an:07162216
Zbl 1441.14077
Heard, Drew
On equivariant and motivic slices
EN
Algebr. Geom. Topol. 19, No. 7, 3641-3681 (2019).
00445208
2019
j
14F42 55P91 18G80 55N20 55P42
motivic homotopy; equivariant homotopy theory; slice filtration; slice spectral sequence
Summary: Let \(k\) be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over \(\operatorname{Spec}(k)\) with the \(C_2\)-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of \( \boldsymbol{MGL} \) and \(M\mathbb{R} \), and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of \(M\mathbb{R}\) are even in the sense of \textit{M. A. Hill} and \textit{L. Meier} [Algebr. Geom. Topol. 17, No. 4, 1953--2011 (2017; Zbl 1421.55002)], and give a computation of the slice spectral sequence converging to \(\pi_{*,*}\boldsymbol{BP}\langle n \rangle/2\) for \(1 \le n \le \infty \).
Zbl 1421.55002