The Unstable Slice Filtration

TL;DR

This paper develops an unstable analogue of Voevodsky's slice filtration in motivic homotopy theory using birational invariants, leading to slices that are better behaved as homotopy fibers.

Contribution

It introduces a new unstable slice filtration constructed via birational invariants, bridging stable and unstable motivic homotopy categories.

Findings

Constructed an unstable slice filtration using birational invariants.

Slices appear as homotopy fibers, improving their properties in the unstable setting.

Established an equivalence between orthogonal components and birational motivic categories.

Abstract

The main goal of this paper is to construct an analogue of Voevodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the existence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy categories constructed in \cite{Pelaez:2011fk}. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting.