Splitting vector bundles and {\mathbb{A}}^{1}-fundamental groups of higher dimensional varieties

# Splitting vector bundles and A1-fundamental groups of higher dimensional varieties

Aravind Asok
Department of Mathematics
University of Southern California
Los Angeles, CA 90089-2532
asok@usc.edu
Aravind Asok was partially supported by National Science Foundation Awards DMS-0900813 and DMS-0966589.
###### Abstract

We study aspects of the -homotopy classification problem in dimensions and, to this end, we investigate the problem of computing -homotopy groups of some -connected smooth varieties of dimension . Using these computations, we construct pairs of -connected smooth proper varieties all of whose -homotopy groups are abstractly isomorphic, yet which are not -weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension.

Projectivizations of vector bundles give rise to -fiber sequences, and when the base of the fibration is an -connected smooth variety, the associated long exact sequence of -homotopy groups can be analyzed in detail. In the case of the projectivization of a rank vector bundle, the structure of the -fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank , the -fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher -homotopy groups is influenced by an appropriately defined higher obstruction class.

] ]

## 1 Introduction

The purpose of this paper is to study some aspects of the problem of classifying -connected smooth proper varieties of a fixed dimension up to -homotopy equivalence; this problem was introduced and discussed in [AM11]. We focus here on varieties having dimension and, in particular, projective bundles over a smooth -connected base. For -connected smooth varieties of dimension over an algebraically closed field having characteristic , in [AM11, §5] we showed the homotopy classification is largely governed by the -fundamental sheaf of groups (see Section 2 for a definition); henceforth we abbreviate -fundamental sheaf of groups to -fundamental group.

In dimensions , it is possible to describe all -connected smooth proper varieties over an algebraically closed field, and then to compute all the corresponding -fundamental groups (again, see [AM11, §5]). In contrast, in dimension , there seems to be no reasonable description of the collection of all -connected smooth proper -folds: by [AM11, Corollary 2.3.7], stably rational smooth proper complex -folds are -connected, and stably rational varieties are already quite mysterious (see also [AM11, Example 2.3.4]).

This paper is motivated in part by the fact that it is still, in a sense, possible to provide a homotopy classification of all closed -manifolds, even though we cannot answer the question of which finitely generated groups appear as fundamental groups of such manifolds [Tho69, Swa74, Hen77]; see Remark 5.9 for more discussion of this point. We consider the question: what invariants are necessary for -homotopy classification in dimensions greater than two?

Lens spaces provide examples of closed -manifolds that are homotopy inequivalent yet whose homotopy groups are abstractly isomorphic (see Remark 5.1 for some quick recollections on lens spaces). Therefore, if one believes that the analysis of -homotopy theory of smooth proper -connected -folds is in any way similar to the analysis of homotopy theory of closed -manifolds, then it seems reasonable to expect that knowledge of -homotopy sheaves of groups (henceforth, -homotopy groups) alone is not sufficient to distinguish all -homotopy types in dimension . In support of this expectation, we establish the following result.

###### Theorem 1 (See Theorems 5.4, 5.5 and Remark 5.6).

Suppose is an integer . There exist pointed -connected smooth proper -folds and that fail to be -weakly equivalent, yet for which is (abstractly) isomorphic to for all .

The examples arise from arguably the next simplest class of varieties beyond projective spaces: projective bundles of direct sums of line bundles over projective spaces. Voevodsky showed that integral motivic cohomology rings (in particular, Chow cohomology rings) are unstable -homotopy invariants, and we distinguish the -homotopy types of the examples in the previous theorem by direct computation of motivic cohomology rings. The bulk of the work is devoted to studying -homotopy groups of projectivizations of vector bundles over -connected base schemes to establish the existence of the abstract isomorphism of the theorem statement.

By careful choice of the projective bundles, -homotopy groups of degree can be controlled in a rather straightforward fashion. On the other hand, the -fundamental group of a positive dimensional smooth proper -connected variety is always non-trivial [AM11, Proposition 5.1.4]. Very few computations of -fundamental groups exist, and much of the paper is devoted to providing such computations for projective bundles over an -connected smooth base. Our computations can be viewed as providing a “relative” version of Morel’s fundamental computations [Mor12, §7.3] of “low-degree” -homotopy groups of ; just as in topology, higher -homotopy sheaves of are isomorphic to -homotopy sheaves of spheres and are therefore extremely complicated in general.

Before describing the computation, we mention another motivation for this investigation. An old question of Schwarzenberger asks about the existence of non-split small rank vector bundles on projective spaces (see, e.g., [MFK94, p. 227] or [OSS80, §4.4] for discussion of this question). A precise form of this question is the conjecture of Hartshorne asserting that a rank vector bundle on splits so long as ; see [Har74, Conjecture 6.3], though no counterexamples are known even for . Of course, if this conjecture is true, then the -homotopy type of the projectivization of any rank vector bundle on projective space of large dimension is indistinguishable from the -homotopy type of the projectivization of a split bundle. At the very least, this circle of ideas suggests that the splitting behavior of a vector bundle influences the -homotopy type, and this belief is borne out in our computations.

Just as in topology, we can study the -homotopy groups of projective space bundles using the theory of -fiber sequences, via techniques pioneered by F. Morel and extended by M. Wendt [Mor12, Wen11]. Indeed, given such a bundle, there is an associated long exact sequence relating the -homotopy groups of the base, fiber and total space; see Corollary 3.22. Our main goals are thus twofold. First, we want to provide explicit descriptions of the connecting homomorphisms; this is accomplished by study of the -Postnikov tower (see Definition 2.11). Second, we want to solve the resulting extension problem to determine the group structure on the -fundamental group. Proposition 3.4 shows that projective bundles on -connected smooth varieties over algebraically closed fields of characteristic are automatically projectivizations of vector bundles, so discussion can be simplified by restricting to this case.

Proposition 4.4 and Theorem 4.16 demonstrate that if a vector bundle splits, then the -homotopy groups of the associated projective bundle are very restricted. In particular, the -fundamental group of the projectivization of a split vector bundle is a split extension of the -fundamental group of the base by the -fundamental group of the fiber (i.e., projective space), and we completely describe the group structure on this extension. Furthermore, the higher -homotopy groups of a split bundle are simply a product of the -homotopy groups of the base and fiber.

In contrast to the split case, a certain obstruction class, which we refer to as the Euler class (see Definition 4.8), intercedes in the structure of the -fundamental group of a potentially non-split bundle: the -fundamental group of the total space is an extension of the -fundamental group of the base by a quotient of the -fundamental group of the fiber. Moreover, this extension of sheaves of groups is no longer obviously split. We show that, if our Euler class is trivial (which always happens, e.g., if the -connected base variety is , ) then i) the -fundamental group of the total space is again a split extension of the -fundamental group of the base by the -fundamental group of the fiber, and ii) we can write down the group structure on the extension.

While is quite simple for , Morel showed [Mor12, §7.3], the -fundamental sheaf of groups of is a highly non-trivial (non-abelian!) sheaf of groups. As a consequence, the study of -fundamental groups of projectivizations of rank bundles is much more complicated than the case of higher rank bundles. We recall Morel’s description of in Theorem 2.29.

The outlines of our approach are already visible in the study of homotopy groups of -bundles over closed connected manifolds; we review this study as motivation at the beginning of §4. The following statement summarizes our calculations in the situation where the -connected base variety is a projective space (though more general results are established in the text).

###### Theorem 2 (See Theorems 4.6 and 4.16).

Suppose is a rank vector bundle on , and , and fix an identification . If , then there are isomorphisms

If , and furthermore is trivial (see Definition 4.8), then there is a split short exact sequence of the form

 1⟶πA11(P1)⟶πA11(P(E))⟶Gm⟶1;

the class of the extension is determined by the image of in .

###### Remark 3.

In [Mor12, Theorem 8.14], Morel introduces an Euler class, defined via obstruction theory, that provides the only obstruction to splitting a rank vector bundle on a smooth affine scheme of dimension . The Euler class we define is conceived in the same spirit: it provides an obstruction to splitting a vector bundle up to the --(co)skeleton of the base (see Definition 2.11) in a manner independent of the dimension of (or whether is affine), though at the expense of introducing some -connectivity hypotheses. It can be shown that our Euler class coincides with Morel’s Euler class, but we have not provided a detailed verification here. In a sense, our discussion is classical (cf. [MS74, Chapter 12]), with some additional contortions necessitated in order to avoid imposing orientability hypotheses.

Regarding computations of -homotopy groups, this work suggests two natural questions. First, in the case where is a rank vector bundle on an -connected smooth variety and is non-trivial, the theorem above is definitely not correct; we explain this in more detail in Example 4.20. What does the -fundamental group look like in this case? Second, providing a description of higher -homotopy groups of projective bundles will require more effort. Proposition 4.4 provides some statements in the case of split vector bundles and Lemma 4.26 gives a statement regarding possibly non-split vector bundles in the presence of strong -connectivity hypotheses on the base variety (satisfied, for example, if the base variety is a high-dimensional projective space). Recent computations of higher -homotopy sheaves of punctured affine spaces (see [AF12b, AF12a, AF12c]) open the door to providing a more detailed description of the next obstruction in the case of -bundles, and the first “interesting” obstruction in the case of -bundles.

### Overview of sections

Section 2 recalls a number of general facts from -homotopy theory and -algebraic topology, especially aspects related to the theory of the -fundamental sheaf of groups. The subject we consider is quite young an in an attempt to keep the paper reasonably self-contained, we have chosen to repeat statements of results. Section 3 collects a number of facts about specific -fiber sequences related to classifying spaces for and , which will be necessary for analyzing the various obstructions that arise. Finally, the main results are proven in Sections 4 and 5. We refer the reader to the introduction to each section for a more detailed list of contents.

### Acknowledgements

We wish to thank Fabien Morel for many discussions related to the -homotopy classification problem, and Brent Doran for discussions about the theory of variation of GIT quotients; taken together, these discussions provided the motivation for the construction of the examples required for Theorem 1. We thank Marc Levine and Christian Haesemeyer for discussions about notions of motivic degree and Euler classes. We thank Matthias Wendt for discussions regarding his work and a number of insightful comments on a draft of this paper. We also thank Lukas-Fabian Moser for making preliminary versions of his results available to us, and Ben Williams for discussions about obstructions. Finally, we thank one of the referees of a previous version of this paper for providing an extremely detailed report with a number of comments and corrections that certainly improved its form and substance.

## 2 Review of some A1-algebraic topology

In this section, we review some basic facts from -homotopy theory and -algebraic topology. The material in this section will be used throughout the paper. Much of what we write here can be found in the Morel’s beautiful work [Mor12], but we have provided some proofs of results stated but not proven in ibid.

### A1-homotopic preliminaries

Fix a field ; in the body of the text is assumed to have characteristic if no alternative hypotheses are given. Write for the category of schemes separated, smooth and finite type over . The word sheaf we will always mean Nisnevich sheaf on . We write for the category of simplicial Nisnevich sheaves on ; objects of the category will be referred to as -spaces, or simply as spaces if is clear from context. Sending a smooth scheme to the representable functor and taking the associated constant simplicial sheaf (i.e., all face and degeneracy maps are the identity map) determines a fully faithful functor . Spaces will generally be denoted by calligraphic letters, while schemes will be denoted by capital roman letters; we use the composite functor just mentioned to identify with its essential image in and we use the same notation for both a scheme and the associated space. Write for the category of pointed -spaces, i.e., pairs consisting of a -space and a morphism .

We write for the (Nisnevich) simplicial homotopy category and for its pointed analog; each of these categories is the homotopy category of a model structure on or . For more details regarding the precise definitions, we refer the reader [MV99, §2 Definition 1.2]. The set of morphisms between two spaces in or will be denoted , though when considering pointed homotopy classes of maps, the base-point will be explicitly specified unless it is clear from context. For a pointed space , we write for the simplicial suspension operation and for the space , where is a simplicially fibrant model of . As usual, simplicial suspension is left adjoint to (simplicial) looping.

We use the Morel-Voevodsky -homotopy category and its pointed analog ; these categories are constructed as a (left) Bousfield localization of either or ; see [MV99, §3 Definition 2.1]. In particular, the category is equivalent to the subcategory of consisting of -local objects, and the inclusion of the subcategory of -local objects admits a left adjoint called the -localization functor. There is a choice of such a functor, for which we write , that commutes with formation of finite products (one model of the functor is given in [MV99, §2 Lemma 3.20], and one replaces in this lemma by the Godement resolution functor via [MV99, §2 Theorem 1.66]). We write for morphisms in or and explicitly indicate the base-point in the latter case.

An important fact regarding the -local model structure is that it is proper; in particular, pullbacks of -weak equivalences along -fibrations are again -weak equivalences; see [MV99, §2 Theorem 3.2]. We will also import various definitions from classical homotopy theory (e.g., regarding connectivity) to the simplicial or -homotopy category by prepending either the word simplicial or the symbol ; when we do not give precise definitions, the terms are defined in analogy with their classical topological counterparts.

If is a space, the sheaf of simplicial connected components, denoted , is the Nisnevich sheaf on associated with the presheaf . Similarly, the sheaf of -connected components of , denoted is the Nisnevich sheaf on associated with the presheaf . A space is simplicially connected if the canonical morphism is an isomorphism, and -connected if the canonical morphism is an isomorphism. We recall the following result, which provides many examples of -connected varieties, and suffices to establish -connectedness of most of the examples in the remainder of the paper.

###### Proposition 2.1.

If is a field, and is a smooth scheme admitting an open cover by open sets isomorphic to affine space (with non-empty intersections), then is -connected.

If is a pointed space, then is the Nisnevich sheaf on associated with the presheaf , and is the Nisnevich sheaf on associated with the presheaf (here, we take pointed homotopy classes of maps). These are sheaves of groups for , and sheaves of abelian groups for . For notational compactness, we will often suppress base-points when writing -homotopy groups when the space under consideration is either -connected or if the choice of base-point is clear from context.

A presheaf on is -invariant if for every the canonical map is a bijection. A sheaf of groups (possibly non-abelian) is strongly -invariant if the cohomology presheaves are -invariant for . A sheaf of abelian groups is strictly -invariant if the cohomology presheaves are -invariant for . The main structural properties of the sheaves are summarized in the following fundamental results due to Morel.

###### Theorem 2.2 ([Mor12, Theorem 6.1 and Corollary 6.2]).

If is a pointed space, then for any integer the sheaves are strongly -invariant.

###### Theorem 2.3 ([Mor12, Theorem 5.46]).

If is a perfect field, a strongly -invariant sheaf of abelian groups on is strictly -invariant.

###### Notation 2.4.

We write for the category of strongly -invariant sheaves of groups, and for the category of strictly -invariant sheaves of groups.

### Simplicial homotopy classification of torsors

Let be a Nisnevich sheaf of groups. Let denote the C̆ech construction of the epimorphism , and let denote the (Nisnevich) sheaf quotient for the diagonal (right) action of on . The space , which is the simplicial classifying space for , has a canonical base point, which we denote in the sequel. We write for the set of Nisnevich locally trivial -torsors over , i.e., triples consisting of a (right) -space , a morphism equivariant for the trivial right action on , and isomorphism of Nisnevich sheaves . The terminology classifying space is justified by the following result.

###### Theorem 2.5 ([Mv99, §4 Propositions 1.15 and 1.16]).

If is a Nisnevich sheaf of groups, then for any space there is a canonical bijection

 [X,BG]s∼⟶H1Nis(X,G).

Moreover, for any integer , and any smooth scheme , the group is isomorphic to if and is trivial if .

If is a linear algebraic group viewed as a Nisnevich sheaf of groups and is a smooth scheme, the set studied above can actually be identified with the set of Nisnevich locally trivial -torsors on in the usual sense. For our purposes, it suffices to observe that a Nisnevich locally trivial -torsor on in the usual sense gives rise to a -torsor on in the sense above by means of the Yoneda embedding.

###### Corollary 2.6.

If is a Nisnevich sheaf of groups, then .

###### Proof.

By the unstable -connectivity theorem [MV99, §3 Corollary 3.22] there is an epimorphism

 πs0(BG)⟶πA10(BG).

The sheaf is, by definition, the Nisnevich sheafification of . To check triviality of this sheaf, it suffices to check triviality over stalks. By Theorem 2.5 and the discussion just prior to the statement, if is a Henselian local scheme, corresponds to the set of isomorphism classes of Nisnevich locally trivial -torsors over . However, Nisnevich locally trivial -torsors over are trivial. ∎

###### Remark 2.7.

One immediate consequence of the above discussion is the fact that a Nisnevich sheaf of groups is strongly -invariant if and only if the classifying space is -local; we use this observation repeatedly in the sequel. Using Theorem 2.2, the inclusion of the subcategory of strongly -invariant sheaves of groups into the category of Nisnevich sheaves of groups admits a left adjoint defined by : this left adjoint creates finite colimits, e.g., amalgamated sums.

If is a (simplicially) fibrant model of , then any element of can be represented by a morphism of simplicial sheaves . If is simplicially connected, we can choose a base-point making the aforementioned morphism a morphism of pointed simplicial sheaves. Thus, any -torsor on a simplicially connected space can be represented by a pointed morphism for an appropriate choice of base-point. In the sequel, will be an -connected smooth scheme, in which case the -localization is a simplicially connected space by the unstable -connectivity theorem [MV99, §3 Corollary 3.22].

###### Corollary 2.8.

If is a Nisnevich sheaf of groups, there is a canonical simplicial weak equivalence .

###### Proof.

For any smooth scheme there is an adjunction

 [ΣisU+,RΩ1sBG]s∼⟶[Σi+1sU+,BG]s.

By Theorem 2.5, if the presheaf on the right hand side is precisely , and if it is trivial. In other words, after sheafification, the morphism of spaces is a simplicial weak equivalence. ∎

### Generalities on fiber sequences

We fix some results about -fiber sequences; this material is taken from [Hov99, §6.2]. In any pointed model category, there is a notion of loop object. In the setting of -homotopy theory, if is a pointed -fibrant space, then the simplicial loop space is precisely the model categorical notion of loop space.

If is an -fibration between pointed -fibrant objects (equivalently, by [MV99, §2 Proposition 2.28], simplicially fibrant and -local objects), let be the fiber of this map, and be the inclusion of the fiber. The general formalism of model categories gives an action of on , specified functorially. Here is the construction. Given a space , an element of can be represented by a morphism (where is the simplicial interval). If is a morphism, by composition we get a morphism . Let be a lift in the diagram

where is the inclusion at . One then defines , where is the unique map such that . By [Hov99, Theorem 6.2.1], this construction gives a well-defined right action of on . In this situation, there is a boundary morphism defined as the composite

 (2.1) δ:Ω1sB⟶F×Ω1sB⟶F,

where the first map is the product of the inclusion of the base-point and the identity map, and the second map is the action map just constructed.

###### Definition 2.9.

An -fiber sequence is a diagram of pointed spaces together with a right action of on that is isomorphic in to a diagram where is an -fibration of (pointed) -fibrant spaces with (-homotopy) fiber together with the action of on discussed above.

Fiber sequences in topology give rise to long exact sequences in homotopy groups; this result can be generalized to the context of an arbitrary pointed model category. Applying this observation in the context of -homotopy theory the next result is a consequence of (the dual of) [Hov99, Proposition 6.5.3] together with a sheafification argument. (Note: a corresponding result holds for fiber sequences in the simplicial homotopy category as well.)

###### Lemma 2.10.

If is an -fiber sequence, then there is a long exact sequence in -homotopy sheaves

 ⋯⟶πA1i+1(Z)δ∗⟶πA1i(X)⟶πA1i(Y)⟶πA1i(Z)⟶⋯,

where is the map on -homotopy sheaves induced by the morphism of Equation 2.1, the sequence terminates with (here, the expression “long exact sequence” is modified in a standard fashion for or where the constituents are groups or pointed sets).

### Postnikov towers in A1-homotopy theory

###### Definition 2.11.

An -Postnikov tower for a pointed space consists of a sequence of pointed spaces , together with pointed maps , and pointed maps having the following properties.

• The morphisms are all -fibrations.

• The morphisms induce isomorphisms of sheaves for .

• The sheaves are trivial for .

• The induced map is an -weak equivalence.

###### Theorem 2.12 (Morel-Voevodsky).

An -Postnikov tower exists, functorially in the input space .

###### Sketch of proof..

The proof of this fact is really a construction. The simplicial Postnikov tower is constructed for any space in [MV99, §2 pp. 57-61]; roughly speaking this provides all of the statements in the simplicial homotopy category.

Consider the simplicial Postnikov tower of ; we write for the -th stage of this tower. Morel and Voevodsky show that the map

 LA1X⟶holimi(LA1X(i))

is a simplicial weak equivalence (that is the content of [MV99, Definition 1.31], that the Nisnevich topology satisfies this hypothesis is [MV99, Theorem 1.37]).

By definition, . By Theorem 2.2 and Theorem 2.3, we know that is strongly -invariant and is strictly -invariant for any . Equivalently, is -local, and is -local for any and every . By induction, one can then replace the map by an -fibration without changing the -homotopy fiber. ∎

### Looping and A1-localization

If is a simplicially fibrant (but not necessarily -local space), then there is a canonical pointed map that induces a morphism of spaces

 Ω1sX⟶LA1Ω1sX⟶Ω1sLA1X.

While the first morphism is an -weak equivalence by its very definition, the second morphism is not an -weak equivalence in general. Since , by Theorem 2.2, a necessary condition that this morphism be an -weak equivalence is that is a strongly -invariant sheaf of groups. Morel proves that this condition is also sufficient.

###### Theorem 2.13 ([Mor12, Theorem 6.46]).

The morphism is an -weak equivalence if and only if the sheaf of groups is strongly -invariant.

### A1-covering spaces

One consequence of the existence of the -Postnikov tower is the existence of an -universal covering space and a corresponding collection of results one refers to as -covering space theory. For more details about the constructions below, see [Mor12, §7.1].

###### Definition 2.14.

If is a space, an -cover of is a morphism that has the unique right lifting property with respect to morphisms that are simultaneously -weak equivalences and cofibrations. If is an -connected space, then an -universal cover of is a space that is -connected and -simply connected.

###### Remark 2.15.

By their very definition, -covers are -fibrations. Therefore, they give rise to -fiber sequences. We will use this observation repeatedly in the sequel.

If is a pointed -connected space, then the -Postnikov tower shows that is -weakly equivalent to the chosen base-point, and is -weakly equivalent to . Morel deduces from these observations (see [Mor12, Theorem 7.8]) that the -homotopy fiber of the map is an -universal cover of . More precisely, for any -connected space , an -universal covers exists, and the construction is functorial in the input space.

Using Theorem 2.5 and the fact that is -local, one observes that is a (“Nisnevich locally trivial”) -torsor over . Said differently, admits a free transitive right action of , which one can think of as the action by “deck transformations.”

###### Remark 2.16.

Suppose is an -connected space. Fix a base-point and set . Since the -universal cover is an -simply connected space equipped with a free right action of , functoriality of the Postnikov tower shows that the Postnikov tower of admits a right -action. Keeping track of the action of gives rise to a twisted Postnikov tower of , which will be necessary for doing obstruction theory later. This twisted Postnikov tower is explained in the simplicial setting in [GJ99, VI.4-5].

###### Lemma 2.17.

If is a pointed -cover, then the induced map

 πA1i(Y′)⟶πA1i(Y)

is an epimorphism for and an isomorphism for .

###### Proof.

We can assume without loss of generality that and are -fibrant. In that case, the -homotopy fiber of the map is precisely the actual fiber of this morphism. Let . Since pullbacks of -coverings are -coverings, it follows that is an -covering of . The unique right lifting property then implies that any pointed morphism is -homotopically constant if , and so the canonical morphism is an isomorphism. The result of the claim follows from the long exact sequence in -homotopy sheaves of a fibration. ∎

The following result, which is a straightforward consequence of obstruction theory and the fact that is strongly -invariant (equivalently is -local), provides a universality property of ; see [Mor12, Lemma B.7].

###### Theorem 2.18.

If is a pointed -connected space, and is any strongly -invariant sheaf of groups, then the morphism

 [(X,x),BG]A1⟶HomGrA1k(πA11(X),G)

induced by evaluation on is a bijection.

One consequence of this theorem is the following result that is referred to as the unstable -connectivity theorem: a simplicially -connected space is --connected.

###### Corollary 2.19 ([Mor12, Theorem 6.38]).

Suppose is an integer. If is a pointed simplicially -connected space (i.e., for all ), then is simplicially -connected.

###### Proof.

If is a pointed -connected space, then is pointed and -connected by [MV99, §2 Corollary 3.22]. Therefore, suppose is a pointed and simplicially -connected space. Since is simplicially -connected, it suffices to prove is trivial. Equivalently, it suffices by Theorem 2.18 and the Yoneda lemma to prove that for any strongly -invariant sheaf of groups , the set is trivial. If is strongly -invariant, is -local, so the canonical map is an isomorphism. However, since is simplicially -connected, it follows that the canonical map is a bijection as well (the homomorphisms on the right hand side are taken in the category of Nisnevich sheaves of groups). Since is simplicially -connected, this pointed set is trivial.

There are a number of ways to prove the result for , but all the methods we know implicitly involve Theorem 2.3. We proceed by induction on . Suppose is simplicially -connected and is simplicially -connected. It suffices by the Yoneda lemma to show that if is an arbitrary strictly -invariant sheaf of groups, then is trivial. Indeed, using the fact that is simplicially -connected, one can show that the induced map

 [X,K(A,i)]A1→HomAbA1k(πA1i(X),A)

is a bijection (one argues using obstruction theory; see again [Mor12, Lemma B.7] or [AD09, Theorem 3.30]). In that case, the assumption that be strictly -invariant is equivalent to being -local. Therefore, the map is a bijection, and the set on the left hand side is trivial since is simplicially -connected. ∎

### Properties of the A1-fundamental group

If is a split torus, is strongly -invariant. Indeed, is -invariant since there are no nonconstant maps from to (in fact, is -rigid in the sense of [MV99, §3 Example 2.4]), and is -invariant by homotopy invariance of the Picard group. The following result encodes some facts about the role of torus torsors in -covering space theory.

###### Proposition 2.20.

Assume is a pointed -connected smooth scheme, and let be a split torus.

• There is a canonical isomorphism .

• If is a -torsor with also -connected, then for satisfying , there is a short exact sequence

 1⟶πA11(~X,~x)⟶πA11(X,x)⟶πA10(T)⟶1,and
• there are isomorphisms for all .

###### Proof.

The first statement is a consequence of Theorem 2.18 using the fact that is an abelian sheaf of groups. In that case, the set is the quotient of by the induced conjugation action of , which is trivial since is abelian. The second statement follows immediately from Lemma 2.10 and Theorem 2.5. The third statement follows immediately from Lemma 2.17. ∎

###### Remark 2.21.

An immediate consequence of the first observation is that the -fundamental group of a (strictly positive dimensional) smooth proper -connected scheme is always non-trivial: take and use the fact that the Picard group of a (strictly positive dimensional) smooth proper -scheme is always non-trivial.

### A1-homotopy theory of projective spaces and punctured affine spaces

We now recall some results, due to Morel, regarding the structure of the -homotopy groups of projective space. The results are essentially a breezy overview of [Mor12, §7.3] introducing the main objects we will need in subsequent sections.

The standard open cover of by two copies of with intersection realizes as the colimit of the diagram . The natural map from the homotopy colimit of this diagram to the colimit of this diagram is an -weak equivalence, and yields an -weak equivlence [MV99, §3 Corollary 2.18]. Using induction and a similar open covering by two sets, one shows that is -weakly equivalent to [MV99, §3 Example 2.20]. It follows that is simplicially -connected. By the unstable -connectivity theorem (see Corollary 2.19), it follows that is also --connected.

Morel gives a description of the st -homotopy group of in terms of so-called Milnor-Witt K-theory sheaves (see [Mor06, §4] or [Mor12, §3] for details regarding this theory). One could take the following result as a definition of Milnor-Witt K-theory sheaves, and this point of view will suffice for much of the paper. However, the results of [Mor12, §3] actually provide a concrete description of the sections of this sheaf over fields, which completely determines the sheaf since it is strictly -invariant.

###### Proposition 2.22 ([Mor12, Theorem 6.40]).

For every integer , there is a canonical isomorphism .

Combining Propositions 2.20 and 2.22, one can deduce the following computations of the -homotopy groups of projective spaces; these properties will be used repeatedly in Section 4.

###### Lemma 2.23.

Suppose is an integer . There are isomorphisms

 πA1i(Pn)=⎧⎪⎨⎪⎩Gm if i=10 if 1
###### Remark 2.24.

For the sake of comparison, we note that the above computation bears a formal resemblance to the computation of the homotopy groups of . Indeed, the structure of the group () depends on . If , then . In that case, is non-vanishing only for , in which case it is isomorphic to . If , then the canonical map is a covering space and gives identifications and for . In particular, vanishes in the range .

###### Remark 2.25.

The fact that one can choose to commute with formation of finite products implies that the -th -homotopy group of a product of -connected spaces is the product of the -th -homotopy groups of the constituent factors (one can check the isomorphism on stalks and reduce to the corresponding fact for simplicial sets).

The homotopy colimit description of above gives the identification . Any space of simplicial dimension (e.g., a sheaf of groups) is simplicially fibrant by [MV99, §2 Proposition 1.13]. Using this fact, Corollary 2.8, and the loop-suspension adjunction there is a sequence of bijections

 HomSpck,∙(Gm,G)⟶[Gm,G]s⟶[Gm,Ω1sBG]s⟶[Σ1sGm,(BG,∗)]s

for any Nisnevich sheaf of groups .

If is, in addition, strongly -invariant, there is an identification . Precomposing the (bijective) morphism of Theorem 2.18 with the sequence of maps described in the previous paragraph gives the following result.

###### Lemma 2.26.

If is a strongly -invariant sheaf of groups, then the morphism

 HomSpck,∙(Gm,G)⟶HomGrA1k(πA11(Σ1sGm),G)

described in the previous two paragraphs is a bijection, functorially in .

This identification can be interpreted as saying is the free strongly -invariant sheaf of groups generated by , and for that reason one uses the following notation.

Set .

###### Lemma 2.28.

There is a short exact sequence of sheaves of groups of the form

 1⟶KMW2⟶FA1(1)⟶Gm⟶1

The second to last arrow on the right admits a set-theoretic splitting.

###### Proof.

The usual -torsor is an -covering space and thus gives rise to an -fiber sequence. The first statement then follows immediately by combining Propositions 2.20 and 2.22.

Taking in Lemma 2.26, the identity map corresponds to a pointed morphism of sheaves of sets . Functoriality of the construction in Lemma 2.26 applied to the morphism of strongly -invariant sheaves of groups shows that the composite map is the identity map and thus the claimed morphism is in fact a set-theoretic splitting. ∎

The set-theoretic section gives an isomorphism of sheaves of sets . The group structure on is then specified by a factor set, i.e., a morphism of sheaves (satisfying an appropriate cocycle condition) by means of the formula

 ψ(a,x)ψ(b,y)=(abΦ(xy),xy).

Morel identifies this factor set explicitly. To this end, he constructs a symbol morphism (see [Mor12, Theorem 3.37]) that can be precomposed with the epimorphism that collapses ; we refer to this composite morphism also as the symbol morphism.

###### Theorem 2.29 ([Mor12, Theorem 7.29]).

The sheaf is a central extension of by with corresponding factor set given by the symbol morphism.

### Automorphisms of some A1-homotopy sheaves

When we study group structures on -homotopy sheaves in Section 4, we will need some information about the sheaf of automorphisms of various homotopy sheaves. While most cases that appear are short enough to be treated without disturbing the exposition, the automorphisms of require more care.

The sheaf of endomorphisms of admits a rather explicit description. Indeed, we know that since is the free strongly -invariant sheaf of groups on . The sheaf of morphisms from to admits a particularly nice description in terms of what are called “contractions.” If is a sheaf of pointed sets, set