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CATEGORIES:Algebra and Representation Theory Seminar
SUMMARY:Dimension series and homotopy groups of spheres -
Laurent Bartholdi (GĂ¶ttingen\, Lyon)
DTSTART;TZID=Europe/London:20191219T140000
DTEND;TZID=Europe/London:20191219T150000
UID:TALK136327AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/136327
DESCRIPTION:It has been\, for the last 80 years\, a fundamenta
l problem of group theory to relate the lower cent
ral series and the dimension series introduced by
Magnus. One always has that the nth term of the di
mension series contains the nth lower central subg
roup\, and a conjecture by Magnus\, with false pro
ofs by Cohn\, Losey\, etc.\, claims that they coin
cide\; but Rips constructed\nan example with diffe
rent fourth terms.On the positive\nside\, Sjogren
showed that the quotient of the nth dimension grou
p by the nth lower central subgroup is always a to
rsion group\, of exponent bounded by a function of
$n$. Furthermore\, it was\nbelieved (and falsely
proven by Gupta) that only $2$-torsion may occur.\
n\nIn joint work with Roman Mikhailov\, we prove h
owever that for every\nprime $p$ there is a group
with $p$-torsion in some such quotient.\n\nEven mo
re interestingly\, I will show that these quotient
s are related to the difference between homotopy a
nd\nhomology: our construction is fundamentally ba
sed on the order-$p$ element in the homotopy group
$\\pi_{2p}(S^2)$ due to Serre.\n
LOCATION:MR12
CONTACT:Christopher Brookes
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