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Splitting vector bundles outside the stable range and $$\mathbb A^1$$ – homotopy sheaves of punctured affine spaces. (English) Zbl 1329.14045
The authors establish the following theorem: if $$k$$ is an algebraically closed field of characteristic different from $$2$$, $$X$$ a smooth affine $$4$$-fold over $$k$$ and $$E$$ a rank three vector bundle over $$X$$, then $$E$$ splits off a trivial summand if and only if the third Chern class of $$E$$ (in Chow theory) vanishes. This establishes a low-dimensional case of a conjecture of Murthy.
The proof is a crafty application of $$\mathbb{A}^1$$-algebraic topology, as initiated by Morel. In analogy with classical topology, there is a “space” (simplicial sheaf) $$\mathrm{BGL}_n$$ such that for a smooth affine variety $$X$$, the set of weak $$\mathbb{A}^1$$-homotopy classes $$[X, \mathrm{BGL}_n]_{\mathbb{A}^1}$$ is in natural bijection with the set of isomorphism classes of rank $$n$$ vector bundles over $$X$$. There is a map $$\mathrm{BGL}_{n-1} \to \mathrm{BGL}_n$$ corresponding to adding a trivial bundle, and so the question of splitting off a trivial bundle is converted into a lifting problem. This can be tackled by obstruction theory. Again in analogy with classical topology, there is an appropriate type of fibre sequence $$\mathbb{A}^n \setminus 0 \to \mathrm{BGL}_{n-1} \to \mathrm{BGL}_n$$. Thus the splitting problem is intimately related to the unstable homotopy sheaves of $$\mathbb{A}^n \setminus 0 \simeq S^{n-1} \wedge \mathbb{G}_m^{\wedge n}$$. As established by F. Morel [$$\mathbb A^1$$-algebraic topology over a field. Lecture Notes in Mathematics 2052. Berlin: Springer (2012; Zbl 1263.14003)], one has $$\underline{\pi}_d^{\mathbb{A}^1} (\mathbb{A}^n \setminus 0) = 0$$ for $$d < n-1$$ and $$\underline{\pi}_{n-1}^{\mathbb{A}^1} (\mathbb{A}^n \setminus 0) = \underline{K}_n^{MW}$$. This is where the primary (Euler class) obstruction lives. The authors explicitly compute $$\underline{\pi}_{3}^{\mathbb{A}^1} (\mathbb{A}^3 \setminus 0)$$ (and a host of other unstable homotopy sheaves), which allows them to completely control the question of rank three vector bundles over affine four-folds (since the higher homotopy sheaves are irrelevant for dimension reasons). This way their theorem is reduced to a (not entirely trivial) computation.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 55S35 Obstruction theory in algebraic topology 13C10 Projective and free modules and ideals in commutative rings 19A13 Stability for projective modules 19D45 Higher symbols, Milnor $$K$$-theory
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