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Relations between slices and quotients of the algebraic cobordism spectrum. (English) Zbl 1209.14019
Let \(MGL\) be the algebraic cobordism spectrum. V. Voevodsky [Int. Press Lect. Ser. 3, No. I, 3–34 (2002; Zbl 1047.14012)] introduced the slice filtration and formulated a conjecture known as the slice conjecture [Conjecture(5)]. This conjecture describes the slice filtration using motivic Eilenberg-Mac Lane spectrum amd the coefficients of the complex cobordism \(MU_{*}.\) The author discusses the relation between the slice conjecture and quotients of \(MGL.\) Let \({ L}_{*}\) be the graded Lazard ring. In motivic homotopy theory the filtration on \(MGL\) obtained by dividing out by ideals of \(L_{*}\) generated by the \(x_{i},\) or more precisely ideals of \(L_{*}\) consisting of elements of degree greater than a given bound, is conjecturally the slice filtration. The author proves that if this holds on the level zero then it holds for all levels. The situation for rational slices and \(K\)-theory is also described.

14F42 Motivic cohomology; motivic homotopy theory
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
19E20 Relations of \(K\)-theory with cohomology theories
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