrams in the Homotopy Category}, Proc. Amer. Math. Soc. (1984), 456-460.
%\bibitem{Evens} L. Evens, \textit{The Cohomology of Groups}, Oxford Science Publications, Oxford, New York, Tokyo, Clarendon Press 1991.
\bibitem{Farjoun} E.\ D.\ Farjoun, \textit{Two Completion Towers for Generalized Homology}, Contemporary Mathematics Volume 265 (2000).
\bibitem{Farjoun2} E.\ D.\ Farjoun, \textit{Pro-nilpotent representation of homology types}, Proceedings of the American Mathematical Society, Volume 38, Number 3, May 1973, 657--660.
%\bibitem{Gorenstein} D. Gorenstein, \textit{Finite Groups}, Chelsea Publishing Company, New York, 1980.
\bibitem{Hall} M. Hall, \textit{The theory of groups}, Macmillan (1959).
%\bibitem{Hertweck} M. Hertweck, \textit{A counterexample to the isomorphism problem for integral group rings}, Ann. of Math. (2) 154 (2001), no. 1, 115-138.
%\bibitem{JMO} S. Jackowski, J. McClure, B. Oliver, \textit{Homotopy Classification of Self-Maps of BG via G-actions}, The Annals of Mathematics, 2nd Ser., Vol. 135, No. 1.(Jan., 1992), pp. 183-226.
%\bibitem{KP} Daniel S. Kahn, Stewart B. Priddy, \textit{The transfer and stable homotopy theory}, Math. Prod. Camb. Phil. Soc. (1978), 83, 103.
%\bibitem{L} Ian Leary, \textit{$p$-groups are not determined by their integral cohomology groups}, Bull. London. Math. Soc. 27 (1995) 585-589.
% \bibitem{dgmp} A. Diaz, A. Glesser, Nadia Mazza, S. Park, \textit{Glauberman's and Thompson's theorems for fusion systems}, Proc. Amer. Math. Soc. 137 (2009), 495-503.
%\bibitem{KanThurston}C.\ Kan, W.\ Thurston, \textit{
%Every connected space has the homology of a $K(\pi,1)$},
%Topology 15 (1976), no. 3, 253--258. 
%\bibitem{Ian+Radu} I.\ Leary, R.\ Stancu, \textit{Realising fusion systems}, Algebra Number Theory 1.1, (2007), 17-34.

\bibitem{IvanovMikhailov1} S.\ O.\ Ivanov, R.\ Mikhailov, \textit{On a problem of Bousfield for metabelian groups}, Adv.\ Math.\ 290 (2016), 552--589. 
\bibitem{IvanovMikhailov2} S.\ O.\ Ivanov, R.\ Mikhailov, \textit{On lenghts of $H\mathbb{Z}-$localization towers}, preprint.
%\bibitem{anewfam} M.\ Leip, N.\ Seeliger, \textit{Signalizer functors for group models, existence of classifying spaces and applications to the fundamental group}, arXiv:1105.3403.
%\bibitem{LL} R.\ Levi, A.\ Libman, \textit{Existence and uniqueness of classifying spaces for fusion systems over discrete $p$-toral groups}, preprint.
%\bibitem{LO} R.\ Levi, B.\ Oliver, \textit{Construction of 2-local finite groups of a type studied by Solomon and Benson}, Geometry and Topology 6 (2002), 917--990.
% \bibitem{LC} A. Libman, N. Castellana,
% \textit{Wreath products and representations of $p-$local finite groups}
% , Adv. Math. 221 (2009), no.4 , 1302-1344.
%   `qA$221$ (2009), no. 4, $1302?1344$.
%\bibitem{LS} A. Libman, N. Seeliger, \textit{Homology decompositions and groups inducing fusion systems}, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, 167--187.
%\bibitem{LS}A.\ Libman, N.\ Seeliger, \textit{Homology Decompositions and Groups Inducing Fusion Systems}, Homotopy, Homology and Applications, Vol 14(2012), No. 2, pp. 167--187.
%\bibitem{LV} A. Libman, A. Viruel, \textit{On the homotopy type of the non-completed classifying space of a $p$-local finite group}, Forum Mathematicum 21 (2009), 723-757.



% Libman, A. (2006). 'The normaliser decomposition of p-local finite groups'. Algebraic and Geometric Topology, vol 6, pp. 1267-1288.


%\bibitem{IntroMarkus} M.\ Linckelmann, \textit{Introduction to Fusion Systems}, in: Group Representation Theory, EPFL Press, Lausanne, 2007, pp. 79--113.
%\bibitem{SMcL} S. MacLane, \textit{Categories for the Working Mathematician}, 2nd edition, Graduate Texts in Mathematics, vol. 5, Springer-Verlag New York, Inc., 1998.
%\bibitem{MKS} W. Magnus, A. Karrass, D. Solitar, \textit{Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations}, Interscience Publishers, A division of John Wiley and Sons Inc. New York, London, Sydney, 1966.
\bibitem{Malcev} A.\ L.\ Malcev, \textit{Nilpotent groups without torsion}, Jzv. Akad. Nauk. SSSR
, Math. 13 (1949), 201--212.
\bibitem{MartinoPriddy} J.\ Martino, S.\ Priddy, \textit{Unstable homotopy classification of $BG\pcom$}, Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 137 (2004) 321--347.
\bibitem{Massey} William S. Massey, \textit{Algebraic topology: an introduction}, Graduate Texts in Mathematics, Springer-Verlag, New York, 1977.
%bibitem{Oliver} B.\ Oliver, \textit{Existence and uniqueness of linking systems: Chermak`s proof via obstruction theory}, arXiv:1203.6479.
 %\bibitem{MP1} B.\ Oliver, \textit{Equivalences of classifying spaces completed at odd primes}, Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 137, 2004, 321--347.
%\bibitem{MP2} B.\ Oliver, \textit{Equivalences of classifying spaces completed at the prime 2}, Mem.\ Amer.\ Math.\ Soc.\ 180 (2006), no.\ 848, vi+102pp.
%bibitem{O4} B.\ Oliver, \textit{ Extensions of linking systems and fusion systems}, Trans. Amer. Math. Soc. 362 (2010), 5483--5500.
%\bibitem{Sejong} S.\ Park, \textit{Realizing a fusion system by a single finite group}, Arch. Math. 94 (2010), 405--410.
%\bibitem{P} L. Puig, \textit{Structure locale dans les groupes finis}, Bull. Soc. Math. France 47 (1976).
%
\bibitem{Quillen} D.\ G.\
 Quillen, \textit{Rational homotopy theory
 }, Annals of Math. 90
 (1969), 205--295.
%\bibitem{Ragnarsson} K.\ Ragnarsson, \textit{Classifying spectra of saturated fusion systems}, Algebr. Geom. Topol. 6 (2006), 195--252.
%\bibitem{Robinson1} G.\ Robinson, \textit{Amalgams, blocks, weights, fusion systems and finite simple groups}, Journal of Algebra 314 (2007), 912--923.
%\bibitem{Robinson2} G.\ Robinson, \textit{Reduction (Mod q) of Fusion System Amalgams}, Trans. Amer. Math. Soc. 363 (2011), 1023--1040.
%\bibitem{gmffs} N.\ Seeliger, \textit{Group models for fusion systems}, Topology and its Applications 159 (2012), no. 12, 2845--2853.
\bibitem{pgoodpbad} N.\ Seeliger, \textit{A few examples of $p$-good and $p$-bad classifying spaces}, 	arXiv:1411.7490.
\bibitem{Sullivan} D.\ Sullivan, \textit{Geometric topology, part I: localization, periodicity and Galois symmetry}, MIT (1970).
%bibitem{Serre} J.\ P.\ Serre, \textit{Trees}, Springer-Verlag Berlin Heidelberg New York, 1980.
\end{thebibliography}
Dr.\ Nora Seeliger, Department of Mathematics, University of Haifa, Faculty of Natural Sciences, Science and Education Building, Room 615, Abba Khoushy Avenue 199, 3498838 Haifa, ISRAEL.\\Email:
nseelige@campus.haifa.ac.il
\end{document}
1703.05754.extracted_bib
{AKO M.\ Aschbacher, R.\ Kessar, B.\ Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series: 31, Cambridge University Press, 2011. %
1703.05754.extracted_bib
{BGH C.\ Benson, J.\ Grodal, E.\ Henke, Group cohomology and control of $p$-fusion, arXiv:1210.1564. %
1703.05754.extracted_bib
{BW D.\ Benson, C.\ Wilkerson, Finite simple groups and Dickson invariants, Homotopy Theory and its Applications, ed. Adem, Milgram and Ravenel, Contemp. Math. 188, 31--42, American Math. Society, 1995.
1703.05754.extracted_bib
{Bousfield1 A.\ K.\ Bousfield, Homological localization towers for groups and $\pi -$modules, Mem.\ Amer.\ Math.\ Soc, no. 186, 1977.
1703.05754.extracted_bib
{Bousfield A.\ K.\ Bousfield, On the $p-$adic completions of nonnilpotent spaces, Transactions of the American Mathematical Society, Volume 331, Number 1, May 1992, 335--359.
1703.05754.extracted_bib
{BK A.\ K.\ Bousfield, D.\ M.\ Kan, Homotopy Limits, Completions and Localizations, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
1703.05754.extracted_bib
{BK2 A.\ K.\ Bousfield, D.\ M.\ Kan, The core of a ring, J.\ Pure Applied Algebra 2 (1972), 73--81. %
1703.05754.extracted_bib
{BCGLO1 C.\ Broto, N.\ Castellana, J.\ Grodal, B.\ Oliver, Subgroup families controlling $p-$local finite groups, Proc., London Math. Soc. 91 (2005), 325--354. %
1703.05754.extracted_bib
{BCGLO2 C. Broto, N. Castellana, J. Grodal, B. Oliver, Extensions of $p-$local finite groups, Transactions of the American Mathematical Society, 359 (2007), 3791-3858. %
1703.05754.extracted_bib
{BLO1 C.\ Broto, R.\ Levi B.\ Oliver, Homotopy Equivalences of $p-$Completed Classifying Spaces of Finite Groups, Invent. Math. 151 (2003), 611--664. %
1703.05754.extracted_bib
{BLO2 C.\ Broto, R.\ Levi, B.\ Oliver, The Homotopy Theory of Fusion Systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779--856. %
1703.05754.extracted_bib
{BLO3 C.\ Broto, R.\ Levi, B.\ Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups, Geometry and Topology 11 (2007) 315--427. %
1703.05754.extracted_bib
{BLO4 C.\ Broto, R.\ Levi, B.\ Oliver, A Geometric Construction of Saturated Fusion Systems, Contemp. Math. 399, 2006, 11--39. %
1703.05754.extracted_bib
{CE H.\ Cartan, S.\ Eilenberg, Homological Algebra, Princeton University Press, Princeton, New Jersey, 1956. %
1703.05754.extracted_bib
{Chermak A.\ Chermak, Fusion systems and localities, preprint 2011.
1703.05754.extracted_bib
{CurtisReiner C.\ W.\ Curtis, I.\ Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, New York, 1962. %
1703.05754.extracted_bib
{McCleary J.\ McCleary, A user's guide to spectral sequences, Cambridge studies in Advanced Mathematics 58, second edition, Cambridge University Press, Cambridge, UK, 2001. %
1703.05754.extracted_bib
{Dwyer W. G. Dwyer, Classifying Spaces and Homology Decompositions, Homotopy Theoretic Methods inn Group Cohomology, Advanced Courses in Mathematics CRM Barcelona, Birkh?user Verlag, Basel-Boston-Berlin, 2001.
1703.05754.extracted_bib
{DwyerFarjounRavenel W.\ G.\ Dwyer, E.\ D.\ Farjoun, and D.\ C. Ravenel, Bousfield localizations of classifying spaces of nilpotent groups, Proceedings of the American Mathematical Society, Volume 127, Pages 1855--1861. %
1703.05754.extracted_bib
{DK W. G. Dwyer and D. M. Kan, Centric Maps and Realization of Diagrams in the Homotopy Category, Proc. Amer. Math. Soc. (1984), 456-460. %
1703.05754.extracted_bib
{Evens L. Evens, The Cohomology of Groups, Oxford Science Publications, Oxford, New York, Tokyo, Clarendon Press 1991.
1703.05754.extracted_bib
{Farjoun E.\ D.\ Farjoun, Two Completion Towers for Generalized Homology, Contemporary Mathematics Volume 265 (2000).
1703.05754.extracted_bib
{Farjoun2 E.\ D.\ Farjoun, Pro-nilpotent representation of homology types, Proceedings of the American Mathematical Society, Volume 38, Number 3, May 1973, 657--660. %
1703.05754.extracted_bib
{Gorenstein D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.
1703.05754.extracted_bib
{Hall M. Hall, The theory of groups, Macmillan (1959). %
1703.05754.extracted_bib
{Hertweck M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154 (2001), no. 1, 115-138. %
1703.05754.extracted_bib
{JMO S. Jackowski, J. McClure, B. Oliver, Homotopy Classification of Self-Maps of BG via G-actions, The Annals of Mathematics, 2nd Ser., Vol. 135, No. 1.(Jan., 1992), pp. 183-226. %
1703.05754.extracted_bib
{KP Daniel S. Kahn, Stewart B. Priddy, The transfer and stable homotopy theory, Math. Prod. Camb. Phil. Soc. (1978), 83, 103. %
1703.05754.extracted_bib
{L Ian Leary, $p$-groups are not determined by their integral cohomology groups, Bull. London. Math. Soc. 27 (1995) 585-589. %
1703.05754.extracted_bib
{dgmp A. Diaz, A. Glesser, Nadia Mazza, S. Park, Glauberman's and Thompson's theorems for fusion systems, Proc. Amer. Math. Soc. 137 (2009), 495-503. %
1703.05754.extracted_bib
{KanThurstonC.\ Kan, W.\ Thurston, %Every connected space has the homology of a $K(\pi,1)$, %Topology 15 (1976), no. 3, 253--258. %
1703.05754.extracted_bib
{Ian+Radu I.\ Leary, R.\ Stancu, Realising fusion systems, Algebra Number Theory 1.1, (2007), 17-34.
1703.05754.extracted_bib
{IvanovMikhailov1 S.\ O.\ Ivanov, R.\ Mikhailov, On a problem of Bousfield for metabelian groups, Adv.\ Math.\ 290 (2016), 552--589.
1703.05754.extracted_bib
{IvanovMikhailov2 S.\ O.\ Ivanov, R.\ Mikhailov, On lenghts of $H\mathbb{Z-$localization towers, preprint. %
1703.05754.extracted_bib
{anewfam M.\ Leip, N.\ Seeliger, Signalizer functors for group models, existence of classifying spaces and applications to the fundamental group, arXiv:1105.3403. %
1703.05754.extracted_bib
{LL R.\ Levi, A.\ Libman, Existence and uniqueness of classifying spaces for fusion systems over discrete $p$-toral groups, preprint. %
1703.05754.extracted_bib
{LO R.\ Levi, B.\ Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry and Topology 6 (2002), 917--990. %
1703.05754.extracted_bib
{LC A. Libman, N. Castellana, % Wreath products and representations of $p-$local finite groups % , Adv. Math. 221 (2009), no.4 , 1302-1344. % `qA$221$ (2009), no. 4, $1302?1344$. %
1703.05754.extracted_bib
{LS A. Libman, N. Seeliger, Homology decompositions and groups inducing fusion systems, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, 167--187. %
1703.05754.extracted_bib
{LSA.\ Libman, N.\ Seeliger, Homology Decompositions and Groups Inducing Fusion Systems, Homotopy, Homology and Applications, Vol 14(2012), No. 2, pp. 167--187. %
1703.05754.extracted_bib
{LV A. Libman, A. Viruel, On the homotopy type of the non-completed classifying space of a $p$-local finite group, Forum Mathematicum 21 (2009), 723-757. % Libman, A. (2006). 'The normaliser decomposition of p-local finite groups'. Algebraic and Geometric Topology, vol 6, pp. 1267-1288. %
1703.05754.extracted_bib
{IntroMarkus M.\ Linckelmann, Introduction to Fusion Systems, in: Group Representation Theory, EPFL Press, Lausanne, 2007, pp. 79--113. %
1703.05754.extracted_bib
{SMcL S. MacLane, Categories for the Working Mathematician, 2nd edition, Graduate Texts in Mathematics, vol. 5, Springer-Verlag New York, Inc., 1998. %
1703.05754.extracted_bib
{MKS W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Interscience Publishers, A division of John Wiley and Sons Inc. New York, London, Sydney, 1966.
1703.05754.extracted_bib
{Malcev A.\ L.\ Malcev, Nilpotent groups without torsion, Jzv. Akad. Nauk. SSSR , Math. 13 (1949), 201--212.
1703.05754.extracted_bib
{MartinoPriddy J.\ Martino, S.\ Priddy, Unstable homotopy classification of $BG\pcom$, Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 137 (2004) 321--347.
1703.05754.extracted_bib
{Massey William S. Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics, Springer-Verlag, New York, 1977. %bibitem{Oliver B.\ Oliver, Existence and uniqueness of linking systems: Chermak`s proof via obstruction theory, arXiv:1203.6479. %
1703.05754.extracted_bib
{MP1 B.\ Oliver, Equivalences of classifying spaces completed at odd primes, Math.\ Proc.\ Camb.\ Phil.\ Soc.\ 137, 2004, 321--347. %
1703.05754.extracted_bib
{MP2 B.\ Oliver, Equivalences of classifying spaces completed at the prime 2, Mem.\ Amer.\ Math.\ Soc.\ 180 (2006), no.\ 848, vi+102pp. %bibitem{O4 B.\ Oliver, Extensions of linking systems and fusion systems, Trans. Amer. Math. Soc. 362 (2010), 5483--5500. %
1703.05754.extracted_bib
{Sejong S.\ Park, Realizing a fusion system by a single finite group, Arch. Math. 94 (2010), 405--410. %
1703.05754.extracted_bib
{P L. Puig, Structure locale dans les groupes finis, Bull. Soc. Math. France 47 (1976). %
1703.05754.extracted_bib
{Quillen D.\ G.\ Quillen, Rational homotopy theory , Annals of Math. 90 (1969), 205--295. %
1703.05754.extracted_bib
{Ragnarsson K.\ Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006), 195--252. %
1703.05754.extracted_bib
{Robinson1 G.\ Robinson, Amalgams, blocks, weights, fusion systems and finite simple groups, Journal of Algebra 314 (2007), 912--923. %
1703.05754.extracted_bib
{Robinson2 G.\ Robinson, Reduction (Mod q) of Fusion System Amalgams, Trans. Amer. Math. Soc. 363 (2011), 1023--1040. %
1703.05754.extracted_bib
{gmffs N.\ Seeliger, Group models for fusion systems, Topology and its Applications 159 (2012), no. 12, 2845--2853.
1703.05754.extracted_bib
{pgoodpbad N.\ Seeliger, A few examples of $p$-good and $p$-bad classifying spaces, arXiv:1411.7490.
1703.05754.extracted_bib
{Sullivan D.\ Sullivan, Geometric topology, part I: localization, periodicity and Galois symmetry, MIT (1970). %bibitem{Serre J.\ P.\ Serre, Trees, Springer-Verlag Berlin Heidelberg New York, 1980.