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015 Smooth models of motivic spheres

and the clutching construction

Aravind Asok∗ Brent Doran† Jean Fasel

Abstract

We study the representability of motivic spheres by smooth varieties. We show that cer-tain explicit “split” quadric hypersurfaces have theA1-homotopy type of motivic spheres overthe integers and that theA1-homotopy types of other motivic spheres do not contain smoothschemes as representatives. We then study some applications of these representability/non-representability results to the construction of new exoticA

1-contractible smooth schemes. Then,we study vector bundles on even dimensional “split” quadrichypersurfaces by developing analgebro-geometric variant of the classical construction of vector bundles on spheres via clutch-ing functions.

Contents1 Introduction 1

2 Geometric models of motivic spheres 5

3 A1-contractible subvarieties of quadrics 10

4 Clutching functions for principal G-bundles on algebraic spheres 13

1 Introduction

This note is concerned with several results about spheres inthe Morel-VoevodskyA1-homotopytheory [MV99], also known as motivic spheres. First, we make precise the idea that (split) smoothaffine quadric hypersurfaces are motivic spheres. Second, we try to better understand the situationsin which motivic spheres admit models as smooth schemes. Finally, we use homotopic ideas in anattempt to understand how to explicitly construct all vector bundles on split smooth affine quadrics.We now explain our results more precisely.

Given a commutative ringk, the Morel-VoevodskyA1-homotopy categoryH (k) is constructedby first enlarging the categorySmk of schemes smooth overSpec k to a categorySpck of spacesoverSpec k and then performing a categorical localization. The category Spck has all small limitsand colimits, but the functorSmk → Spck does not preserve all colimits that exist inSmk, e.g.,quotients inSmk that exist need not coincide with quotients computed inSpck.

∗Aravind Asok was partially supported by National Science Foundation Award DMS-1254892.†Brent Doran was partially supported by Swiss National Science Foundation Award 200021138071.

1

2 1 Introduction

The resulting homotopy theory of schemes bears a number of similarities to the classical homo-topy category. For example, there are two objects ofSpck that play a role analogous to that playedby the circle in classical homotopy theory. On the one hand, one can view the circle asI/0, 1whereI is the unit interval[0, 1]. If we viewA

1 as analogous to the unit interval, we can form thequotient spaceA1/0, 1 (pointed by the image of0, 1 in the quotient), and we call this spaceS1s . This quotient does exist as a singular scheme (i.e., a nodalcurve). However, it is nota priori

clear whetherS1s is isomorphic inH (k) to a smooth scheme. On the other hand, if we emphasize

the fact thatS1 has a group structure, then we could also think ofGm (pointed by1) as a version ofthe circle. Of course,Gm is a smoothk-scheme.

General motivic spheres are obtained by taking smash products (this notion makes sense inSpck because of existence of colimits) of copies ofS1

s andGm [MV99, §3.2]; by convention, wesetS0

s to be the disjoint union of two copies ofSpeck. While in classical topology spheres areby construction smooth manifolds, the example ofS1

s shows that, in stark contrast, the followingquestion inA1-homotopy theory has no obvious answer.

Question 1. Which motivic spheresSis ∧G∧jm have theA1-homotopy type of a smooth scheme?

Throughout this paper, we consider the smooth affine quadrichypersurfaces

Q2m−1 := Spec k[x1, . . . , xm, y1, . . . , ym]/〈∑

i

xiyi − 1〉, and

Q2m := Spec k[x1, . . . , xm, y1, . . . , ym, z]/〈∑

i

xiyi − z(1 + z)〉.

One immediately verifies that all of these hypersurfaces arein fact smooth overSpeck.Note thatQ0 is the subscheme ofA1 given by the two points0, 1 and thus coincides withS0

s .The hypersurfaceQ2m−1 is well-known to beA1-weakly equivalent toAm \ 0 (by projection ontox1, . . . , xm) and the latter isA1-weakly equivalent to the motivic sphereSm−1

s ∧ G∧mm [MV99,

Example 2.20]. Our first result provides an explicit description of theA1-homotopy type of theother family of quadrics, thus answering Question1 in a collection of cases.

Theorem 2 (See Theorem2.2.5). Let k be a commutative unital ring andn ≥ 0 an integer. Thereare explicitA1-weak equivalences

Qn ∼A1

Sm−1s ∧G∧m

m if n = 2m− 1 and

Sms ∧G∧mm if n = 2m

of spaces overSpeck.

Remark3. The interesting part of Theorem2 is the case wheren = 2m, and the result was knownpreviously form = 1, 2 by special geometric considerations. On the other hand, theS1-stablehomotopy types of the quadricsQ2m were known over fields having characteristic unequal to2 byunpublished work of Morel [Mor08], and Dugger and Isaksen [DI08]. We emphasize that Theorem2 is an unstableresult. In contrast to the corresponding proof forn odd, it uses the Nisnevichtopology in an essential way, via the Morel-Voevodsky homotopy purity theorem.

While Theorem2 tells us that some motivic spheresdo have theA1-homotopy type of smoothschemes, the following result shows that not all motivic spheres have this property.

3 1 Introduction

Proposition 4 (See Proposition2.3.1). If i, j are integers withi > j, the spheresSis ∧G∧jm do not

have theA1-homotopy type of smooth (affine) schemes.

Classically,Sn admits an open cover by two contractible subspaces homemorphic to Rn (via

stereographic projection), namelySn \ N andSn \ S, whereN andS are the North and Southpoles. The intersection of these two open sets is homeomorphic to the total space of a line bundleover Sn−1 and, in particular, homotopy equivalent toSn−1. Pursuing the analogy between theclassical spheres and the quadrics further, one can ask whether analogous covers exist in algebraicgeometry. We begin by establishing the existence ofA

1-contractible open subschemes coveringQ2n; the main results are summarized in the following theorem (see Subsection2.1 and Remark3.1.2for detailed explanations of the connections with [AD07]).

Theorem 5(See Theorem3.1.1and Corollary3.2.3). Supposek is a commutative unital ring. WriteX2m for the open subscheme ofQ2m defined as the complement ofx1 = · · · = xm = 0, z = −1.

1. The varietyX2m is A1-contractible overSpec k.

2. Form ≥ 3,X2m cannot be realized as a quotient of an affine space by a (scheme-theoretically)free action of a unipotentk-group scheme.

The identification of theA1-homotopy type of spheres has a number of applications. Pursingthe analogy with the classical geometry ofSn further, recall that the open cover ofSn describedabove also yields the “clutching construction” which provides a bijection between pointed homo-topy classes of mapsSn−1 → GLn(R) and isomorphism classes of real vector bundles onSn. Wepursue an analogous construction in algebraic geometry. The open subschemeX2m has an “oppo-site” defined as the complement ofx1 = · · · = xm = z = 0; this subscheme isA1-contractible aswell (if 2 is a unit ink, thenX2m and its opposite are isomorphic algebraic varieties). Whenm = 1,the intersection ofX2m and its opposite is isomorphic toA1 ×Gm. Form ≥ 2, the situation ismore complicated as the intersection ofX2m and its opposite is a quasi-affine but not affine variety,so every regular function on the intersection extends toQ2m itself. As a consequence, consider-ing the intersection ofX2m and its opposite does not produce a reasonable analog of the clutchingconstruction.

Instead, we produce two variants (abstract and concrete) ofthe clutching construction (see The-orems4.1.6and4.3.6). The abstract clutching construction works rather generally and, roughlyspeaking, yields a surjection from the set of “naive”A

1-homotopy classes of pointed maps fromQ2m−1 toGLr to the set of isomorphism classes of rankr vector bundles onQ2m; this result holdsoverZ (or, more generally, any base ring that is smooth over a Dedekind domain with perfect residuefields).

The concrete clutching construction refines the abstract clutching construction and shows howto produce explicitGLm-valued1-cocycles on a suitable open cover ofQ2m realizing any rankrvector bundle, at least under suitable additional restrictions on the basek. More precisely, considerthe open affine subschemesV 0

2m := Dz ⊂ Q2m andV 12m := D1+z ⊂ Q2m. The subschemesV 1

2m

andV 02m arenotA1-contractible. Nevertheless, there is an explicit morphism ψm : V 0

2m ∩ V12m →

Q2m−1 such that the vector bundle onQ2m attached to a pointed morphismf : Q2m−1 → GLr isobtained by gluing copies of the trivial bundle of rankr onV i

2m alongV 02m ∩V

12m via the composite

mapf ψm. One consequence of this construction is the following result, which is perhaps ofindependent interest.

4 1 Introduction

Theorem 6(See Theorem4.3.6(2)). If k is a regular ring, then for every integern ≥ 1, there existsan explicit matrixβm : Q2m−1 → GLm, constructed by Suslin, such that the rankn vector bundleonQ2m obtained by gluing trivial bundles via the automorphismβm ψm yields a generator of therank1 freeK0(k)-moduleK0(Q2m).

These results have a number of applications which will be developed elsewhere, but which wemention here for the sake of context. Naive homotopy classesof maps from a smooth affinek-schemeX to Q2n appear in problems related to complete intersection ideals[Fas15, §2]. On theother hand, the set of naiveA1-homotopy classes of maps from a smooth affine scheme toQ2n

was identified, at least ifk is an infinite field having characteristic unequal to2, with [X,Q2n]A1

in [AHW15b, Theorem 4.2.2]. These ideas will be put together with the results established herein [AF15]. Among other things, the set[X,Q2m]A1 is a “motivic cohomotopy group” (providedthe dimension ofX is ”small” compared tom): it can be equipped with a functorial abelian groupstructure and this additional structure can be used to studyvarious classical problems, e.g., relatedto comparison of “Euler class groups.”

Note on prerequisites

We close this introduction with a warning: the prerequisites required to read this paper are non-uniform. The proofs of Theorem2 and (the first part of) Theorem5 use only basic algebraicgeometry, standard results about cofiber sequences in pointed model categories [Hov99, Chapter6] and knowledge of the basic aspects of [MV99] (see the preliminaries for further discussion re-garding relaxation of hypotheses on the base); we have attempted to make them as accessible aspossible. The proofs of Proposition4 and the second part of Theorem5 are slightly more algebro-geometrically involved: the first requires some knowledge of properties of motivic cohomology[MVW06] while the second requires knowledge about Cousin complexes for coherent sheaves. Theresults of Section4 on the clutching construction require some familiarity with some of the resultsof [AHW15a, AHW15b] (which we review), together with some ideas from [Mor12]. The proof ofTheorem6 requires familiarity with some results of Suslin.

Acknowledgements

Some of the results claimed in this paper were announced in talks by first and second named authorsas early as 2007. For example, the original proof of Theorem2 we envisioned, specifically thedescription ofA1-homotopy types ofQn for n even≥ 6, was an elementary geometric argumentthat was supposed to work overSpecZ; this argument contained a gap. The authors would also liketo thank Paul Balmer, Dan Isaksen, Christian Haesemeyer, and Fabien Morel, for useful discussionsover the course of the project and Matthias Wendt for a numberof useful comments on a first draftof this note. Finally, we thank Marc Hoyois for helpful discussions about theA1-homotopy categoryover non-Noetherian base schemes.

Preliminaries/Notation

Throughout the paper the phrase “k is a ring” will always mean “k is a commutative unital ring”.We writeSmk for the category of schemes that are separated, finite type and smooth overSpeck,

5 2 Geometric models of motivic spheres

andSpck, i.e., the category of “spaces overk”, is the category of simplicial presheaves onSmk. Wewrite H Nis

s (k) (resp. H Niss,• (k)) for the Nisnevich local homotopy category (as in [MV99, §2.1])

andH (k) (resp.H•(k)) for the (pointed) Morel-VoevodskyA1-homotopy category [MV99].The assumptions here require further discussion ifk is not Noetherian of finite Krull dimension,

which are the explicit assumptions onk used in[MV99] in the construction of the Morel-VoevodskyA1-homotopy category. In contrast to [MV99], but following [AHW15a], we define the Nisnevich

topology onSmk to be the topology generated by finite families of etale mapsUi → X admit-ting a splitting sequence by finitely presented closed subschemes, in the sense of [MV99, §3, p.97]. This definition, which is equivalent to the “standard” definition [MV99, §3 Definition 1.2]if k is Noetherian by [MV99, §3 Lemma 1.5], is studied in more detail in [Lur11, §1]. We re-fer the reader to [Hoy14, Appendix C] for a “good” version of theA1-homotopy category over aquasi-compact, quasi-separated base scheme. We note here that arbitrary affine schemes are quasi-compact and quasi-separated. In addition, in the construction of theA1-homotopy category whenk is not Noetherian, one imposes Nisnevich descent (not hyperdescent) as discussed in [AHW15a,§3.1] andA1-invariance.

As usual, smooth schemes are, via the Yoneda embedding, identified with their correspondingrepresentable (simplicially constant) presheaves. IfX ∈ Spck, we writeX+ for the pointed spaceX

∐Spec k, pointed by the disjoint copy ofSpeck. If Y → X is a closed immersion of smooth

scheme, we writeνY/X for the normal bundle toY in X. In that case,Th(νY/X) is the Thom spaceof the vector bundleνY/X as in [MV99, §3 Definition 2.16].

Suppose(X , x) and (Y , y) are pointed spaces. We use the notationΣisX := Sis ∧ X forsimplicial suspension, and suppress the base-point. Likewise, we writeRΩ1

sX for the derivedsimplicial loops ofX : i.e., first apply Nisnevich fibrant replacement and then apply Ω1

s. We set[(X , x), (Y , y)]s := Hom

H Niss,• (k)((X , x), (Y , y)), [(X , x), (Y , y)]A1 := HomH•(k)((X , x), (Y , y))

[X ,Y ]s := HomH Niss

(k)(X ,Y ) and [X ,Y ]A1 := HomH (k)(X ,Y ). Finally, if (X , x) is a

pointed space, we writeπA1

i (X , x) for the Nisnevich sheaf associated with the presheafU 7→HomH•(k)(S

is ∧ U+, (X , x)) (such sheaves are typically denoted with boldface symbols).

2 Geometric models of motivic spheres

This section is devoted to establishing Theorem2 from the introduction. The argument proceedsby induction. Subsection2.1 gives an elementary geometric argument showing thatQ2 is A

1-weakly equivalent toP1, and a related (mostly elementary) argument showing thatQ4 isA1-weaklyequivalent to(P1)∧2; the first computation is the base-case for the induction argument. Subsection2.2contains the proof of the main result, which is Theorem2.2.5. Finally, Subsection2.3establishessome non-geometrizability results for motivic spheres.

2.1 On theA1-homotopy types ofQ2 and Q4

Proposition 2.1.1. If k = Z, then there is a (pointed)A1-weak equivalenceQ2∼→ P

1.

Proof. We can identifyQ2 as the quotient ofSL2 by its maximal torusT := diag(t, t−1) ∼= Gm

acting by right multiplication; this follows from a straightforward computation of invariants. IfB

6 2.2 On theA1-homotopy type ofQ2n, n ≥ 3

is the linear algebraic group of upper triangular matrices with determinant1, then the closed im-mersion group homomorphismT → B induces a morphism of homogeneous spacesSL2/T →SL2/B. This morphism of homogeneous spaces is Zariski locally trivial with fibers isomorphicto A

1 (use the standard open cover ofP1 by two copies ofA1), in particular anA1-weak equiva-

lence. To conclude, we observe that the quotientSL2/B is isomorphic toP1, and we can make thismorphism a pointed morphism by picking anyk-point in SL2/T and looking at the image of thisk-point inP

1.

There are two “natural” (pointed)A1-weak equivalences ofQ2 → P1 that we know. In the

previous proof, theA1-weak equivalence we wrote down arose from the structure ofQ2 as a ho-mogeneous space, but as we now explain this seems to be an “exceptional” low-dimensional phe-nomenon. Ifk is a ring in which2 is invertible, the varietiesQ2n are all isomorphic to homogeneousspaces for suitable orthogonal groups, but this seems falseif 2 is not invertible ink.

The other “natural” (pointed)A1-weak equivalence arises as follows. The locus of points wherex1 = 0 andz = −1 is a closed subscheme ofQ2 isomorphic toA1; the open complement of thisclosed subscheme is isomorphic toA

2. The normal bundle to this embedding ofA1 comes equipped

with a prescribed trivialization, and we can use homotopy purity together withA1-contractibility ofA2 to construct anotherA1-weak equivalence ofQ2 with P

1. We now explain how to identify theA1-homotopy type ofQ4 using the method just sketched.

Proposition 2.1.2. If k = Z, then there is a (pointed)A1-weak equivalenceQ4∼→ P

1∧2.

Proof. LetE2 be the closed subscheme ofQ4 defined byx1 = x2 = 0 andz = −1. Observe thatE2 is isomorphic toA2. The normal bundle toE2 is trivial, and we fix a choice of trivialization. LetX4 := Q4 \E2. The homotopy purity theorem gives a cofiber sequence of the form

X4 −→ Q4 −→ Th(νE2/Q4) −→ · · · .

The choice of trivialization determines an isomorphismTh(νE2/Q4) ∼= P

1∧2 ∧ (E2)+. The map

E2 → Speck is anA1-weak equivalence, it follows thatTh(νE2/Q4

) ∼= P1∧2 in H•(k). We

showed in [AD08, Corollary 3.1 and Remark 3.3] thatX4 is A1-contractible overSpecZ (indeed,

it is the base of a Zariski locally trivial morphism with total spaceA5 and fibers isomorphic toA1). Properness of theA1-local model structure [MV99, §2 Theorem 3.2] guarantees that pushouts

of A1-weak equivalences along cofibrations areA1-weak equivalences. Applying this fact to the

diagram∗ ← X4 → Q4, one concludes that the mapQ4 → Th(νE2/Q4) is anA1-weak equivalence

and the proposition follows by combining the stated isomorphisms.

Remark2.1.3. The varietyQ4 is isomorphic to the varietyHP1 studied in [PW10, §3]. The varietywe callX4 appears in the “cell decomposition” ofHP1 described in [PW10, Theorem 3.1] (it is thevariety they callX0).

2.2 On theA1-homotopy type ofQ2n, n ≥ 3

The “octahedral” axiom in a pretriangulated category

If C is a pointed model category, then recall that one can understand the homotopy cofiber of acomposite in terms of the homotopy cofibers of the constituents by means of the following result,

7 2.2 On theA1-homotopy type ofQ2n, n ≥ 3

which is sometimes called the “octahedral” axiom, by analogy with a corresponding result for stablemodel categories [Hov99, Proposition 6.3.6].

Proposition 2.2.1. Suppose given a sequence of maps in a pointed model categoryC of the form:

Xu−→ Y

v−→ Z.

If U = hocofib(u), V = hocofib(uv), W = hocofib(v), then there is a commutative diagram ofthe form

Xu //

id

Y //

v

U

r

Xuv //

u

Z //

id

V

s

Yv // Z //W

where the rows and the third column are cofiber sequences, andthe mapsr and s are the mapsinduced by functoriality of the homotopy cofiber construction.

The induction step

Consider the quadricQ2n introduced in the introduction overZ. The open subschemeUn defined bythe non-vanishing ofxn is isomorphic toA2n−1×Gm (with coordinates(x1, . . . , xn−1, y1, . . . , yn−1, z)on the first factor andxn on the second factor). The closed complement of this open subscheme,which we will callZn, is isomorphic toQ2n−2×A

1, with coordinatesx1, . . . , xn−1, y1, . . . , yn−1, zon the first factor andyn on the second factor. The subvarietyZn has a distinguished point “0” cor-responding tox1 = · · · = xn = y1 = · · · = yn = z = 0. Moreover, the normal bundle toZn inQ2n is a line bundle equipped with a chosen trivialization (coming fromxn).

The closed subscheme ofUn defined byx1 = · · · = xn−1 = y1 = · · · = yn−1 = z = 0 isisomorphic toGm with coordinatexn and the inclusion map

Gm −→ Un

is a cofibration andA1-weak equivalence (it even admits a retraction). The closedsubscheme ofQ2n defined byx1 = · · · = xn−1 = y1 = · · · = yn−1 is isomorphic toQ2. Imposing further theconditionyn = 0, the resulting variety is isomorphic to two copies ofA

1 (corresponding toz = 0andz = −1). Imposing yet further the conditionz = 0 gives a subvariety ofQ2n isomorphic toA1

with coordinatexn. Note that, by construction, the intersection of this copy of the affine line withZn is precisely the point “0”, i.e., there is a pullback square (of smooth schemes) of the form:

0 //

A1

Zn // Q2n.

Moreover, the natural map from the pullback of the normal bundle toZn in Q2n to the normalbundle of0 in A

1 is an isomorphism compatible with the specified trivializations.

8 2.2 On theA1-homotopy type ofQ2n, n ≥ 3

It follows from the construction of the purity isomorphism [MV99, §3 Theorem 2.23] that, givena pullback square as in the previous paragraph, the diagram

A1/Gm

//

Th(ν0/A1)

Q2n/Un // Th(νZn/Q2n)

is commutative inH•(k) (cf. [Voe03a, Lemma 2.1]). Moreover, the specified trivializations yieldA1-weak equivalences of the formTh(ν0/A1) ∼= P

1 andTh(νZn/Q2n) ∼= P

1∧(Zn)+ by [MV99, §3Proposition 2.17.2] and their compatibility ensures that the diagram

Th(ν0/A1) //

P1

Th(νZn/Q2n) // P

1∧(Zn)+

commutes (this time in the category of pointed spaces). Furthermore, under these identifications, wecan make the right vertical map in the diagram very explicit.Indeed, consider the mapS0

s → (Zn)+sending the base-point ofS0

s to the base-point of(Zn)+ and the point1 of S0s to the point “0” in Zn

described above; the right hand vertical map is theP1-suspension of this map.

Now, we apply Proposition2.2.1to the composition

(Gm)+u−→ (A1)+

v−→ (Q2n)+.

Observe that the cofiber ofv is Q2n pointed with the image of“0”, while the cofiber ofu is, bymeans of the purity isomorphism,P1. It remains to identify the cofiber of(Gm)+ → (Q2n)+,which is the content of the next lemma.

Lemma 2.2.2. If k = Z, then the cofiber of the map(Gm)+ → (Q2n)+ isA1-weakly equivalent toP1∧(Q2n−2)+.

Proof. The projection mapZn → Q2n−2 is anA1-weak equivalence, it follows that it induces anA1-weak equivalenceP1∧(Zn)+

∼→ P

1∧(Q2n−2)+. On the other hand, by the purity isomorphism,the spaceP1∧(Zn)+ isA1-weakly equivalent toQ2n/Un. Since the mapGm → Un described aboveis anA1-weak equivalence and cofibration and since the induced mapGm → Q2n is a cofibration,it follows that the induced map of cofibers

Q2n/Gm → Q2n/Un

is anA1-weak equivalence. Moreover, the same statement remains true for the cofibers of the

pointed morphisms(Gm)+ → (Q2n)+ and(Un)+ → (Q2n)+.

Combining Proposition2.2.1and Lemma2.2.2, we deduce the following result.

Corollary 2.2.3. If k = Z, then for any integern ≥ 1, there is a cofiber sequence of the form

P1 −→ P

1∧(Q2n−2)+ −→ Q2n −→ · · · .

9 2.3 Non-geometric motivic spheres

To finish, we provide an alternative identification of the cofiber ofP1 → P1∧(Q2n−2)+. Recall

that this map is induced byP1-suspension of a mapS0s → (Zn)+. We prove a more general

statement about smashing this map with the identity map.

Proposition 2.2.4. Assume(X , x) is a pointed space, andX+ is X with a disjoint base-point at-tached. Letι : S0

s → X+ be the map that sends the base-point ofS0s to the (disjoint) base-

point of X+ and the non-base-point tox ∈ X (k). If (Y, y) is a pointed space, then the map

Y ∼= S0s∧Y

ι∧id−→ X+∧Y fits into a split cofiber sequence of the form

Yι∧id−→ X+∧Y −→ X∧Y.

Proof. The splitting is given by the mapX+ → S0s that collapses the non base-point component of

X+ to the non-base-point ofS0s . In particular, the mapι∧id is a cofibration. The homotopy cofiber

of this map is then computed by the actual quotient.The quotientX+∧Y is, by definition, the quotientX+ × Y by X+ ∨ Y, where the latter is the

disjoint union ofX+×y andSpeck×Y. IdentifyX+×Y as(X×Y)∐(Spec k×Y). We describe the

quotient in two steps. First, we collapseSpec k×Y to the base-point and then the image ofX+× yto the base-point. We conclude that there is an isomorphism of spacesX+∧Y ∼= (X ×Y)/(X × y).To conclude, we simply observe thatX × Y/(X × y) modulo the image ofx× Y isX∧Y.

Theorem 2.2.5.For any base ringk, there is an isomorphism inH•(k) of the formQ2n∼→ P

1∧n.

Proof. Since both sides of the weak equivalence are preserved by base-change, it suffices to provethe result fork = Z. In that case, by Proposition2.1.1, we know thatQ2

∼→ P

1 in H•(k). Com-bining Proposition2.2.4and Corollary2.2.3, we conclude that for everyn ≥ 2 that there is anisomorphism inH•(k) of the formQ2n

∼= Q2n−2∧P1. The result follows by a straightforward

induction.

2.3 Non-geometric motivic spheres

Proposition 2.3.1. If k is a base ring, and ifi > j are integers, thenSis ∧G∧jm does not have the

A1-homotopy type of a smooth scheme.

Proof. Assume to the contrary thatX is a smoothk-scheme having theA1-homotopy type ofSis ∧G

∧jm . By picking a a geometric point ofSpeck, we can, without loss of generality, assumek is an

algebraically closed field, in particular perfect. Then, byA1-representability of motivic cohomology

(see, e.g., [Voe03b, §2]), we know that for arbitrary integersp, q

Hp,q(X,Z) = [X,K(Z(q), p)]A1 .

Next, observe that by the cancellation theorem [Voe10, Corollary 4.10] and the (simplicial) suspen-sion isomorphism, there is a non-trivial morphismSis ∧G

∧jm → K(Z(j), i+ j), giving a non-trivial

class inH i+j,j(X,Z). However, ifX is a smooth scheme, andi > j, then this latter group mustvanish by [MVW06, Theorem 19.3].

10 3A1-contractible subvarieties of quadrics

Remark2.3.2. One can obtain another more “homotopic” proof of the proposition at the expenseof introducing slightly different terminology; we freely use the conventions of [AF14a] and someterminology from [Mor12]. Consider an Eilenberg-Mac Lane spaceK(KM

j , i), i.e., a space withexactly1 non-vanishingA1-homotopy sheaf in degreej, in which degree it is isomorphic to theunramified Milnor K-theory sheafKM

j . If X is a smooth scheme, the explicit Gersten resolution ofKMj shows thatH i(X,KM

j ) vanishes fori > j. On the other hand, there is a non-trivial (pointed)

morphismSis∧G∧jm → K(KM

j , i). Indeed, by [Mor12, Theorem 6.13], we haveπA1

i,j (K(KMj , i))

∼=

(KMj )−j and by applying, e.g., [AF14a, Lemma 2.7] we deduce(KM

j )−j ∼= Z.

The following conjecture summarizes our expectations regarding representability of the remain-ing motivic spheres by smooth schemes.

Conjecture 2.3.3. If k is a base ring, and ifi, j ∈ N with i < j − 1, the sphereSis ∧G∧jm does not

have theA1-homotopy type of a smoothk-scheme.

3 A1-contractible subvarieties of quadrics

A smoothk-schemeX is A1-contractible if the structure mapX → Spec k is an isomorphism in

H (k), i.e., anA1-weak equivalence. The affine spaceAnk is A1-contractible by construction of

theA1-homotopy category. AnexoticA1-contractible smooth schemeis anA1-contractible smoothscheme that is not isomorphic to an affine space. In [AD07] the first two authors constructed manyexoticA1-contractible smooth schemes as quotients of an affine spaceby a free action of the additivegroupGa (or, more generally, free actions of unipotent groups on affine spaces). All of the examplesconstructed in [AD07] could be realized as open subschemes of affine schemes with complement ofcodimension≤ 2.

LetEn ⊂ Q2n be the closed subscheme defined byx1 = · · · = xn = 0 andz = −1. ObservethatEn ∼= A

n and has codimensionn in Q2n. Set

X2n := Q2n \En.

It is straightforward to check thatX2∼= A

2, and we recalled in the proof of Proposition2.1.2thatX4 is A

1-contractible. The main results of this section are Theorem3.1.1, which shows thatX2n

is A1-contractible forn ≥ 1, and Corollary3.2.3, which establishes thatX2n is not a unipotent

quotient of affine space forn ≥ 3.

3.1 On theA1-contractibility of X2n

If k = C, one can check thatX2m(C) is a contractible complex manifold. OverC, the varietyQ2m is isomorphic to the variety defined by

∑2m+1i=1 x2i = 1. Wood explains [Woo93, §2] how to

identify the last variety with the tangent bundle of a sphere. Under this isomorphism, the varietyEm can be identified with the tangent space at a point. The complement ofEm is then a contractibletopological space diffeomorphic toR4m. Here is the generalization of this result toA1-homotopytheory.

Theorem 3.1.1. If k is a base ring, then the varietyX2n is A1-contractible overSpec k.

11 3.2 A1-contractibles that are not unipotent quotients

Proof. Once more, since both sides are preserved by base-change, itsuffices to prove the resultwith k = Z. We continue with the notation of Subsection2.2. The proof is essentially a repeatof the Proof of Theorem2.2.5, so we will explain only the changes required. Consider the closedsubvarietyZn ⊂ Q2n defined byxn = 0. We saw thatZn ∼= Q2n−2×A

1 and that the normal bundleto Zn comes equipped with a specified trivialization. Observe that, by constructionEn ⊂ Zn andtherefore we see thatX2n := Q2n \En containsUn := Q2n \Zn as an open subscheme. Moreover,the subvarietyZn \ En is isomorphic toX2n−2 × A

1.Recall thatUn ∼= A

2n−1×Gm with coordinatesx1, . . . , xn−1, y1, . . . , yn−1, z on the first factorandxn on the last factor. The closed subschemeGm → Un is defined byx1 = · · · = xn−1 =y1 = · · · yn−1 = 0 andz = 0. The intersection ofX2n andQ2, viewed as a subvariety ofQ2n byimposingx1 = · · · = xn−1 = y1 = · · · = yn−1 = 0 is isomorphic toA2. The copy ofA1 ⊂ Q2

identified by further imposing the conditionsyn = z = 0 is disjoint fromEn and therefore containedin X2n. Note that the point “0” ofQ2n is contained inX2n.

We therefore have a diagram of the form

Gmu−→ A

1 v−→ X2n.

We understand the cofiber of this composite map using Proposition 2.2.1again. In particular, re-peating the proofs of Lemma2.2.2and Corollary2.2.3with all instances ofQ2i replaced byX2i,we deduce the existence of a cofiber sequence of the form

P1 ι−→ (X2n−2)+∧P

1 −→ X2n −→ · · ·

Now, applying Proposition2.2.4, we conclude that the cofiber ofι is A1-weakly equivalent to

X2n−2∧P1. The induction hypothesis guarantees thatX2n−2 isA1-contractible, so the smash prod-

uct X2n−2∧P1 is A

1-contractible as well. SinceX2n−2∧P1 is A

1-weakly equivalent toX2n, itfollows thatX2n isA1-contractible.

Remark3.1.2. Theorem3.1.1gives an alternative proof of Theorem2.2.5, this time repeating theargument of Proposition2.1.2. In this case, the mapQ2n → Th(νEn/Q2n

) obtained by collaps-ing X2n to a point and then using the homotopy purity theorem is anA

1-weak equivalence byproperness of theA1-local model structure, as explained in the proof of Proposition 2.1.2. Theexplicit trivialization of the normal bundle toEn that arises by its presentation as a (connectedcomponent of a) complete intersection defined by a regular sequence ofQ2n yields an isomorphismTh(νEn/Q2n

) ∼= P1∧n ∧ (En)+, and the projection mapEn → Speck then yields anA1-weak

equivalenceQ2n∼→ P

1∧n that is perhaps slightly more explicit than the one arising in the proof ofTheorem2.2.5.

3.2 A1-contractibles that are not unipotent quotients

We will now see that form ≥ 3, the varietiesX2m studied in Theorem3.1.1cannot be realizedas quotients of affine space by free actions of unipotent groupsU . We begin by proving a general“excision” style result (Theorem3.2.1) for (Zariski) cohomology with coefficients in a unipotentgroup; the main result then follows from this excision result when applied in the special case ofdegree1 cohomology (Corollary3.2.2). Indeed, the degree1 cohomology group in question can be

12 3.2 A1-contractibles that are not unipotent quotients

identified as the (pointed) set ofU -torsors on the schemeX. We warn the reader that, contrary to“understood” prior conventions, the letterU in this section stands for a unipotent group, as opposedto an open subscheme; we refer the reader to [Bor91, §15 (Definition 15.1)] for general informationon split unipotent groups.

Theorem 3.2.1(Excision for unipotent groups). Let d ≥ 3 be an integer. SupposeX is a regularscheme over a field andj : W → X is an open immersion whose closed complement has codi-mension≥ d. SupposeU is a (not necessarily commutative) split unipotent group. The restrictionmap

j∗ : H1Zar(X,U) −→ H1

Zar(W,U)

is a pointed bijection, i.e., everyU -torsor onW extends uniquely to aU -torsor onX.

Proof. SinceU is split, by definition it admits an increasing filtration by normal subgroups1 =U0 ⊂ U1 ⊂ · · ·Un = U with subquotientsUi+1/Ui isomorphic toGa. We first show that, workinginductively with respect to the dimension ofU , it suffices to prove the result in the caseU = Ga.

Indeed, suppose we know the result holds true for all unipotent groups of dimensionn and allopen immersions of regular schemes with closed complement of codimensiond ≥ 3. In that case,supposeU has dimensionn + 1 and we are given aU -torsorϕ : P → W . We know that thereis a normal subgroupU ′ of U such thatU/U ′ ∼= Ga. In that case, the quotient mapU → Ga

gives rise to aGa-torsorW ′ := P ×U Ga → W ; note thatW ′ is again regular. The pullback ofϕalongW ′ →W gives aU ′-torsor onW ′. Now, again by induction, we also know that theGa-torsorW ′ →W extends (uniquely) to aGa-torsorX ′ → X; again,X ′ is regular. Moreover, working overa Zariski trivialization ofX ′ → X, we conclude thatW ′ ⊂ X ′ is open with closed complementhaving codimension≥ 3 as well. However, the induction hypothesis guarantees thattheU ′-torsoroverW ′ extends uniquely to aU ′-torsor onX ′, and this extension provides the required extensionof ϕ overW .

Now, assumeU = Ga. In this case, one knows that by definitionHq(X,Ga) = Hq(X,OX ).We will show that the map

j∗ : Hq(X,OX ) −→ Hq(W,OW )

induced by pull-back alongj is an isomorphism forq ≤ d− 2.To prove the last fact, we use the Cousin complex forOX as discussed in, e.g., [Har66, IV.2].

Under the assumption thatX is regular, the Cousin complex provides an injective resolution ofOX(see especiallyibid. p. 239). IfX(p) denote the set of codimensionp points inX (recall this meansdimOX,x = p), thep-th term of the Cousin complex is

∐

x∈X(p)

(ix)∗(Hpx(OX)).

Sincej is an open immersion, it is an affine morphism and the Leray spectral sequence forjdegenerates to yield isomorphismsj∗ : Hq(W,OW )

∼→ Hq(X, j∗OW ). Adjunction gives rise to a

morphismOX → j∗OW . The push-forward of the Cousin complex forOW to X by j remains aflasque resolution ofj∗OW onX. Since the inclusionW → X is, by definition, an isomorphismon points of codimensiond − 1, it follows that the cokernel of the induced morphism of Cousincomplexes, which provides a flasque resolution of the cone ofthe mapOX → j∗OX , only depends

13 4 Clutching functions for principal G-bundles on algebraic spheres

on points of codimension≥ d and the isomorphism of the theorem statement follows immediatelyfrom the long exact sequence in cohomology.

Corollary 3.2.2. If X is a regular affine scheme andW ⊂ X is an open subscheme whose comple-ment has codimensiond ≥ 3, then every torsor under a split unipotent group overW is trivial. Inparticular, the varietiesX2m cannot be realized as quotients of affine space by free actions of splitunipotent groups.

Proof. By Serre’s vanishing theorem [Gro61, Theoreme 1.3.1], we know that all higher cohomologyof a quasi-coherent sheaf on an affine scheme vanishes. By Theorem3.2.1, we conclude that ifWis as in the the theorem statement, and ifU is a (split) unipotent group, thenH1(W,U) = ∗, soeveryU -torsor overW is trivial. In particular, the total space of anyU -torsor overW is of the formW × U , which is itself a quasi-affine variety that is not affine.

Corollary 3.2.3. If k is a base ring, andm ≥ 3 is an integer, thenX2m is not a quotient of an affinespace by the free action of a unipotent group.

Proof. Pick an algebraically closed fieldF such thatSpec k has aSpecF -point. Assume thatX2m

is a quotient ofAn by the free action of a unipotent group. By base change, we conclude that theunipotent group is necessarily smooth. Base-change toSpecF then allows us to assume that theunipotent group in question is also split. Form ≥ 3, X2m has codimensionm ≥ 3 in Q2m, so theresult then follows immediately from Corollary3.2.2.

Question 3.2.4.If Z → X2n is a Jouanolou device, thenZ is an affineA1-contractible variety. IsZ ∼= A

m for some integerm?

Remark3.2.5. If k is a field having characteristicp > 0, work of N. Gupta [Gup14, Gup13] showsthat there exist smooth affinek-varieties of every dimensiond ≥ 3 that are stably isomorphic toaffinek-space, yet which are not isomorphic to affinek-space. Such varieties are necessarily exoticsmooth affineA1-contractible varieties. At the moment, we do not have any examples of exoticsmooth affineA1-contractible varieties over a fields having characteristic 0. Nevertheless, it seemsreasonable to ask: ifk is an arbitrary field, is every smooth affineA1-contractiblek-variety a retractof an affinek-space?

4 Clutching functions for principal G-bundles on algebraic spheres

In this section, we provide an algebro-geometric analog of the clutching construction of vectorbundles on even-dimensional spheres. As an application, weprovide an explicitvector bundleyielding a generator ofK0(Q2n) for n ≥ 1. Forn = 1, 2, such explicit generators are given byclassical geometric constructions (Hopf bundles). Forn ≥ 3, while it is easy to write down virtualvector bundle representatives, writing down explicit vector bundles is more complicated.

The main results of this section are Theorem4.1.6, which provides a version of the clutchingconstruction for any smooth affine schemeY whose (simplicial) suspension has theA

1-homotopytype of a smooth affine scheme. Making the abstract clutchingconstruction explicit depends on thechoice of weak equivalenceΣ1

sY with an affine schemeX. In the special case whereY = Q2n−1 in

14 4.1 The abstract clutching construction

Theorem4.2.3, we construct an explicit weak equivalenceΣ1sQ2n−1 → Q2n, at least under suitable

additional hypotheses. Finally in Theorem4.3.6, we combine these two observations to obtain aversion of the clutching construction that gives explicit cocycle representatives of vector bundles.

4.1 The abstract clutching construction

We begin by providing an abstract analog of the clutching construction in algebraic geometry.Let us begin by quickly reviewing the topological situation, where the clutching construction issometimes phrased as follows. SupposeM is a pointed finite CW complex, and consider the (re-duced) suspensionΣM . The usual identificationG ∼= ΩBG together with the loop-suspensionadjunction yield a canonical bijection between sets of pointed homotopy classes of maps of theform [M,G]• ∼= [ΣM,BG]•. By the representability theorem for principalG-bundles, the forget-ful map [ΣM,BG]• → [ΣM,BG] from based to free homotopy classes yields a surjection from[ΣM,BG]• to the set of isomorphism classes of principalG-bundles onΣM . If G is furthermoreconnected, thenBG is necessarily1-connected, then the set of pointed homotopy classes of maps[M,G]• is actually in bijection with the set of isomorphism classesof principalG-bundles onΣM .

By Theorem2.2.5, we find ourselves in an analogous situation. Indeed, we knowthatQ2n∼=

P1∧n in H•(k) for n ≥ 0. Using the standard identificationQ2n−1

∼= Σn−1s G∧n

m , we can thenconclude thatQ2n is A

1-weakly equivalent toΣ1sQ2n−1 if n ≥ 1, though providing an explicit

A1-weak equivalence of this form is more complicated. More abstractly, we are given a (pointed)

smooth affine schemeY such thatΣ1sY has theA1-homotopy type of a smooth affine scheme.

(If k is a regular ring, using Jouanolou-Thomason devices, it suffices to assume thatΣ1sY has the

A1-homotopy type of a smooth scheme).

We would like to link principalG-bundles on a smooth affine model ofΣ1sY with “clutching

functions” onY itself. As Σ1sY only has theA1-homotopy type of a smooth affine scheme, we

appeal to the results of [AHW15a, AHW15b] onA1-representability results for principalG-bundles

(which themselves extend results of F. Morel [Mor12] and M. Schlichting [Sch15]). We break theargument into three pieces along the lines of what is sketched above: (i) adjunction, to obtain a bi-jection ofA1-homotopy classes of maps, (ii) connectivity statements, to obtain a suitable surjectivitystatement, and (iii) representability, to connectA

1-homotopy classes with actual geometric objects(i.e., morphisms on the one side and bundles on the other). These three pieces are put together inthe main result of this section, Theorem4.1.6.

We begin by recalling some notation. Recall that a presheafF onSmk is calledA1-invariant onaffinesif, for any smooth affinek-schemeX, the mapF(X) → F(X × A

1) induced by pullbackalong the projectionX × A

1 → X is a bijection. WriteSingA1(−) for the singular construc-

tion of [MV99, p. 87]; briefly, this space is the diagonal of the bisimplicial object obtained byconsidering the internal homHom(ƥ

k,X ). Write RZar for a Zariski fibrant replacement func-tor on the category of simplicial presheaves. With this notation, we can recall the results from[AHW15a, AHW15b] that we need.

Lemma 4.1.1.Supposek is a base ring,G is a smoothk-group scheme and assume that the presheafH1

Nis(−, G) onSmk is A1-invariant on affines.

1. The spacesRZarSingA1BG andRZarSing

A1G are Nisnevich local andA1-invariant.

15 4.1 The abstract clutching construction

2. For any smooth affinek-algebraA, the mapsSingA1BG(A) → RZarSing

A1BG(A) and

SingA1G(A)→ RZarSing

A1G(A) are weak equivalences.

3. For any smooth affinek-algebraA, the canonical mapSingA1RΩ1

sBG(A)→ RΩ1sSing

A1BG(A)

is a weak equivalence.

Proof. The first two statements are contained in [AHW15b, Theorems 2.2.5 and 2.3.2]. The thirdstatement follows from [AHW15b, Corollary 2.1.2].

We begin by establishing the “adjunction” part of the clutching construction.

Lemma 4.1.2.Supposek is a base ring,G is a smoothk-group scheme and assume that the presheafH1

Nis(−, G) on Smk is A1-invariant on affines. If(Y, y) is a (pointed) smooth affine scheme such

thatΣ1sY has theA1-homotopy type of a smooth affine scheme, then for any (pointed) smooth affine

model(X,x) of Σ1sY there is a pointed bijection

[(Y, y), (G, 1)]A1 −→ [(X,x), (BG, ∗)]A1 .

Proof. By Lemma4.1.1(1), we know thatRZarSingA1G is Nisnevich local andA1-invariant, i.e.,

A1-fibrant [MV99, §2 Proposition 3.19]. By Lemma4.1.1(2), we know that the mapSingA

1G →

RZarSingA1G is a Nisnevich local weak equivalence. In particular, we know that SingA

1(G) is

A1-local and thus that[(Y, y), (G, 1)]A1

∼= [(Y, y), (SingA1(G), 1)]s.

The evident mapG → RΩ1sBG is a Nisnevich local weak equivalence, e.g., by [AHW15b,

Lemma 2.2.2(iii)]. It follows that the induced mapSingA1G → SingA

1RΩ1

sBG, which can bewritten as a homotopy colimit of Nisnevich local weak equivalences, is again a Nisnevich lo-cal weak equivalence. On the other hand, Lemma4.1.1(3) allows us to conclude that the mapSingA

1RΩ1

sBG → RΩ1sSing

A1BG is a Nisnevich local weak equivalence. Combining these two

observations, we deduce that the induced mapSingA1G → RΩ1

sSingA1BG is a Nisnevich local

weak equivalence. Together with the conclusion of the previous paragraph, we deduce that

[(Y, y), (SingA1(G), 1)]s −→ [(Y, y),RΩ1

sSingA1BG]s

is a bijection.By the using the loop-suspension adjunction, we conclude that [(Y, y),RΩ1

sSingA1BG]s ∼=

[Σ1s(Y, y),Sing

A1BG]s. However, arguing as in the first paragraph, we also know thatSingA

1BG

is A1-local, i.e.,[Σ1

s(Y, y),SingA1BG]s ∼= [Σ1

s(Y, y),SingA1BG]A1 = [Σ1

s(Y, y), BG]A1 . Thus,combining all these identifications, we conclude that[(Y, y), (G, 1)]A1

∼→ [Σ1

s(Y, y), BG]A1 . Now,sinceΣ1

sY has theA1-homotopy type of a smooth schemeX (and explicitly writing base-points forBG), we conclude that

[(Y, y), (G, 1)]A1 = [(X,x), (BG, ∗)]A1 ,

which is precisely what we wanted to show.

Next, we establish the analog of the surjectivity and bijectivity statements in the topologicalclutching construction.

16 4.1 The abstract clutching construction

Lemma 4.1.3.Supposek is a base ring,G is a smoothk-group scheme and assume that the presheafH1

Nis(−, G) on Smk is A1-invariant on affines. If(X,x) is a pointed smooth affine scheme, then

the evident map[(X,x), (BG, ∗)]A1 → [X,BG]A1 is surjective; ifG is furthermoreA1-connected,then this map is bijectictive.

Proof. As in the Proof of Lemma4.1.2we know thatSingA1G andSingA

1BG areA1-local and

thatSingA1G → RΩ1

sSingA1BG is a Nisnevich local weak equivalence. Therefore, we conclude

there are identifications of homotopy sheavesπA1

0 (G) = π0(SingA1G) ∼= π0(RΩ1

sSingA1BG) ∼=

π1(SingA1BG) = π

A1

1 (BG).The spaceBG is always0-connected by construction, and therefore by [MV99, §2 Corollary

3.22] it is alwaysA1-connected. In particular,SingA1BG is always0-connected. SinceX is a

pointed smooth scheme, any morphismX → RZarSingA1BG can be pointed and this yields the

surjectivity statement. IfG isA1-connected, then we similarly conclude thatBG isA1-1-connected.In that case, the map of the theorem statement is a bijection by [AF14a, Lemma 2.1].

Remark4.1.4. The spaceBGLn is notA1-1-connected in general, and thus the map from pointedto free homotopy classes isnot a bijection in general; see [AF14a, Corollary 4.12] for an explicitexample to see that injectivity can indeed fail in this case.On the other hand, the set of pointedhomotopy classes of maps[(X,x), (BG, ∗)]A1 also has a “concrete” description in terms of pairsconsisting of aG-torsor P on X equipped with a trivialization ofx∗P; we will not need thisdescription.

Now, we give a concrete description of elements of[(Y, y), (G, 1)]A1 in terms of pointed mor-phisms of smooth schemesY → G. SupposeY = SpecA, whereA is a k-algebra, a choiceof k-point y of Y corresponds to a homomorphismy : A → k. This homomorphism induces asurjective morphismresy of simplicial groupsresy : G(A[∆•])→ G(k[∆•]), and we set

SingA1G(Y, y) := ker(resy) : G(A[∆

•]) −→ G(k[∆•]).

Equivalently,SingA1G(Y, y) is the (pointed) subsimplicial set ofSingA

1G(Y ) consisting of mor-

phismsY ×∆n → Gwhose restriction toy×∆n are constant with image1 ∈ G, which explainsthe choice of terminology.

Lemma 4.1.5.Supposek is a base ring,G is a smoothk-group scheme and assume that the presheafH1

Nis(−, G) onSmk is A1-invariant on affines. Assume(Y, y) is a (pointed) smooth affine scheme.

1. The mapπ0(SingA1G(Y, y)) → [(Y, y), (G, 1)]A1 is a bijection, i.e.,[(Y, y), (G, 1)]A1 coin-

cides with the set of naive homotopy classes of pointed maps fromY toG.2. There is a pointed bijection[Y,BG]A1

∼= H1Nis(Y,G).

Proof. For Point (1), we proceed as follows. The mapSingA1G→ RZarSing

A1G is a morphism of

pointed simplicial presheaves and upon evaluation at any smooth affine schemeY is a weak equiva-lence. We can assume thatRZar commutes with finite limits (it has a model given by the “Godementresolution” [MV99, §2 Theorem 1.66]). Therefore,RZarSing

A1G is again a simplicial group and

RZarresy : RZarSingA1G(Y ) → RZarSing

A1G(k) is a homomorphism; this homomorphism is

surjective becauseresy admits a set-theoretic section corresponding to the structure morphism. WesetFy := ker(RZarresy).

17 4.1 The abstract clutching construction

By functoriality, we obtain the following commutative diagram:

SingA1G(Y, y) //

SingA1G(Y )

resy// SingA

1G(k)

Fy // RZarSingA1G(Y )

RZarresy// RZarSing

A1G(k).

The two right vertical maps in this diagram are weak equivalences by Lemma4.1.1(2). On the otherhand, asresy andRZarresy are both surjective group homomorphism, they are fibrations, with fibersisomorphic to the respective kernels, i.e., the diagram is amorphism of homotopy fiber sequences[Cur71, Lemma 3.2]. Therefore, we conclude that the leftmost vertical mapSingA

1G(Y, y) → Fy

is also a (pointed) weak equivalence.On the other hand,π0(Fy) coincides with the quotient of the group of pointed morphismsY →

RZarSingA1G by the homotopy relation. Then, sinceRZarSingA

1G is Nisnevich local andA1-

fibrant by Lemma4.1.1(1), we conclude that[(Y, y), (G, 1)]A1 coincides withπ0(Fy). By the resultsof the previous paragraphs, we conclude thatπ0(Fy) ∼= π0(Sing

A1G(Y, y)), which is precisely what

we wanted to show.Point (2) follows immediately from [AHW15b, Theorem 2.2.5(ii)].

The simplicial weak equivalenceG ∼= RΩ1sBG yields, by adjunction, a canonical morphism

Σ1sG→ BG. If (Y, y) is a pointed smooth scheme, a pointed morphismf : Y → G then determines

a morphismclG(f) : Σ1sY → Σ1

sG → BG. If Σ1sY has theA1-homotopy of a smooth schemeX,

then fixing anA1-weak equivalenceΣ1sY∼= X, the elementclG(f) determines an element of

[X,BG]A1 by forgetting the base-point. The resulting assignment factors through naiveA1-weakequivalence and determines a “clutching” function

clG : π0(SingA1G(Y, y)) −→ [X,BG]A1 .

Regarding this clutching function, we have the following result, which puts everything above to-gether.

Theorem 4.1.6. Supposek is a base ring,G is a smoothk-group scheme and assume that thepresheafH1

Nis(−, G) on Smk is A1-invariant on affines. IfY is a (pointed) smooth affine scheme

such thatΣ1sY has theA1-homotopy type of a smooth affine scheme, then the assignmentf 7→

clG(f) yields a surjective “clutching” function of the form

clG : π0(SingA1G(Y, y)) −→ H1

Nis(Σ1sY,G);

this assignment is functorial in bothG andY , and bijective ifG is A1-connected. Moreover, the

A1-invariance on affines hypothesis is satisified if:(i) G isGLn, SLn or Sp2n, andk is smooth over a Dedekind domain with perfect residue fields,

or(ii) k is an infinite field, andG is an isotropic reductivek-group (in the sense of[AHW15b,

Definition 3.3.4]);and theA1-connectedness hypothesis is satisfied if, e.g.,G = SLn or Sp2n.

18 4.2 Mayer-Vietoris cofiber sequences and an explicit weak equivalence

Proof. For the first statement, simply combine Lemmas4.1.2, 4.1.3and4.1.5. Points (i) and (ii)are a consequence of [AHW15a, Theorem 5.2.1] and [AHW15a, Theorems 3.3.1, 3.3.2 and 3.3.6].TheA1-connectedness ofSLn or Sp2n if k is infinite follows from existence of elementary matrixfactorizations combined with [Mor05, Lemma 6.1.3].

Remark4.1.7. One criticism of Theorem4.1.6is that, given an explicit pointed mapf : Y → G,we have only “abstractly” produced the bundleclG(f) as anA1-homotopy class of maps. Whilewe will see this is sufficient to make interesting existence statements, to “concretely” produce abundle, say in terms of a suitableG-valued1-cocycle on a smooth affine modelX of Σ1

sY , requiresspecifying anA1-weak equivalenceΣ1

sY∼= X.

4.2 Mayer-Vietoris cofiber sequences and an explicit weak equivalence

We now turn our attention to producing anexplicitA1-weak equivalenceQ2n∼→ Σ1

sQ2n−1; this isaccomplished in Theorem4.2.3below. We begin with some preliminaries on Mayer-Vietoris cofibersequences. Suppose(X , x) and(Y , y) are pointed spaces. WriteC(X ) for the usual cone onX ,i.e., X ∧∆1

s, where the latter is pointed by1. If f : X → Y , the map collapsingY to a pointyields a morphism of diagrams:

C(X )

=

Xoof

//

=

Y

C(X ) Xoo // ∗.

The induced map of homotopy pushouts along the rows yields a morphismC(f) −→ Σ1sX , func-

torial in all the inputs.Now, suppose we are given a smooth schemeX and a Nisnevich distinguished square of the

form

U ×X Vj′

//

V

ψ

Uj

// X.

Let us assume that(U ×X V )(k) is non-empty and fix a base-point; we pointU , V andX by theimage of this base-point under the various maps. In that case, the morphismψ : V/(U ×X V ) →X/U induced byψ is an isomorphism by [MV99, §3 Lemma 1.6]. On the other hand, since allhorizontal morphism in the diagram are cofibrations, the quotients V/(U ×X V ) andX/U are

models for the homotopy cofibers ofj andj′. Thus,ψ−1

provides a weak equivalenceC(j) →C(j′). On the other hand, there is a mapC(j′) → Σ1

s(U ×X V ) by the discussion of the previousparagraph. Therefore, one obtains a composite morphism of the form

X −→ C(j′)∼−→ C(j) −→ Σ1

s(U ×X V );

we use this connecting morphism below.

19 4.2 Mayer-Vietoris cofiber sequences and an explicit weak equivalence

Example4.2.1. If X is a smooth scheme, andX = U ∪V is a Zariski cover ofX by two open sub-schemes such that(U ∩V )(k) is non-empty, then the construction just described yields amorphism

X −→ Σ1s(U ∩ V ).

As we will see below, the assumption(U ∩ V )(k) 6= ∅ is a non-trivial restriction.

Let us now specialize the constructions just made takingX = Q2n. Before proceeding, recallthat the schemeQ2n−1 has a standard base-point(1, 0, . . . , 0, 1, 0, . . . , 0). Consider the two prin-cipal open subschemesV 0

2n := Dz andV 12n := D1+z. The intersectionV 0

2n ∩ V12n is the principal

open subschemeDz(1+z) ⊂ Q2n on which the2n sections

(x1/z, . . . , xn/z, y1/(1 + z), . . . , yn/(1 + z))

satsify the equation defining the quadricQ2n−1. We thus obtain a morphismψn : V 02n ∩ V

12n →

Q2n−1.

Remark4.2.2. The set(V 02n ∩ V

12n)(k) canbe empty if, e.g.,k = F2.

If (V 02n ∩ V

12n)(k) is non-empty, then we can pick a base-point here. Up to post-composing

with an automorphism ofQ2n−1, we can assume that this base-point is mapped to the standardbase-point ofQ2n−1. Therefore, we obtain a morphismΣ1

s(V02n ∩ V

12n) −→ Σ1

sQ2n−1. Then, theconnecting homomorphism in the Mayer-Vietoris cofiber sequence as in Example4.2.1composedwith the morphism of the previous sentence yields a morphism

ϕn : Q2n −→ Σ1s(V

02n ∩ V

12n) −→ Σ1

sQ2n−1.

Regarding this composite morphism, we have the following result:

Theorem 4.2.3.If the basek is an (infinite) perfect field, then, for any integern ≥ 2, the morphismϕn : Q2n → Σ1

sQ2n−1 is anA1-weak-equivalence.

Remark4.2.4. The careful reader will note that the results of Morel to which we appeal only requirek to be perfect. However, the precise results to which we appeal, specifically [Mor12, Theorem5.46] and [Mor12, Theorem 6.1] implicitly make use of [Mor12, Lemma 1.15], which is a form ofa presentation lemma of Gabber. Morel does not provide a proof for this statement; the publishedversion of Gabber’s presentation lemma [CTHK97] requires that base field is infinite and so wehave made this assumption as well.

Proof. In the proof, we appeal to the results of[Mor12] and this is the reason we require the base tobe an (infinite) perfect field. The assumption thatk is infinite also guarantees that(V 0

2n ∩ V12n)(k)

is non-empty. By Theorem2.2.5, we have an explicit weak-equivalenceQ2n → (P1)∧n. Inparticular, by [Mor12, Corollary 6.38],Q2n is A

1-(n − 1)-connected. On the other hand, weknow thatQ2n−1 ≃ Sn−1 ∧ G∧n

m by [MV99, §3 Example 2.20]. Sincen ≥ 2, Σ1sQ2n−1 is

at leastA1-1-connected, and it follows from [AF14a, Lemma 2.1] that for any pointed space(X , x) the map[(X , x),Σ1

sQ2n−1]A1 → [X ,Σ1sQ2n−1] is a bijection (this bijection allows us

to be sloppy with pointed and free homotopy classes below). Combining this observation with theconclusion of [Mor12, Corollary 6.43], which computes[(Q2n, 0),Σ

1sQ2n−1]A1 , we conclude that

20 4.2 Mayer-Vietoris cofiber sequences and an explicit weak equivalence

[Q2n,Σ1sQ2n−1] ∼= KMW

0 (k). Then, it suffices then to show that the class of the morphismϕncorresponds to an invertible element ofKMW

0 (k) under the previous identifications.Observe that any morphismf : Q2n → Σ1

sQ2n−1 yields a function

[Σ1sQ2n−1, Q2n]A1 −→ [Q2n, Q2n]A1

by precomposing withf . Identifying both terms withKMW0 (k), this map is precisely the multipli-

cation by the class off in [Q2n,Σ1sQ2n−1]A1 = KMW

0 (k).Next, we claim that[Q2n, Q2n]A1

∼→ Hn

Nis(Q2n,KMWn ). Indeed, we knowQ2n is A1-(n −

1)-connected andπA1

n (Q2n) ∼= KMWn by [Mor12, Theorem 6.40]. Therefore, we conclude that

there is anA1-weak equivalenceQ(n)2n∼= K(KMW

n , n) between then-th stage of theA1-Postnikovtower (see [AF14b, Theorem 6.1]) ofQ2n, and the Eilenberg-Mac Lane spaceK(KMW

n , n). The

functorial morphismQ2n → Q(n)2n thus yields a homomorphism

[Q2n, Q2n]A1 −→ HnNis(Q2n,K

MWn ).

By [AF14a, Lemma 4.5] and the suspension isomorphism, for any strictly A1-invariant sheafA

there are isomorphisms of the form

H iNis(Q2n,A) ∼= H i

Nis(Σ1sQ2n−1,A) ∼= H i−1

Nis (Q2n−1,A) = 0

if i > n. Thus, arguing as in [AF14b, Proposition 6.2], this vanishing implies that the map[Q2n, Q2n]A1 → [Q2n, Q

(n)2n ]A1

∼= [Q2n,K(KMWn , n)]A1 is a bijection. Reading through the above,

one has also deduced that[Σ1sQ2n−1, Q2n]A1

∼→ Hn−1

Nis (Q2n−1,KMWn ).

To establish the theorem, it suffices to prove thatϕn : Q2n → Σ1sQ2n−1 yields an isomorphism

Hn−1Nis (Q2n−1,K

MWn ) −→ Hn

Nis(Q2n,KMWn ).

Now, by construction, the morphism induced byϕn is the composite

Hn−1Nis (Q2n−1,K

MWn ) −→ Hn−1

Nis (V 02n ∩ V

12n,K

MWn ) −→ Hn

Nis(Q2n,KMWn )

where the first morphism is the pull-back along the mapϕn : V 02n ∩ V

12n → Q2n−1 and the second

morphism is the connecting homomorphism in the Mayer-Vietoris exact sequence. The result thenrests on the explicit cohomological computations performed in Lemmas4.2.5, 4.2.6and4.2.7.

We recall a few consequences of [Mor12, Corollary 5.43] concerning cohomology with coeffi-cients in the sheavesKMW

n , which will be helpful in performing the necessary computations. Forany smooth schemeX, the canonical mapH i

Zar(X,KMWn )

∼→ H i

Nis(X,KMWn ) is an isomorphism,

and the groupsH iZar(X,K

MWn ) can be computed using an explicit flasque resolution; the degreem

term of this complex takes the form⊕

x∈X(m)

(KMWn−m(k(x))⊗Z[k(x)×] Z[Λ

∗x])

whereΛx = ∧mmx/m2x, mx is the maximal ideal in the local ringOX,x andΛ∗

x is the set of nonzeroelements in the (one dimensional)k(x)-vector spaceΛx (usually, one considers the dual ofmx/m

2x

but this makes no difference in our analysis [Mor12, §5.1] and we drop it to lighten the notation).In particular, in the lemmas below, cohomology could be taken either with respect to the Zariski orNisnevich topologies.

21 4.3 The concrete clutching construction

Lemma 4.2.5. For anyn ≥ 1, the groupHn−1Zar (Q2n−1,K

MWn ) is a freeKMW

0 (k)-module of rankone generated by the class of the cycle[xn]⊗x1∧ . . .∧xn−1 in KMW

1 (k(s))⊗Z[k(s)×]Z[Λ∗s]) where

s is the complete intersection given by the equationsx1 = . . . = xn−1 = 0 (and s is the genericpoint if n = 1).

Proof. See [Fas11, §3.3].

Lemma 4.2.6.For anyn ≥ 1, the groupHnZar(Q2n,K

MWn ) is a freeKMW

0 (k)-module of rank onegenerated by the class of the cycle〈1〉 ⊗ x1 ∧ . . . ∧ xn in KMW

0 (k(t)) ⊗Z[k(t)×] Z[Λ∗t ]) wheret is

the closed subscheme defined by the vanishing locus of the sectionsx1, . . . , xn, 1 + z.

Proof. By Theorem3.1.1, we know that the open subschemeX2n ⊂ Q2n is A1-contractible and

has closed complementEn ≃ An. It suffices then to use the long exact sequence associated with

this embedding, together with homotopy invariance of cohomology with coefficients in Milnor-WittK-theory sheaves to obtain the result.

Lemma 4.2.7. The composite

Hn−1Zar (Q2n−1,K

MWn ) −→ Hn−1

Zar (V 02n ∩ V

12n,K

MWn ) −→ Hn

Zar(Q2n,KMWn )

induced byϕn maps the generator ofLemma4.2.5to 〈(−1)n〉 times the generator ofLemma4.2.6.

Proof. Pulling-back the generator of Lemma4.2.5to V 02n ∩ V

12n we find the cycle[xn/z]⊗ x1/z ∧

. . . ∧ xn−1/z. Its image under the boundary mapHn−1Zar (V 0

2n ∩ V12n,K

MWn ) → Hn

1+z(V02n,K

MWn )

is the cycle〈1〉⊗x1/z ∧ . . .∧xn/z in KMW0 (k(t))⊗Z[k(t)× ] Z[Λ

∗t ]). As z = −1 in k(t), the above

cycle corresponds to〈(−1)n〉⊗x1∧. . .∧xn, which can be seen as an element ofHn1+z(Q2n,K

MWn ).

The result follows.

4.3 The concrete clutching construction

We now apply the results of the abstract clutching construction with Y = Q2n−1 andX = Q2n∼=

Σ1sY . Assumek = Z momentarily. In that case, Theorem4.1.6yields a surjective function

clGLm: π0(Sing

A1GLm(Q2n−1, 1)) −→ Vm(Q2n).

In particular, all rankm vector bundles onQ2n are obtained from pointed morphisms from thequadric toGLm.

We are particularly interested in the clutching function inthe case wherem = n. The weakequivalenceQ2n

∼= P1∧n of Theorem2.2.5, together withA1-representability of algebraic K-theory

[MV99, §4 Theorem 3.13], determines an isomorphism of reduced K-theory K0(Q2n) ∼= K0(Z) ∼=Z. In fact if k is (Noetherian) regular, then we conclude thatK0(Q2n) is a freeK0(k)-module ofrank1.1Our goal, accomplished in Theorem4.3.4, is to describe an explicitvector bundlegeneratorfor thisK0(k)-module.

1Note: these kinds of representability statements continueto hold if k is not Noetherian. As observed in [Hoy14,Appendix C], algebraic K-theory is representable in the version of H (k) we use. Likewise, the proof of geometricrepresentability (by the Grassmannian) goes through in this setting; one may modify [ST15, Theorem 4] as necessitatedby [ST15, Remark 2] to check it whenk = Z and then use base-change to treat the general case.

22 4.3 The concrete clutching construction

Example4.3.1. For small values ofn, the generator ofK0(Q2n) are given by classical geometricconstructions. Forn = 1, the generator is simply the class ofO(1), viewed as an element ofPic(Q2). Alternatively, this line bundle is the line bundle associated with Hopf mapη : Q3 → Q2,which is aGm-torsor. Similarly, forn = 2, a generator is given by a Hopf bundle. Indeed,consider the associated vector bundle to theSL2-torsor corresponding with the motivic Hopf mapν : Q7 → Q4. The mapν can can be defined as follows: given a pair of2 × 2-matrices(M1,M2)such thatdetM1 − detM2 = 1, send(M1,M2) to (M1M2,detM2). A corresponding generatorcan also be given forQ8 in terms of the motivic Hopf mapσ, which can be defined using splitoctonion algebras.

Remark4.3.2. For n ≥ 3, it is easy to write down generators using the isomorphismK0(Q2n) ∼=G0(Q2n) and taking the pushforwardi∗[OQ2n\X2n

], but an explicit projective resolution of this classyields only a virtual vector bundle representative.

If x = (x1, . . . , xn) andy = (y1, . . . , yn) be elements of a ringA such thatxyt = 1. Attachedto such a pair, Suslin inductively defined matricesαn(x,y) ∈ GL2n−1 [Sus77, §5]. The universalsuch example corresponds to a morphismαn : Q2n−1 → GL2n−1 . By means of elementary rowoperations, the matrixαn can be reduced to an invertiblen×n-matrix (non-uniquely). Equivalently,the morphismQ2n−1 → GL2n−1 lifts (non-uniquely) up to (naive)A1-homotopy to a morphismβn : Q2n−1 → GLn [Sus77, p. p. 489 Point (b)].

Remark4.3.3. A more topologically inclined reader may prefer the following description of thematricesαn. Atiyah-Bott and Shapiro constructed an explicit generator of the topological K-theorygroupK0(S

n) in terms of Clifford algebras [ABS64]. Suslin’s matrixα can be thought of as analgebraic version of Clifford multiplication; this has been worked out in more detail in [Chi15].The morphismsαn (resp.βn) are closely related to algebraic K-theory. Indeed, for anyintegeri,there is an isomorphism of reduced K-theoryKi(Q2n−1) ∼= Ki−1(Z) using the fact thatQ2n−1

∼=Σn−1s G∧n

m andA1-representability of algebraicK-theory. Suslin showed [Sus82, Theorem 2.3] thatthis isomorphism is given by multiplication byαn (or, equivalently,βn); see [AF14c, §3] for morediscussion of Suslin matrices in the context ofA

1-homotopy theory.

Theorem 4.3.4. If k is a (Noetherian) regular base ring, then for everyn ≥ 1, clGLn(βn) deter-

mines a rankn vector bundle onQ2n; the stable isomorphism class of this vector bundle yields agenerator ofK0(Q2n).

Proof. The first statement follows from Theorem4.1.6(i). Indeed,clGLn(βn) gives a vector bundle

onQ2n overSpecZ, which we may then pullback toQ2n overSpec k.For the second statement, recall that under these hypotheses K0(Q2n) is a freeK0(k)-module

of rank1. By functoriality, it suffices to produce a generator overSpecZ. In that case, we appealto functoriality of the clutching construction for the mapGLn → GL, the fact that the clutchingconstruction is given by suspension and appeal to Suslin’s results. Indeed, it suffices to show thatclGL(βn) is a generator ofK0(Q2n). However, Suslin showed that the mapβn : Q2n−1 → GLis a generator of[Q2n−1, GL]A1

∼= Z; this is contained in [Sus82, Theorem 2.3], but see [AF14c,Theorem 3.4.1] for a translation in this language. The statement then follows from the suspen-sion isomorphism in K-theory together with the identification [Q2n−1, GL]A1

∼= [Q2n, BGL]A1∼=

K0(Q2n).

23 4.3 The concrete clutching construction

Remark4.3.5. The results of [AF14c, Proposition 3.3.3] together with obstruction theory argumentscan be used to show that the generators ofK0(Q2n) described in Theorem4.3.4admit reductionsof structure group in certain situations: ifn ≡ 0 mod 4, then the vector bundleclGLn

(βn) admitsa reduction of structure group to the split orthogonal group, while if n ≡ 2 mod 4, then it admitsa reduction of structure group to the symplectic group.

Now, using Theorem4.2.3, we show how to refine Theorem4.1.6to obtain explicit cocyclesrepresenting vector bundles onQ2n from the associated clutching functions. To this end, assumethatk is an (infinite) perfect field, and consider the commutative diagram

V 02n

V 02n ∩ V

12n

//oo

V 12n

∗ Q2n−1oo // ∗.

The induced map of homotopy pushouts gives a reinterpretation of the mapϕn and thus is anA1-weak equivalence by Theorem4.2.3. Moreover, the inducedA1-weak equivalenceQ2n →

Σ1sQ2n−1 factors through a morphismQ2n → Σ1

s(V02n ∩ V

12n) → Σ1

sQ2n−1. The following result,which can be viewed as a refinement of some of the results in [NRS10, §6], is our concrete versionof the clutching construction in the case ofQ2n.

Theorem 4.3.6. Assumek is a field, andn, r ≥ 2 are integers andG is an isotropic reductivek-group (in the sense of[AHW15b, Definition 3.3.4]).

1. If k is infinite and perfect, given a pointedk-morphismf : Q2n−1 → G, theG-torsor clG(f)onQ2n attached tof is obtained, up to isomorphism, by gluing the trivialG-torsor onV 0

2n

with a trivial G-torsor onV 12n along the functionf ψn.

2. For arbitrary k, the rankn vector bundle onQ2n corresponding toclGLn(βn) is that associ-

ated withβn ψn by the construction of the previous point.

Proof. Here the assumption thatk is infinite is used in conjunction with the assumption thatG isisotropic to appeal to Theorem4.1.6(ii). Considering the diagram above, point (1) follows fromthefunctoriality assertion in Theorem4.1.6combined with Theorem4.2.3. In more detail, given a vec-tor bundle on[Q2n, BG]A1 , we can lift it to a pointed homotopy class[(Q2n, 0), (BG, ∗)]A1 . Sincethe weak equivalenceQ2n → Σ1

sQ2n−1 factors throughV 02n∩V

12n, every element of[(Q2n, 0), (BG, ∗)]A1

corresponds to an element of[Σ1sQ2n−1, (BG, ∗)]A1 and thus determines an element of[Σ1

s(V02n ∩

V 12n), (BG, ∗)]A1 . The adjoints to these elements are precisely a pointed mapf : Q2n−1 → G and

the composite mapf ψn.The clutching construction proceeds by composingΣ1

sf with the canonical morphismΣ1sG →

BG. The factorization throughV 02n ∩ V

12n in the previous paragraph, then corresponds, by contem-

plating the diagram before the statement, to gluing two homotopically constant mapsV i2n → ∗ →

BG, i.e., trivialG-torsors onV i2n along the morphismf ψn as claimed.

For Point (2), begin by observing that the functionclGLn(βn) over the fieldk is obtained by

base-change from a morphism of schemes overSpecZ. In particular, the functionclGLn(βn) over

k is obtained by extending scalars from a function defined overthe prime fieldF ⊂ k, whichis perfect. Now, since an algebraic closureF of the prime field is infinite and perfect, Point (2)follows by combining Theorem4.3.4with the conclusion of Point (1). However, by Galois descent,

24 REFERENCES

there is an equivalence between vector bundles overF andGal(F /F )-invariant cocycles overF .Therefore, the vector bundle corresponding toclGLn

(βn) over F descends to a vector bundle overF . Pulling back this bundle tok yields the required vector bundle.

Remark4.3.7. Note that Point (2) makes sense even if the intersectionV 02n ∩ V

12n is empty. More

generally, using faithfully flat descent techniques, it is possible to buildG-torsors over fields thatarise from clutching functions given byk-morphismsf : Q2n−1 → G defined over perfect subfieldsF ⊂ k. For example, ifk is a finite field, everyG-torsor overQ2n under an isotropic reductiveG isdescribed by means of a clutching functionQ2n−1 → G. Lacking any applications of the result inthis form at the moment, we leave the proof of this fact to the reader.

Example4.3.8. Unwinding the definitions, the Suslin matrixQ3 → SL2 corresponds to the identitymapSL2 → SL2 under the isomorphism ofQ3 with SL2 given by sending(x1, x2, y1, y2) to the

2×2-matrix,

(x1 x2−y2 y1

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Corollary 4.3.9. If k is a (Noetherian) regular base ring, then for everyn ≥ 2, theA1-contractiblek-schemeX2n has a non-trivial vector bundle.

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A. Asok, DEPARTMENT OFMATHEMATICS, UNIVERSITY OF SOUTHERNCALIFORNIA , 3620 S. VERMONT AVE.KAP 104, LOS ANGELES, CA 90089-2532, UNITED STATES; E-mail address:asok@usc.edu

B. Doran, DEPARTEMENTMATHEMATIK , EIDGENOSSISCHETECHNISCHEHOCHSCHULE, RAMISTRASSE101,8092 ZURICH SWITZERLAND ; E-mail address:brent.doran@math.ethz.ch

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