BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Laurent Bartholdi (Georg-August Universität zu Göttingen) DTSTART:20210513T150000Z DTEND:20210513T160000Z DTSTAMP:20260423T021329Z UID:GOThIC/28 DESCRIPTION:Title: Dimension series and homotopy groups of spheres\nby Laurent Bartholdi (Georg-August Universität zu Göttingen) as part of GOThIC - Ischia Onlin e Group Theory Conference\n\n\nAbstract\nThe lower central series of a gro up $G$ is defined by $\\gamma_1=G$ and $\\gamma_n = [G\,\\gamma_{n-1}]$. T he "dimension series"\, introduced by Magnus\, is defined using the group algebra over the integers: $$\\delta_n = \\{g: g-1\\text{ belongs to the $ n$-th power of the augmentation ideal}\\}.$$\n\nIt has been\, for the last 80 years\, a fundamental problem of group theory to relate these two seri es. One always has $\\delta_n\\ge\\gamma_n$\, and a conjecture by Magnus\, with false proofs by Cohn\, Losey\, etc.\, claims that they coincide\; bu t Rips constructed an example with $\\delta_4/\\gamma_4$ cyclic of order 2 . On the positive side\, Sjogren showed that $\\delta_n/\\gamma_n$ is alwa ys a torsion group\, of exponent bounded by a function of $n$. Furthermore \, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.\n\nIn joint work with Roman Mikhailov\, we prove however that ever y torsion abelian group may occur as a quotient $\\delta_n/\\gamma_n$\; th is proves that Sjogren's result is essentially optimal.\n\nEven more inter estingly\, we show that this problem is intimately connected to the homoto py groups $\\pi_n^(S^m)$ of spheres\; more precisely\, the quotient $\\del ta_n/\\gamma_n$ is related to the difference between homotopy and homology . We may explicitly produce $p$-torsion elements starting from the order-$ p$ element in the homotopy group $\\pi_{2p}(S^2)$ due to Serre.\n LOCATION:https://researchseminars.org/talk/GOThIC/28/ END:VEVENT END:VCALENDAR