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The simplicial suspension sequence in \(\mathbb A^1\)-homotopy. (English) Zbl 1365.14027
The article under review consists of a detailed study of certain aspects of unstable \(\mathbb{A}^1\)-homotopy theory. Among the contributions are:
An axiomatisation of the unstable connectivity property.
A James-style model for simplicial loops on a simplicial suspension.
The introduction of Whitehead products and the EHP sequence as a computational tools in unstable motivic homotopy theory.
Each of these contributions is of fundamental importance to the subject.
The authors use all of these tools to compute certain unstable motivic homotopy sheaves of spheres. For example they show that if \(i \geq 0\) and the base is a field of characteristic zero containing an algebraically closed field, then \[ \pi^{\mathbb{A}^1}_{4+i+5\alpha}(S_s^{3+i} \wedge \mathbb{G}_m^{\wedge 3}) = \mathbb{Z}/24. \]

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
55Q15 Whitehead products and generalizations
55Q20 Homotopy groups of wedges, joins, and simple spaces
55Q25 Hopf invariants
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