Approximations by finite fusion systems

Abstract: Fix a prime \(p\), and let \(\mathbb{Z}/p^\infty\) be the union of the cyclic \(p\)-groups \(\mathbb{Z}/p < \mathbb{Z}/p^2 < \mathbb{Z}/p^3 < \cdots\). Equivalently, \(\mathbb{Z}/p^\infty \cong \mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}\). A group \(S\) is discrete \(p\)-toral if it contains a normal subgroup \(S_0 \unlhd S\) of \(p\)-power index such that \(S_0 \cong (\mathbb{Z}/p^\infty)^r\) for some \(r \ge 0\).

A saturated fusion system over a discrete \(p\)-toral group \(S\) is a category whose objects are the subgroups of \(S\), and whose morphisms are homomorphisms between the subgroups satisfying certain axioms. If \(G\) is a group and \(S \le G\) is a discrete \(p\)-toral subgroup, then the fusion system of \(G\) over \(S\) is the category \(\mathcal{F}_S(G)\) whose objects are the subgroups of \(S\) and whose morphisms are those homomorphisms induced by conjugation by elements of \(G\). If \(G\) is a finite group and \(S \in \operatorname{Syl}_p(G)\), or if \(G\) is a compact Lie group and \(S\) is a maximal discrete \(p\)-toral subgroup (always unique up to conjugacy), then \(\mathcal{F}_S(G)\) is saturated. Similarly, if \(G\) is a locally finite group all of whose \(p\)-subgroups are discrete \(p\)-toral, then it has a maximal \(p\)-subgroup \(S\) that is ``weakly Sylow'', and \(\mathcal{F}_S(G)\) is a saturated fusion system.

Every saturated fusion system \(\mathcal{F}\) over a discrete \(p\)-toral group \(S\) has a classifying space \(B\mathcal{F}\). If \(\mathcal{F} = \mathcal{F}_S(G)\) where \(G \ge S\) lies in one of the three cases listed above, then one can show that \(B\mathcal{F} \simeq BG^\wedge_p\) (the \(p\)-completion of the classifying space of \(G\)). Alex Gonzalez’s approximation theorem says that for each saturated fusion system \(\mathcal{F}\) over an infinite discrete \(p\)-toral group \(S\), there is a sequence of finite fusion subsystems \(\mathcal{F}_0 \le \mathcal{F}_1 \le \mathcal{F}_2 \le \cdots \le \mathcal{F}\) over finite \(p\)-subgroups \(S_0 \le S_1 \le S_2 \le \cdots \le S\) such that \[ \mathcal{F} = \bigcup_{i=0}^{\infty} \mathcal{F}_i \quad\text{and}\quad B\mathcal{F} \simeq \left(\operatorname{hocolim}(B\mathcal{F}_i)\right)^\wedge_p . \]

We note the following two important consequences of this result, both already known for finite fusion systems but not in the general case: \begin{itemize} \item (Stable elements theorem) For each saturated fusion system \(\mathcal{F}\) over a discrete \(p\)-toral group \(S\), \(H^*(B\mathcal{F}; \mathbb{F}_p) \cong \lim_{\mathcal{F}} H^*(-; \mathbb{F}_p)\). This follows easily from the approximation theorem, together with the finite case.

\item (Mapping spaces) For each pair of discrete \(p\)-toral groups \(Q\), \(S\) and each saturated fusion system \(\mathcal{F}\) over \(S\), each connected component of the mapping space \(\operatorname{map}(BQ, B\mathcal{F})\) has the homotopy type of \(BC_{\mathcal{F}}(P)\) for some \(P \le S\). This was already known in certain special cases (in addition to the finite case), but not in general. \end{itemize}

The key idea when proving the approximation theorem is to approximate \(B\mathcal{F}\) by the \(B\mathcal{F}_m\) in a way similar to how an algebraic group \(G(\overline{\mathbb{F}}_q)\) over the algebraic closure of \(\mathbb{F}_q\) (for a prime \(q\)) is approximated by the finite subgroups \(G(\mathbb{F}_{q^m})\). Just as \(G(\mathbb{F}_{q^m}) = C_{G(\overline{\mathbb{F}}_q)}(\psi^m)\), the fixed subgroup of a certain automorphism \(G(\psi^m) \in \operatorname{Aut}(G(\overline{\mathbb{F}}_q))\) where \(\psi^m \in \operatorname{Aut}(\overline{\mathbb{F}}_q)\) is a field automorphism, one defines \(\mathcal{F}_m = C_{\mathcal{F}}(\Psi^{p^m})\): the fusion subsystem fixed by \(\Psi^{p^m}\) for \(\Psi \in \operatorname{Aut}(\mathcal{F})\) having certain properties. This was the underlying idea, going back to Ran Levi, Assaf Libman, and others, but it wasn’t at all obvious how to fill in the details needed to show that \(B\mathcal{F}\) is the homotopy colimit of the spaces \(B\mathcal{F}_m\).

algebraic topologycategory theory

Audience: researchers in the topic

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