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SUMMARY:Bob Oliver (Université Paris 13)
DTSTART:20260511T103000Z
DTEND:20260511T113000Z
DTSTAMP:20260525T015459Z
UID:BilTop/139
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BilTop/139/"
 >Approximations by finite fusion systems</a>\nby Bob Oliver (Université P
 aris 13) as part of Bilkent Topology Seminar\n\nLecture held in SA 141.\n\
 nAbstract\nFix a prime \\(p\\)\, and let \\(\\mathbb{Z}/p^\\infty\\) be th
 e union of the cyclic \\(p\\)-groups \\(\\mathbb{Z}/p < \\mathbb{Z}/p^2 < 
 \\mathbb{Z}/p^3 < \\cdots\\). Equivalently\, \\(\\mathbb{Z}/p^\\infty \\co
 ng \\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\\).\n\nA saturated fusion system over a discre
 te \\(p\\)-toral group \\(S\\) is a category whose objects are the subgrou
 ps of \\(S\\)\, and whose morphisms are homomorphisms between the subgroup
 s 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 subg
 roups of \\(S\\) and whose morphisms are those homomorphisms induced by co
 njugation 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.
 \n\nEvery 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 th
 e three cases listed above\, then one can show that \\(B\\mathcal{F} \\sim
 eq BG^\\wedge_p\\) (the \\(p\\)-completion of the classifying space of \\(
 G\\)). Alex Gonzalez’s approximation theorem says that for each saturate
 d fusion system \\(\\mathcal{F}\\) over an infinite discrete \\(p\\)-toral
  group \\(S\\)\, there is a sequence of finite fusion subsystems \\(\\math
 cal{F}_0 \\le \\mathcal{F}_1 \\le \\mathcal{F}_2 \\le \\cdots \\le \\mathc
 al{F}\\) over finite \\(p\\)-subgroups \\(S_0 \\le S_1 \\le S_2 \\le \\cdo
 ts \\le S\\) such that\n\\[\n\\mathcal{F} = \\bigcup_{i=0}^{\\infty} \\mat
 hcal{F}_i\n\\quad\\text{and}\\quad\nB\\mathcal{F} \\simeq \\left(\\operato
 rname{hocolim}(B\\mathcal{F}_i)\\right)^\\wedge_p .\n\\]\n\nWe note the fo
 llowing two important consequences of this result\, both already known for
  finite fusion systems but not in the general case:\n\\begin{itemize}\n\\i
 tem (Stable elements theorem) For each saturated fusion system \\(\\mathca
 l{F}\\) over a discrete \\(p\\)-toral group \\(S\\)\, \\(H^*(B\\mathcal{F}
 \; \\mathbb{F}_p) \\cong \\lim_{\\mathcal{F}} H^*(-\; \\mathbb{F}_p)\\). T
 his follows easily from the approximation theorem\, together with the fini
 te case.\n\n\\item (Mapping spaces) For each pair of discrete \\(p\\)-tora
 l groups \\(Q\\)\, \\(S\\) and each saturated fusion system \\(\\mathcal{F
 }\\) over \\(S\\)\, each connected component of the mapping space \\(\\ope
 ratorname{map}(BQ\, B\\mathcal{F})\\) has the homotopy type of \\(BC_{\\ma
 thcal{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.\n\\e
 nd{itemize}\n\nThe key idea when proving the approximation theorem is to a
 pproximate \\(B\\mathcal{F}\\) by the \\(B\\mathcal{F}_m\\) in a way simil
 ar to how an algebraic group \\(G(\\overline{\\mathbb{F}}_q)\\) over the a
 lgebraic closure of \\(\\mathbb{F}_q\\) (for a prime \\(q\\)) is approxima
 ted by the finite subgroups \\(G(\\mathbb{F}_{q^m})\\). Just as \\(G(\\mat
 hbb{F}_{q^m}) = C_{G(\\overline{\\mathbb{F}}_q)}(\\psi^m)\\)\, the fixed s
 ubgroup 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 \\(\\m
 athcal{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 hom
 otopy colimit of the spaces \\(B\\mathcal{F}_m\\).\n
LOCATION:https://researchseminars.org/talk/BilTop/139/
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