# K-theory of the integers and the Kummer-Vandiver conjecture

## February 7 at 11:15 – 12:45 CET

Lorenzo Mantovani: Suspended Tits buildings

Georg Tamme: K4(Z) is the trivial group

Talk 9: Suspended Tits buildings (07.02. Lorenzo Mantovani)

This talk covers the results of [Rog00, §5, §6]. Explain the explicit identifications of the poset rank filtration and its subquotients for stable apartments [Prop. 5.1, Prop. 5.4]. Introduce Tits buildings and explain the relation between the Tits buildings of a PID and its fraction field [Lem. 6.1]. Maybe arrange with the subsequent talk’s speaker to cover some material from [§7] in order to alleviate their job.

Talk 10: K4(Z) is the trivial group (07.02. Georg Tamme)

This talks covers the results of [Rog00, §7, §8]. Introduce the component filtration of stable

buldings [Def. 7.1] and explain (as much as time permits) the associated spectral sequence

E1 s,t = Ht(GLk(R); Zs) ⇒ Hs+t( ̄Fk K(R)). Finally, explain what we can conclude about the rank filtration spectral sequence for K(Z) modulo the Serre subcategory of finite 2-groups [(8.4)]. Compute the low degrees of the spectrum homology H∗(K(Z)) of the spectrum K(Z) [Thm. 8.5] and subsequently the vanishing of K4(Z) [Thm. 8.6].