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$$\mathbb A^1$$-algebraic topology over a field. (English) Zbl 1263.14003
Lecture Notes in Mathematics 2052. Berlin: Springer (ISBN 978-3-642-29513-3/pbk; 978-3-642-29514-0/ebook). x, 259 p. (2012).
This book deals with $$\mathbb{A}^1$$-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field, in which the affine line is imposed to be contractible. It is a natural sequel to to the paper by F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)].
A main technical result contained in this text is the precise nature of the $$\mathbb{A}^1$$-homotopy sheaves of $$\mathbb{A}^1$$-connected pointed spaces. Using these results the author introduces the $$\mathbb{A}^1$$-homology sheaves of a space and then proves several results, which are the analogues of classical theorems in algebraic topology such as the Brouwer degree theorem and Hurewicz theorem.
Let $$Sm_k$$ be the category of smooth schemes over a field $$k$$. A sheaf of groups $${\mathcal G}$$ on $$Sm_k$$ in the Nisnevich topology is said to be strongly $$\mathbb{A}^1$$-invariant if for any $$X\in Sm_k$$ the map $H^i_{Nis}(X,{\mathcal G})\to H^i_{Nis}(X\times \mathbb{A}^1,{\mathcal G})$ induced by the projection $$\mathbb{A}^1\times X\to X$$ is a bijection , for $$i= 0,1$$.
$${\mathcal G}$$ is said to be strictly $$\mathbb{A}^1$$-invariant if the above map is a bijection for all $$i\in\mathbb{N}$$. A basic example of strictly $$\mathbb{A}^1$$-invariant sheaves are Voevodsky’s $$\mathbb{A}^1$$-invariant sheaves with transfers.
The notion of strong $$\mathbb{A}^1$$-invariance given here is new: there are important examples of non-commutative strongly $$\mathbb{A}^1$$-invariant sheaves groups , e.g., $$\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)$$. The computation given in 7.3 of the book $$\pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)$$ shows that there is a central extension of sheaves of groups $0\to \mathbb{K}^{MW}_2\to \pi^{\mathbb{A}^1}_1(\mathbb{P}^1_k)\to \mathbb{G}_m\to 0,$ where, for a field $$F$$, $$K^{MW}_*(F)$$ is the Milnor-Witt $$K$$-theory of $$F$$ and $$\mathbb{K}^{MW}_n$$ is the sheaf of Milnor-Witt $$K$$-theory in weight $$n$$, so that, if $$X$$ is irriducible with function field $$F$$, one has $$\mathbb{K}^{MW}_n(X)\subset K^{MW}_n(F)$$.
The $$\mathbb{A}^1$$-fundamental group sheaves play a fundamental role in the understanding of $$\mathbb{A}^1$$-connected projective smooth varieties over $$k$$. In this context a version of Hurewicz theorem is the following
Theorem 0.1. Let $${\mathcal X}$$ be a pointed $$\mathbb{A}^1$$-connected space. Then the Hurewicz morphism $\pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X})$ is the initial morphism from the sheaf of groups $$\pi^{\mathbb{A}^1}_1({\mathcal X})$$ to a strictly $$\mathbb{A}^1$$-invariant sheaf of Abelian groups. This means that, given a strictly $$\mathbb{A}^1$$-invariant sheaf of Abelian groups $$M$$ and a morphism of sheaves of groups $$\pi^{\mathbb{A}^1}_1({\mathcal X})\to M$$, it factors uniquely trough $$\pi^{\mathbb{A}^1}_1({\mathcal X})\to H^{\mathbb{A}^1}_1({\mathcal X})$$.

MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F42 Motivic cohomology; motivic homotopy theory
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