Minami-Webb splittings and some exotic \(p\)-local finite groups
The goal of ``homology decomposition'' is to present, up to \(p\)-completion in the sense of \textit{A. K. Bousfield} and \textit{D. M. Kan} [Homotopy Limits, Completions and Localizations. Lectures Notes in Mathematics 304. Springer (1972; Zbl 0259.55004)], a classifying space of a group by means of the classifying spaces of its proper subgroups. The main papers in this subject are \textit{S. Jackowski, J. McClure} and \textit{B. Oliver}'s work in [Topology 31, No. 1, 113--132 (1992; Zbl 0754.55014) and Ann. Math. (2) 135, No. 1, 183--226 (1992; Zbl 0758.55004), ibid. No. 2, 227--270 (1992; Zbl 0771.55003)] and \textit{W. G. Dwyer}'s achievements [Topology 36, No. 4, 783--804 (1997; Zbl 0872.55014), Sharp homology decomposition for classifying spaces of finite groups. Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Symp. Pure Math. 63, 197--220 (1998; Zbl 0891.55021)]. The introduction of \(p\)-local finite groups by \textit{C. Broto, R. Levi} and \textit{B. Oliver} in [J. Am. Math. Soc. 16, No. 4, 779--856 (2003; Zbl 1033.55010)] re-opened the subject. At present the goal is to decompose the classifying space of a \(p\)-local finite group by means of classifying spaces of finite groups [see e.g. Broto, Levi and Oliver's paper and a paper (to appear) of \textit{A. Libman}]. Every finite group \(G\) gives rise to a \(p\)-local finite group \((S,\mathcal{F},\mathcal{L})\), where \(S\) is a Sylow \(p\)-subgroup of \(G\). A \(p\)-local finite group is called ``exotic'' if its underlying fusion system \(\mathcal{F}\) cannot be associated with a finite group. The classifying space of a \(p\)-local finite group \((S,\mathcal{F},\mathcal{L})\) is by definition the \(p\)-completed nerve \({|\mathcal{L}|}_{p}\) of the linking system \(\mathcal{L}\). In the non-exotic case it is homotopy equivalent to \(BG_{p}^{\wedge}\). A consequence of \textit{N. Minami}'s theorem 6.5 in [Trans. Am. Math. Soc. 351, No. 11, 4481--4513 (1999; Zbl 0931.55014)] and \textit{P. J. Webb}'s observation [Proposition 2.7.1 in Comment. Math. Helv. 66, No. 1, 34--69 (1991; Zbl 0722.57019)] is the following Minami theorem (or Minami-Webb formula). Theorem 1.1. Let \(G\) be a finite group whose order is divisible by \(p\) and let \(\Delta\) denote the simplicial complex associated to one of the following posets \(\mathcal{P}\): All the non identity \(p\)-subgroups of \(G\) \(\mathcal{E}\): All the non identity elementary abelian \(p\)-subgroups of \(G\) \(\mathcal{R}\): All the non identity \(p\)-radical subgroups of \(G\). Then in the homotopy category of spectra \[ \bigl(\Sigma^\infty_{+} BG\bigr)_{p}^{\wedge}\simeq \sum_{\sigma\in\Delta/G} (-1)^{\dim \sigma} \bigl(\Sigma^\infty_{+} BG_{\sigma}\bigr)_{p}^{\wedge} \] where \(G_{\sigma}\) is the stabilizer of a simplex \(\sigma\in\Delta\). In this theorem, \(\Sigma_{+}^{\infty}X\) is the suspension spectrum of a space \(X\) to which one adds a disjoint base-point. The wedge sum of spectra on the right hand side can be transformed into a genuine equivalence of sums of spectra by transferring the negative terms on the right hand side to the left. The notation \((-)_{p}^{\wedge}\) is used for the \(p\)-completion of spectra. In the present paper the author gives a new proof for Theorem 1.1 and shows that the formal sum of spectra is a consequence of a split exact sequence of spectra associated with \(\Delta\). The author observes that this fact could not be deduced from Minami's original proof because it depens on \textit{P. J. Webb}'s results [Theorem A in Comment. Math. Helv. 62, No. 1, 135--167 (1987; Zbl 0616.20022)]. The author proves Theorem A. Let \((S,\mathcal{F},\mathcal{L})\) be one of the following \(p\)-local finite groups (1) The three exotic examples at \(p=7\) of \textit{A. Ruiz} and \textit{A. Viruel} [Math. Z. 248, No. 1, 45--65 (2004; Zbl 1062.20022)]. (2) The familiy of exotic examples described in the second row of the table in Example 9.3 of Broto, Levi and Oliver's work [op.cit]. Then, in the homotopy category of spectra, the \(p\)-completed suspension spectrum \(\Sigma^{\infty}_{+}{|\mathcal{L}|}_{p}\) is an alternating sum, that is, negative terms are allowed, of \(p\)-completed suspension spectra \(\Sigma^{\infty}_{+}BG_{\sigma}\) of classifying space of finite groups. The key lemma of the paper is Theorem B below. For subgroups \(H,K\) of a group \(G\), \(G/H\boxtimes G/K\) denotes the \(G\)-subset of \(G/H\times G/K\) consisting of the elements whose isotropy group is non-trivial. If \(\Delta\) is the simplicial complex associated to a \(G\)-poset \(\mathfrak{X}\) then \(\Delta/G\) is said to be finite if the set of simplices of \(\Delta\) has a finite number of \(G\)-orbits. Theorem B. Let \(S\) be a finite subgroup of \(G\) (possible infinite) and let \(\Delta\) be the simplicial complex associated to a \(G\)-poset \(\mathfrak{X}\). Suppose that \(\Delta/G\) is finite and that 1. \(G\) acts on \(\Delta\) with finite isotropy groups. 2. If \(H\) is an isotropy group of a simplex of \(\Delta\) then the number of elements in the fibres of \(G/S\boxtimes G/H\rightarrow G/H\) is a unit in \(\mathbb{Z}_{(p)}\). 3. The fixed-point subspace \(\Delta^Q\) is contractible for every non identity \(Q\leq S\), and 4. \(\widetilde{H}_{\ast}(\Delta/G;\mathbb{Z}_{(p)})=0\). Then, in the homotopy category of spectra \[ \bigl(\Sigma^{\infty}_{+}\Delta_{hG}\bigr)_{p}= \sum_{\sigma\in\Delta/G} (-1)^{\dim \sigma} \bigl(\Sigma^\infty_{+}BG_{\sigma} \bigr)_{p}^{\wedge} \] where \(\Delta_{hG}\) denotes the Borel construction \(EG\times_{G}\Delta\). The alternating sum reflects a split exact sequence of the spectra \[ \begin{multlined} 0\rightarrow \bigvee_{\sigma\in \text{sd}_{m}\mathfrak{X}/G}\Sigma^\infty_{+}BG_{\sigma}\rightarrow \bigvee_{\sigma\in \text{sd}_{m-1}\mathfrak{X}/G}\Sigma^\infty_{+}BG_{\sigma}\rightarrow\dots \rightarrow \\ \bigvee_{\sigma\in \text{sd}_{1}\mathfrak{X}/G}\Sigma^\infty_{+}BG_{\sigma} \rightarrow \bigvee_{\sigma\in \text{sd}_{0}\mathfrak{X}/G}\Sigma^\infty_{+}BG_{\sigma}\rightarrow \Sigma^\infty_{+}|\text{sd}\mathfrak{X}|_{hG}\rightarrow 0\end{multlined} \] and the \(p\)-completion can be replaced by localization with respect to the Moore spectrum \(S\mathbb{Z}_{(p)}\). As an immediate corollary to Theorem B the author reproves Minami's Theorem 1.1. The core of the proof of Theorem A consists of showing that for each one of the exotic \(p\)-local finite groups \((S,\mathcal{F},\mathcal{L})\) listed there, one can construct a discrete group \(G\), albeit infinite, which is an amalgamated product of finite groups, together with a \(G\)-poset \(\mathcal{R}\) such that a. \(|\mathcal{L}|\simeq\Delta_{hG}\), where \(\Delta\) is the \(G\)-simplicial complex associated to \(\mathcal{R}\), b. \(G\) contains \(S\) and \(\mathcal{F}\) is the fusion system on \(S\) induced by conjugations in \(G\), c. \(\Delta/G\) is finite and \(\Delta\) is \(G\)-acyclic for the Bredon cohomology theories associated to the Mackey functors \(\mathcal{M}_{h^{j}}\) for all \(p\)-local cohomology theories \(h\). An important ingredient used by the author in the proof of Theorem B is \textit{P. Symond}'s theorem [Comment. Math. Helv. 73, No. 3, 400--405 (1998; Zbl 0911.20020)], known as Webb's conjecture. At the end of this substantial work the author uses again his method to obtain a similar decomposition for the classifying spaces of a family of exotic \(p\)-local finite groups which were constructed by Ruiz-Viruel [op.cit] and Broto-Levi-Oliver [op.cit].
scientific article; zbMATH DE number 5133227
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