The local structure of finite groups and of their classifying spaces

Abstract: Fix a prime ${p}$. We say that two finite groups $G$ and $H$ are ``${p}$-equivalent'' if there is an isomorphism between Sylow $p$-subgroups $S \in Syl_p(G)$ and $T\in Syl_p({H})$ that preserves all $G-$ and ${H}-$conjugacy relations among elements and subgroups of $S$ and $T$. We say that two topological spaces ${X}$ and ${Y}$ are ``${p}$-equivalent'' if there is a third space ${Z}$, and maps $X\to Z \leftarrow Y$ that induce isomorphisms in homology with coefficients in $\mathbb{Z}/p$. (Both of these are equivalence relations.) The main theorem I want to describe says that finite groups ${G}$ and ${H}$ are $p$-equivalent (as groups) if and only if their classifying spaces are ${p}$-equivalent (as spaces).

I will start by defining in more detail classifying spaces of discrete groups and the two kinds of ${p}$-equivalence described above, and also saying a little about the background of the theorem. I then plan to give some examples of finite groups that are ${p}$-locally equivalent but not isomorphic, and say something about ideas that went into the proof of the theorem (carried out by several different people over a period of 10--15 years).

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

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CRM - Séminaire du CIRGET / Géométrie et Topologie

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