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SUMMARY:Ergun Yalcin (Bilkent University)
DTSTART:20210208T103000Z
DTEND:20210208T113000Z
DTSTAMP:20260422T175914Z
UID:BilTop/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BilTop/12/">
 The Dade group of a finite group and dimension functions</a>\nby Ergun Yal
 cin (Bilkent University) as part of Bilkent Topology Seminar\n\nLecture he
 ld in SB-Z11.\n\nAbstract\nIf $G$ is a $p$-group and $k$ is a field of cha
 racteristic $p$\, then the Dade group $D(G)$ of $G$ \nis the group whose e
 lements are the equivalence classes of capped endo-permutation $kG$-module
 s\, \nwhere the group operation is given by the tensor product over $k$. T
 he Dade groups of p-groups have been \nstudied intensively in the last 20 
 years\, and a complete description of the group $D(G)$ has been \ngiven by
  Bouc in terms of the genetic sections of $G$.\n\nFor finite groups the si
 tuation is more complicated. There are two definitions of a Dade group of 
 a finite\ngroup given by Urfer and Lassueur\, however both definitions hav
 e some shortcomings. In a recent work \nwith Gelvin\, we give a new defini
 tion for the Dade group $D(G)$ of a finite group $G$ by introducing a noti
 on \nof Dade $kG$-module as a generalization of endo-permutation modules.\
 n \n\nWe show that there is a well-defined surjective group homomorphism $
 \\Psi$ from the group of super class \nfunctions $C(G\, p)$ to the Dade gr
 oup $D^{\\Omega} (G)$ generated by relative syzygies. Our main theorem \ni
 s the verification that the subgroup of $C(G\,p)$ consisting of the dimens
 ion functions of k-orientable real representations \nof $G$ lies in the ke
 rnel of $\\Psi_G$. In the proof we consider Moore $G$-spaces which are the
  equivariant versions \nof spaces which have nonzero reduced homology in o
 nly one dimension\, and use the techniques \nfrom homological algebra over
  the orbit category.\n \n\nThis is a joint work with Matthew Gelvin.\n
LOCATION:https://researchseminars.org/talk/BilTop/12/
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