Boardman's Whole-Plane Obstruction Group for Cartan-Eilenberg Systems

Abstract

Each extended Cartan-Eilenberg system (H, ∂) gives rise to two exact couples and one spectral sequence. We show that the canonical colim-lim interchange morphism associated to H is a surjection, and that its kernel is isomorphic to Boardman's wholeplane obstruction group W , for each of the two exact couples.

On the motivic Segal conjecture

“…To ensure that $$\operatorname{res}$$ detects a homotopy class, we also need strong convergence in its bidegree. By [9, Theorem 7.3], see also [22, Theorem 3.9], it suffices to know that $$RE_\infty ^{s,t,0} = 0$$ whenever $$t-s = +1$$. By [35, Corollary 6.1] this condition is satisfied for $$S = \operatorname{Spec}k$$, subject to the additional hypothesis for $$\ell =2$$ that $$k$$ has finite virtual cohomological dimension.…”