We start to study the problem of classifying smooth proper varieties over a field k from the
standpoint of A 1-homotopy theory. Motivated by the topological theory of surgery, we …

We study a version of the James model for the loop space of a suspension in unstable A 1–
homotopy theory. We use this model to establish an analog of GW Whitehead's classical …

$${\mathbb A}^1$$ -homotopy Theory and Contractible Varieties: A Survey | SpringerLink
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We show that $\mathbb {A}^ 1$-connectedness of a large class of varieties over a field $ k $
can be characterized as the condition that their generic point can be connected to a $ k …

The essence of Galois theory is to lose information, by passing from a field k, an algebraic
structure with two compatible operations, to a (profinite) group, its absolute Galois group G k …

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the
Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study …

We show that the sheaf of A 1 A^1‐connected components of a reductive algebraic group
over a perfect field is strongly A 1 A^1‐invariant. As a consequence, torsors under such …

In this article we prove that any $\mathbb {A}^ 1$-connected smooth $ k $-variety is
$\mathbb {A}^ 1$-uniruled for any algebraically closed field $ k $. We establish that if a non …

This research article provides a thorough exploration of invariants in birational mapping,
focusing on their properties, applications, and mathematical significance in algebraic …

arXiv:2101.04951v2 [math.AG] 25 May 2022 Page 1 arXiv:2101.04951v2 [math.AG] 25 May
2022 ZEROTH A1-HOMOLOGY OF SMOOTH PROPER VARIETIES JUNNOSUKE KOIZUMI …