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SUMMARY:Dimension series and homotopy groups of spheres - Laurent Barthold
 i (Göttingen\, Lyon)
DTSTART:20191219T140000Z
DTEND:20191219T150000Z
UID:TALK136327@talks.cam.ac.uk
CONTACT:Christopher Brookes
DESCRIPTION:It has been\, for the last 80 years\, a fundamental problem of
  group theory to relate the lower central series and the dimension series 
 introduced by Magnus. One always has that the nth term of the dimension se
 ries contains the nth lower central subgroup\, and a conjecture by Magnus\
 , with false proofs by Cohn\, Losey\, etc.\, claims that they coincide\; b
 ut Rips constructed\nan example with different fourth terms.On the positiv
 e\nside\, Sjogren showed that the quotient of the nth dimension group by t
 he nth lower central subgroup is always a torsion group\, of exponent boun
 ded by a function of $n$. Furthermore\, it was\nbelieved (and falsely prov
 en by Gupta) that only $2$-torsion may occur.\n\nIn joint work with Roman 
 Mikhailov\, we prove however that for every\nprime $p$ there is a group wi
 th $p$-torsion in some such quotient.\n\nEven more interestingly\, I will 
 show that these quotients are related to the difference between homotopy a
 nd\nhomology: our construction is fundamentally based on the order-$p$ ele
 ment in the homotopy group $\\pi_{2p}(S^2)$ due to Serre.\n
LOCATION:MR12
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