Endomorphisms of the projective plane and the image of the Suslin–Hurewicz map

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel–Voevodsky motivic stable homotopy category than in Voevodsky’s derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root. A byproduct of these computations is a proof of Suslin’s conjecture on the Suslin–Hurewicz homomorphism from Quillen to Milnor K-theory in degree four, based on work of Asok et al. (Invent Math 219:39-73, 2020).


Introduction
Automorphisms of geometric objects describe their symmetries, and hence important geometric information. It depends on the context which type of morphisms are considered useful. In the case of a projective space P n over the complex numbers, one may consider its linear automorphisms (a group denoted PGL n+1 (C)), birational automorphisms (the Cremona group Cr n (C)), diffeomorphisms, homeomorphisms, and self-homotopy-equivalences, just to name a few. The Morel-Voevodsky A 1 -homotopy theory provides an interesting way to consider self-homotopy-equivalences of varieties [21]. Although this setup is conceptionally very satisfying, concrete determinations of endomorphisms in the A 1 -homotopy category are hard to come by. For example, the endomor-B Oliver Röndigs oliver.roendigs@uni-osnabrueck.de 1 Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany phism ring of P 1 over a perfect field F in the pointed A 1 -homotopy category is given by the Grothendieck ring of isomorphism classes of symmetric inner product spaces over F with a chosen basis, where the isomorphisms preserve the inner product and have determinant 1 with respect to the chosen bases [17,Remark 7.37].
Stabilization with respect to smashing with a projective line P 1 ∧− provides a simpler categorical setting, the motivic stable homotopy category SH(F), which is still richer than the corresponding derived category of motives [27]. Part of the gain from leaving the unstable realm is an additive (in fact triangulated) structure, whence the set of endomorphisms of any object is always a ring. For example, it is a deep theorem of Morel's that the endomorphism ring of the projective line in the motivic stable homotopy category over a field F is the Grothendieck-Witt ring of symmetric bilinear forms with coefficients in F [15]. The addition in the Grothendieck-Witt ring, whose elements are formal differences of symmetric bilinear forms, is induced by orthogonal sum, and the multiplication by tensor product of forms. By construction, it coincides with the endomorphism ring of the unit for the symmetric monoidal structure given by the smash product. Hence it has to be commutative. This is already different for the projective plane.
In particular, the ring [P 2 , P 2 ] SH(F) is non-commutative if and only if there exists a square in F which does not admit a sixth root, or, equivalently, if the cube map u → u 3 is not surjective on F. Its group of units (which could be called the group of P 1 -stable self-A 1 -homotopy-equivalences of P 2 ) consists of all triples (x 1 , x 2 , x 3 ) ∈ Z⊕Z⊕ F × /(F × ) 6 where either x 1 = ±1 and x 2 = 0, or x 1 = ±1 and x 2 = −x 1 . It is as non-commutative as the endomorphism ring it belongs to. Along the way, the homotopy modules π 1 P 2 and π 2 P 2 will be determined, based on computations in [23] and [22]. These computations provide an ingredient to complete the program Aravind Asok, Jean Fasel and Ben Williams developed in [3] to prove Suslin's conjecture on the Suslin-Hurewicz homomorphism from Quillen to Milnor K -theory in degree four.
Theorem Let F be an infinite field of characteristic different from 2 and 3, and A an essentially smooth local F-algebra. The image of the Suslin This closes the gap between three and five in the set of degrees for which Suslin's conjecture was previously known. See the introduction of [3] for more details on its history, as well as Suslin's original paper [25].

Topology
Let CP n denote complex projective space of complex dimension n. This section contains rather elementary calculations in the classical stable homotopy theory, which determine the endomorphism ring of CP 2 in the stable homotopy category. These calculations are based on stable homotopy groups of spheres π m S in degree m < 6 and the action of the topological Hopf map η : S 3 → CP 1 ∼ = S 2 , whose cofiber is CP 2 , on them. The standard reference here is [26]. As is customary in stable homotopy theory, the notation for a map and its (de)suspensions coincide if the context allows it. The purpose of this section is not to present original results (there aren't any), but instead to document the similarities and differences to the situation in the motivic stable homotopy category.
Choose a basepoint for CP 1 , and hence CP 2 , which will not appear in the notation. The cofiber sequence induces a long exact sequence of stable homotopy groups terminating with π 2 CP 1 = π 2 CP 2 . The induced short exact sequences 0 → π m−2 S/ηπ m−3 S → π m CP 2 → η π m−4 S → 0 express the stable homotopy group of the complex projective plane as an extension of two groups, the subgroup annihilated by η, and the cokernel of multiplication by η, on the respective stable homotopy group of spheres. Since η : π 0 S ∼ = Z → π 1 S ∼ = Z/2Z is surjective, η : π 1 S → π 2 S is an isomorphism, η : π 2 S → π 3 S ∼ = Z/24 is injective, and π 4 S ∼ = π 5 S ∼ = 0, the following table results, without any extension problem to solve.
Here 2Z denotes the abelian group of even integers under addition. The vanishing π 3 CP 2 = 0 implies that the cofiber sequence (2.1) induces a short exact sequence of stable homotopy groups. Hence the abelian group [CP 2 , CP 2 ] is an extension of two free abelian groups, each on one generator. Since π 2 CP 2 ∼ = Z, the short exact sequence (2.2) splits. In order to describe the ring structure, it helps to be more specific. A generator for π 2 CP 2 is the inclusion i : To describe a generator for π 4 CP 2 , observe that there exists a unique map ω : S 4 → CP 2 such that q • ω = 2id S 4 . The short exact sequence (2.2) then implies that every element x ∈ [CP 2 , CP 2 ] can uniquely be expressed as a sum The ring structure is then given as and is in particular commutative. The group of units consists of the following 4 elements: As π 5 CP 2 is cyclic, generated by i • ν, the abelian group [ CP 2 , CP 2 ] is generated by the map Its order can be determined by identifying ω • η ∈ π 5 CP 2 , which is ±6(i • ν), as the Toda which shows that it does not contain the zero element. More precisely, since the composition is multiplication with 2 on a cyclic group with 24 elements, and the homomorphism q * : π 3 S → [ 2 CP 2 , S 3 ] is the projection onto a cyclic group with 12 elements (as one deduces from the action of η on π n S for n ∈ {1, 2}), the homomorphism ω * : For comparison purposes with the motivic situation, it is instructive to look at the real case as well. Let RP n denote real projective space. The cofiber sequence 1 induces a long exact sequence of stable homotopy groups terminating with π 1 RP 1 → π 1 RP 2 . The induced short exact sequences 0 → π m−1 S/2π m−1 S → π m RP 2 → 2 π m−2 S → 0 express the stable homotopy group of the real projective plane as an extension of two groups, the 2-torsion subgroup, and the cokernel of multiplication by 2, on the respective stable homotopy group of spheres. The groups π 3 RP 2 and π 4 RP 2 are both extensions of Z/2 by Z/2. The Toda bracket 2, η, 2 = {η 2 } implies that π 3 RP 2 is given by the nontrivial extension, the extension for π 4 RP 2 turns out to be trivial [ The portion for m < 3 of this table implies that the cofiber sequence (2.4) induces a short exact sequence and hence the abelian group [RP 2 , RP 2 ] is an extension of two groups of order two. As in the computation of π 3 RP 2 , the extension is nontrivial, meaning that id RP 2 ∈ [RP 2 , RP 2 ] is an element of order 4, as already proven in [7]. There cannot be any doubt whatsoever regarding the ring structure of [RP 2 , RP 2 ].

Over a field
Let F be a field, and let SH(F) denote the motivic stable homotopy category of F [27]. For a motivic spectrum E ∈ SH(F) and integers s, w ∈ Z, let π s,w E denote the abelian group [ s,w 1, E], where E is a motivic spectrum and 1 F = 1 is the motivic sphere spectrum. The grading conventions are such that the suspension functor 2,1 = 1+(1) is suspension with P 1 , and 1,0 = 1+(0) = 1 = is suspension with the simplicial circle. Set π s+(w) E := π s+w,w E, and let denote the direct sum, considered as a Z-graded module over the Z-graded ring π 0+( ) 1. The notation π s−( ) E := π s+(− ) E will be used frequently. The strictly A 1 -invariant sheaf obtained as the associated Nisnevich sheaf of U → [ s,w U + , E] for U ∈ Sm F is denoted π s,w E, which gives rise to the homotopy module π s+( ) E. In the following, every occurrence of "π " can be replaced by "π" without affecting the truth of the (suitably reinterpreted) statements. See [15] for the following fundamental result. The Milnor-Witt K -theory of F is denoted K MW (F), or simply K MW , following the convention that the base field or scheme may be ignored in the notation. The definition and some details regarding K MW and modules over it are contained in the Appendix A. Theorem 3.1 implies that for every motivic spectrum E and for every integer s, π s+( ) E has a canonical structure as a graded K MW -module. The conventions dictate that this structure comes with a sign change in the sense that elements in K MW d provide homomorphisms Choose a basepoint for P 1 , and hence P 2 , which will not appear in the notation. Neither will the base field F most of the time. The main cofiber sequence over a field is It induces a long exact sequence of K MW -modules terminating with π 1+( ) P 2 by connectivity [16]. The induced short exact sequences express π m+( ) P 2 as an extension of two K MW -modules, the submodule of π m−2+( −2) 1 annihilated by η, and the cokernel of multiplication by η on π m−1+( −1) 1. This justifies the relevance of the following statement. Theorem 3.2 provides the exactness of the sequence mentioned in [8,Remark 4.3]. It is quite special that the kernel of multiplication by η on K MW is generated by a single element, but then the element η is also quite special. As a consequence of Theorem 3.2, the short exact sequence specializes to an isomorphism π 1+(w−1) 1/ηπ 1+(w−2) 1 ∼ = π 2+(w) P 2 for w > 2. Therefore knowing π 1+( −1) 1/ηπ 1+( −2) 1 is essential. Theorem 3.3 Let F be a field of characteristic not two or three. The unit map 1 → kq induces a surjection π 1+( ) 1/ηπ 1+( −1) 1 → π 1+( ) kq/ηπ 1+( −1) kq whose kernel is K M 2− /12 (generated in = 2) after inverting the exponential characteristic e of F. In particular, the vanishing π 1+(w) kq/ηπ 1+(w−1) kq for w > 1 implies that the nontrivial group of highest weight is π 1+(2) 1[e −1 ]/ ηπ 1+(1) 1[e −1 ] ∼ = Z/12, generated by the image of the Hopf map ν.
Proof This is a consequence of [23,Theorem 5.5] in the formulation given in [22,Theorem 2.5]; see also [24,Theorem 1.1]. First of all, the unit map π 1+( ) 1 → π 1+( ) kq is surjective, whence the same is true for the induced map on the quotients. Set e to be the exponential characteristic of F. Consider the following natural transformation denotes the topological Hopf map. 2 Hence the snake lemma also implies that The same proof shows that Theorem 3.3 is valid in characteristic 3 as well in the sense that the kernel of 1 3 ] after inverting 3. Theorems 3.2 and 3.3 provide sufficient information about the outer terms in the short exact sequence of K MW -modules. Actually the outer terms are K M -modules in a natural way; η acts trivially on these. However, η acts nontrivially on the middle term. The reason is the Toda bracket η, h, η = {6ν, −6ν} from [22, Proposition 4.1].

Lemma 3.4 Let F be a field of characteristic neither
implies by Proposition B.1 that there exists an element h ∈ π 2+(2) P 2 which on the one hand maps to h ∈ η π 0+(0) 1, and on the other hand is such that h •η is the image of 6ν ∈ π 1+(2) 1. Inspecting the short exact sequence (3.3) in weight 3 gives an isomorphism π 1+(2) 1/ηπ 1+(1) 1 ∼ = π 2+(3) P 2 by Theorem 3.2. Hence π 2+(3) P 2 is cyclic of order 12 by Theorem 3.3, with the image i • ν of ν as a generator. It follows that h • η is the unique nonzero element of order two in this group, and this is true for any choice of h lifting h. Inspecting the short exact sequence (3.3) in weight 2 provides In order to describe the extension group more precisely, set A := π 1−( −1) 1/ηπ 1−( ) 1. The extension (3.3) is given by an element in by Theorem 3.2. The short exact sequence where the subscript "K MW " is suppressed. Lemma A.3 applies to provide an isomorphism between the group Ext 1 where the last isomorphism follows from Theorem 3.3. Note that multiplication with η on the K MW -module π 1+( ) 1/ηπ 1+( ) 1 is the zero homomorphism by construction, which simplifies the term appearing in Lemma A.3. Hence There results an exact sequence In particular, the soughtafter element in Ext 1 K MW (2K M , A) classifying the extension in question is determined by the relation h • η = 6(i • ν) and an element in the group Ext 1 K M (2K M , A) depending solely on the K M -module structures of 2K M and A. Remark 3.5 Regarding the unstable situation, the A 1 -fiber sequence of (unstable) A 1 -homotopy sheaves, by [17, Theorem 1.23]. Let π A 1 2+(3) P 2 denote the threefold contraction of π A 1 2 P 2 , which coincides with the Nisnevich sheaf associated with the presheaf X → to the stable homotopy sheaf computed in Lemma 3.4. It sends the generator to m(i • ν), where m is an integer unique up to multiples of 12, and i • ν is the generator of the target. A comparison with the classical topological situation via complex or étale realization, which is possible since the target does not depend on the base field, shows that m = ±2, because the order of the canonical map S 5 → CP 2 is 6 after one suspension [19,Theorem 1.2]. Note that étale realization sends P n to the profinite completion of its complex realization CP n by [5,Theorem 12.9]; see also [10,Theorem 8.4]. The same applies to the maps involved here.
The computations provided by π 1−( −1) P 2 ∼ = K M and Lemma 3.4 suffice to conclude the following statement.
of K MW -modules. In particular, after inverting the exponential characteristic, there is an isomorphism with i • ν • q as a generator, and an isomorphism of abelian groups, with id P 2 and h • q each generating one free summand.
Proof As before, the cofiber sequence (3.1) induces a short exact sequence The short exact sequence for π 1+( ) P 2 specializes to the identification π 1+( ) P 2 ∼ = K M mentioned already above. Since η acts as zero on this K MW -module, the short exact sequence (3.5) of K MW -modules follows. The identity id P 2 hits the canonical generator i ∈ π 1+(1) P 2 . The action of η in the K MW -module structure on [ ( ) P 2 , P 2 ] is then determined by specifiying More precisely, as in the proof of Lemma 3.4 there results a short exact sequence 3 identifies the last extension group as Here the description of π 2+( ) P 2 as a K MW -module from Lemma 3.4 supplies the last isomorphism in this sequence, as well as the first isomorphism mentioned in the statement of the theorem. Hence ηid P 2 = m(i • ν • q) for some m ∈ Z which is unique up to multiples of 6. The element ηid P 2 turns out to be the unique nonzero element of order 2, as the Toda bracket η, h = q • h , η implies. The properties of Toda brackets supply an inclusion which shows that it does not contain zero. This already suffices to conclude. More precisely, since the composition (2) 1 identifies with the inclusion Z/12 → Z/24, as one may deduce from the exact sequence Theorem 3.6 implies that every element x ∈ [P 2 , P 2 ] can be expressed uniquely as a sum It would probably be more honest to think of the integers x 1 , x 2 as the ranks of virtual quadratic forms. In particular, the hyperbolic form "h" corresponds to the integer "2". Using that the composition q • i is the zero map, the ring structure is then given as and in particular is not commutative if 2K M 1 (F)/6 is nonzero. The group of units in [P 2 , P 2 ] consists of the elements Remark 3.7 A different motivic type of endomorphisms of the projective plane occurs in the motivic stable homotopy category SH(P 2 F ) for P 2 F , with unit where the source is a commutative ring by definition. Using [22,Theorem 2.7], the short exact sequence is not isomorphic to the Grothendieck-Witt ring of P 2 F , which is isomorphic to K MW 0 (F) ⊕ K M 0 (F) [29].
K -theory in degree four, exploiting the beautiful work [3]. Unstable homotopy sheaves will occur, as already in Remark 3.5. Let be the smooth affine quadric hypersurface which is weakly equivalent to A n {0} via projection to the first n coordinates [9, Example 2.12 (3)]. The quotient scheme Q 2n−1 /G m with respect to the free action λ · (a, b) := (λa, λ −1 b) is then weakly equivalent via projection to the first n coordinates to P n−1 .
For a unit u ∈ G m , let (u) denote the diagonal matrix whose entries are (u, 1, . . . , 1). The map is compatible with the given G m action on the first factor (and trivial actions on the other factor and SL n ) and sends Q 2n to the identity matrix, the canonical basepoint in SL n . Let ψ n : (1) Q 2n−1 /G m → SL n denote also the induced pointed map, a variant of the map (with the same notation) to GL n constructed in [31,Section 5]. Its complex realization is denoted j n in [19, p. 180], and f SU(n) in the even more classical source [33,Section 4]. It is straightforward to check that the diagram with obvious inclusions as vertical maps commutes.
shows. This strong A 1 -deformation retraction is G m -equivariant, whence also the inclusion {b 2 = 0} → {b 1 = 0} is a strong A 1 -deformation retract. Note that {b 2 = 0} is isomorphic to A 1 . Homotopy purity [21, Theorem 3.2.23] supplies a homotopy cofiber sequence inducing a homotopy cofiber sequence after applying (1) . In particular, since Similarly, the smooth closed subscheme {c 21 = 0} → SL 2 gives rise, via homotopy purity, to a homotopy cofiber sequence which can be related to the homotopy cofiber sequence above as follows. While the map ψ 2 : in which the vertical map on the right hand side is a weak equivalence, because it is induced by the isomorphism ψ 2 : Hence ψ 2 is a weak equivalence as claimed.
The proof of the following statement is essentially a modification of Jean Fasel's unpublished proof for the corresponding statement on symplectic groups; I thank him sincerely for the inspiration.

Proposition 4.2 The inclusion SL n → SL n+1 fits into a homotopy cofiber sequence
Proof A matrix C ∈ SL n has entries denoted c 1,1 , . . . , c 1,n , c 2,1 , . . . , c n,n . The homotopy purity theorem [21, Theorem 3.2.23], applied to the smooth closed subscheme W := {c n+1,1 = . . . = c n+1,n = 0} → SL n+1 (which is a global complete intersection and thus has a trivial normal bundle), supplies a homotopy cofiber sequence: (c n+1,1 , . . . , c n+1,n ). Let X and Y be defined by taking pullbacks where the vertical maps are Zariski locally trivial fibrations with A n−1 as fiber, and hence weak equivalences. An element in Y is a pair (E, d) with E ∈ SL n+1 W and d ∈ A n {0} such that n j=1 e n+1, j · d j = 1. The map As in the proof of Lemma 4.1, the homotopy purity theorem provides a homotopy cofiber sequence Enlarging the subscheme SL n × A n {0} in (4.4) to SL n × A n then provides the desired homotopy cofiber sequence:

Corollary 4.3 In the commutative diagram
in which the top row is a homotopy cofiber sequence and the bottom row is a homotopy fiber sequence, the canonically induced map S n+(n+1) → SL n+1 /SL n is a weak equivalence over Spec(Z).
Proof Proposition 4.2 and the standard homotopy cofiber sequence for Q 2n−1 /G m → Q 2n+1 /G m give a diagram of homotopy cofiber sequences where the vertical map on the right hand side is induced by the inclusion of the identity matrix. The canonical map n+(n+1) (SL n ) + → SL n+1 /SL n is induced by the structure map SL n → Spec(Z), because the pullback square last row of smooth schemes induces a commutative diagram of homotopy purity transformations, where the map sends a matrix in W SL n × (A 1 {0}) to its last diagonal element and hence is induced by the structure map SL n → Spec(Z). Proceeding through the zigzag relating SL n+1 /(SL n+1 W ) with SL n+1 /SL n produced in the proof of Proposition 4.2 provides the statement.
With the help of ψ 3 , the cell structure of SL 3 looks as follows.
in which the vertical maps are the canonical quotient maps and the bottom horizontal map is the canonical inclusion is a homotopy pushout diagram. The result follows.
More generally, the total homotopy cofiber of diagram (4.2) can be determined as follows.
Proof This follows from Proposition 4.2, Corollary 4.3 and a straightforward manipulation of homotopy pushout squares.
In the following, Q 2n−1 /G m will be identified via projection to the first n coordinates with P n−1 , which gives rise to maps such as ψ n : (1) P n−1 (1) Q 2n−1 /G m → SL n in the homotopy category. Recall from [6, Convention 2.3.5] that a map f of pointed motivic spaces is A 1 -n-connected if its homotopy fiber 3 hofib( f ) is A 1 -(n − 1)-connected, which is equivalent to the homomorphism π A 1 j f being an isomorphism for j < n and an epimorphism for j = n. Lemma 4.6 Let F be a field. The inclusion (1) P 1 → (1) P 2 induces the canonical projection The assumption that F is perfect may be imposed by pulling back from a perfect subfield, over which everything in sight is defined. As a suspension of an A 1 -0-connected variety, (1) P n is A 1 -0-connected for all n [6, Lemma 3.3.1]. The determination of π A 1 1 P n from [17, Theorem 7.13 and Theorem 7.29] implies that the map π A 1 1 P 1 → π A 1 1 P n is surjective for all n > 0.
In other words, the inclusion P 1 → P n is A 1 -1-connected for all n > 1. While smashing with G m preserves the simplicial connectivity of a map, it is a priori not clear whether smashing with G m preserves the A 1 -connectivity of a map.  ( (1) B) for every pointed simplicial presheaf B. Choosing appropriate A 1 -naive models for projective spaces -which is possible by [4,Example 4.2.13] -provides a model for the canonical inclusion P 1 → P n such that Sing A 1 (P 1 ) → Sing A 1 (P n ) is simplicially 1-connected, and hence so is The source is stalkwise equivalent to Sing A 1 (SL 2 ) by [ where the last isomorphism follows from [17,Theorem 1.27] (see also [18,Theorem 1]). Thus π A 1 0 (1) Sing A 1 (P 1 ) is a strictly A 1 -invariant sheaf. To prove the same for π A 1 0 (1) Sing A 1 (P n ), let G denote the (simplicial) homotopy fiber of the map (4.5). The aforementioned connectivity of the map (4.5) implies that G is simplicially 0-connected, and hence A 1 -0-connected by [21,Cor. 2.3.22]. Since (1) Sing A 1 (P n ) is the simplicial homotopy fiber of the canonical map G → (1) Sing A 1 (P 1 ), [17,Theorem 6.56] (see also [6, Corollary 2.3.6]) provides an exact sequence In this exact sequence π A 1 1 G is strongly A 1 -invariant by [17, Theorem 6.1], its image in π A 1 1 (1) Sing A 1 (P 1 ) is strongly A 1 -invariant by [8,Theorem 1.6], and moreover abelian, and the cokernel π A 1 0 (1) Sing A 1 (P n ) is then a strictly A 1invariant sheaf by [17,Corollary 6.24]. As a consequence [17,Theorem 6.56] applies to show that (1) To determine π A 1 1 (1) P n for n > 1, Morel's A 1 -Hurewicz theorem [17,Theorem 6.35] implies that the Hurewicz transformation π A 1 1 (1) P n → (1) P n is an isomorphism for all n. For n = 2, the latter can be determined via the cofiber sequence Similarly, for n > 2 the inclusion P n−1 → P n is A 1 -(n −1)-connected, as one may deduce from the homotopy fiber of the "covering" map A n {0} → A n+1 {0} of universal A 1 -coverings. Again smashing with G m preserves the simplicial connectivity of P n−1 → P n . Arguing with A 1 -naive models and the functor Sing A 1 as before provides that (1) The pushout in diagram (4.2) gives rise to a map which factors ψ n+1 : (1) P n → SL n+1 for every n > 0.

Proposition 4.7
Let F be a field. The image of the homomorphism π A 1 n SL n+1 → π A 1 n S n+(n+1) induced by taking the last column of a matrix is isomorphic to the image of the homomorphism π A 1 n (1) P n ∪ (1) P n−1 SL n → π A 1 n S n+(n+1) induced by the canonical quotient map collapsing SL n to the basepoint.
Proof In case n = 1, both the last column map and the quotient map are equivalences, whence the induced homomorphisms are isomorphisms. Let n > 1. Diagram (4.2) and the map θ n+1 defined in (4.6) induce the following commutative diagram of sheaves of A 1 -homotopy groups, in which the vertical homomorphisms on the right hand side are isomorphisms by Corollary 4.3 and by construction. In particular, the image of the homomorphism π A 1 n (1) P n ∪ (1) P n−1 SL n → π A 1 n S n+(n+1) embeds in the image of the homomorphism π A 1 n SL n+1 → π A 1 n S n+(n+1) . To prove that the image of π A 1 n SL n+1 → π A 1 n S n+(n+1) coincides with the image of π A 1 n (1) P n ∪ (1) P n−1 SL n → π A 1 n S n+(n+1) induced by collapsing SL n , it suffices to prove that the homomorphism π A 1 n θ n+1 : π A 1 n (1) P n ∪ (1) P n−1 SL n → π A 1 n SL n+1 is surjective. This in turn follows if the map θ n+1 is A 1 -n-connected. This is the connectivity of its homotopy cofiber cone(θ n+1 ) by Lemma 4.5. Unfortunately this does not necessarily imply that its homotopy fiber hofib(θ n+1 ) is A 1 -(n − 1)-connected. To conclude this nevertheless, start with the case n = 2. Then the map θ 3 in question coincides with ψ 3 : (1) P 2 → SL 3 up to equivalence by Lemma 4.1. Lemma 4.6 implies together with the determination of π A 1 1 SL n from [18, Theorem 1] that π A 1 1 ψ 3 is an isomorphism. There results an exact sequence whence it remains to prove that π A 1 2 ψ 3 is an epimorphism. Consider the commutative diagram induced by diagram (4.2) in which the bottom row is induced by the homotopy fiber sequence and in particular exact. The top row in diagram (4.8) is induced by the homotopy cofiber sequence and the weak equivalence ψ 2 : (1) P 1 → SL 2 from Lemma 4.1, in the sense that the homomorphism π A 1 2 S 2+(3) → π A 1 1 (1) P 1 is the composition of the inverse of π A 1 1 ψ 2 and π A 1 2 S 2+(3) ∼ = π A 1 2 A 3 {0} → π A 1 1 SL 2 . Here the first isomorphism follows from Corollary 4.3. It follows that the upper row in diagram (4.8) is exact at the spots involving π A 1 1 . To conclude exactness at (3) , it remains to prove that the image of π A 1 2 (1) P 2 → π A 1 2 S 2+(3) contains the kernel of π A 1 2 S 2+(3) → π A 1 1 (1) P 1 . By construction of the homotopy cofiber sequence, the latter coincides with the kernel of η : K MW 3 → K MW 2 , which is hK MW 3 by Theorem 3.2. The map (a 0 : a 1 ), (b 0 : b 1 ) → (a 0 b 0 : a 0 b 1 + a 1 b 0 : a 1 b 1 ) induces a commutative diagram h in which the identification of the lower horizontal map can be deduced from P 1 -stabilizing first, then observing that its P 1 -stabilization is in the kernel of multiplication with η on the Grothendieck-Witt ring (hence an integer multiple of h by Theorem 3.2), and finally a degree argument using motivic cohomology or realization, showing that the rank of the integer multiple of h is 2. Morel's Theorem [17, Theorem 1.23] then implies that the image of the homomorphism In particular, the top row in diagram (4.8) is also exact at π A 1 2 S 2+(3) . Together with the isomorphism π A 1 2 ψ 2 from Lemma 4.1, a diagram chase then provides that π A 1 2 ψ 3 is an epimorphism. Diagram (4.2) furthermore implies that ψ n+1 is at least A 1 -2-connected for all n > 1. Invoking the A 1 -van Kampen theorem [30,Theorem 3.10], [17,Theorem 7.12] provides with Lemma 4.6 that π A 1 1 θ n+1 is an isomorphism for all n > 1. Hence θ n+1 : (1) P n ∪ (1) P n−1 SL n → SL n+1 is A 1 -2-connected as well. To reach further, let n > 2 and consider the transformation of homotopy fiber sequences, inducing a transformation of long exact sequences of homotopy groups. The vertical map in the middle of diagram (4.9) is A 1 -(n − 1)-connected. In fact, as the cobase change of the map (1) P n−1 → (1) P n which is A 1 -(n − 1)-connected by Lemma 4.6, it is simplicially (n − 1)-connected. To apply [17,Theorem 6.56] or [6, Corollary 2.3.6] and conclude the desired A 1 -connectivity, it remains to prove that π A 1 0 (1) P n ∪ (1) P n−1 SL n is strongly A 1 -invariant. The simplicial van Kampen theorem [30,Corollary 3.5] provides that the Nisnevich sheaf associated with the presheaf of fundamental groups of Sing A 1 ( (1) P n ) ∪ Sing A 1 ( (1) P n−1 ) Sing A 1 (SL n ) (using A 1 -naiveness provided by [17,Theorem 8.1] for GL n and [4, Example 4.2.13]) coincides with K M 2 . In particular, it is strictly A 1 -invariant. The long exact sequence diagram (4.9) induces on A 1 -homotopy groups then provides epimorphisms π A 1 j+1 S n+(n+1) ∼ = π A 1 j S n+(n+1) → π A 1 j hofib(θ n+1 ) for all j < n. In particular, Morel's A 1 -connectivity for S n+(n+1) provides π A 1 j hofib(θ n+1 ) = 0 for j < n − 1, so that θ n+1 is at least A 1 -(n − 1)connected. To conclude the vanishing of π A 1 n−1 hofib(θ n+1 ), observe that it is a quotient of π A 1 n S n+(n+1) ∼ = K MW n+1 . The A 1 -homotopy groups induced by diagram (4.9) are modules over π A 1 1 SL n ∼ = π A 1 1 ( (1) P n ∪ (1) P n−1 SL n ) ∼ = K M 2 , and all homomorphisms involved are K M 2 -equivariant. However, the action of π A 1 1 SL n on π A 1 j S n+(n+1) is trivial because it factors through π A 1 1 * , as the commutative diagram id of fiber sequences implies. Hence the action of π A 1 1 ( (1) P n ∪ (1) P n−1 SL n ) on the quotient π A 1 j hofib(θ n+1 ) of π A 1 j S n+(n+1) is also trivial. It follows that the relative Hurewicz homomorphism π A 1 n−1 hofib(θ n+1 ) → H A 1 n cone(θ n+1 ) introduced in [2, Section 4.2] is an isomorphism by [2,Theorem 4.2.1]. The latter group is trivial by Lemma 4.5. Hence θ n+1 is A 1 -n-connected.

The composition
is generated by η and 6 as a K MW -module. In particular, in the relevant degree, the image of π A 1 3 SL 4 → π A 1 3 S 3+(4) is contained in the subsheaf ηK MW 5 + 6K MW 4 . Since it also contains this subsheaf, equality follows. This completes the proof.

Corollary 4.9
Let F be an infinite field with characteristic coprime to 6. The inclusion BSL 3 → BSL ∞ induces the following short exact sequence of sheaves:  [14] is expressed as the quotient K M ∼ = K MW /ηK MW . Often the grading will be suppressed from the notation. Its reduction modulo 2 is denoted k M for brevity. If A is a (graded) K MW -module, its degree d part is A d . For any x ∈ K MW d , the kernel and cokernel of multiplication with x on A are denoted x A and A /x A −d . As a warm-up, K M -modules will be treated first.
Proof This follows as an equality of zero modules from [13] if F has characteristic 2. Suppose now that the characteristic of F is different from 2. Since of K M -modules. Let τ ∈ π 1−(1) MZ/2 = h 0,1 denote the class of −1 ∈ F. Then ∂ 2 ∞ (τ ) = {−1} ∈ π 0−(1) MZ = K M 1 by inspecting the short exact sequence (A.1) in the given weight, using that −1 is the unique nontrivial unit of order two in F. Voevodsky's solution of the Milnor conjecture on Galois cohomology [28] (or rather the Rost-Voevodsky solution of the Beilinson-Lichtenbaum conjecture relating motivic and étale cohomology with coefficients in Z/2) provides that multiplication with τ is an isomorphism π 0+( ) MZ/2 ∼ = π 1+( ) MZ/2 of K M -modules. It follows that multiplication with {−1} on K M factors as in which the first three maps are surjective. The result follows.

Lemma A.2 Let A be a K M -module. There is an isomorphism
Proof The short exact sequence induces an exact sequence The short exact sequence

Lemma A.3 Let A be a K MW -module. There is a natural isomorphism
Proof Abbreviate Hom(A, B) := Hom K MW (A, B). The short exact sequence induces an exact sequence The short exact sequence The composition K MW +1 → ηK MW → K MW coincides with multiplication by η, whence naturally Hom(K M , A) = η A 0 . In order to identify Hom(ηK MW , A) in a similar way, observe the existence of a further short exact sequence

Appendix B: Toda brackets
Consider three composable maps is the subset of (pointed) homotopy classes maps which can be obtained this way. It depends only on the homotopy classes of α, β, γ . This definition provides a secondary composition on the level of homotopy categories and is sufficiently general to apply in manifold cases, for example in any pointed simplicial model category. In case [ D, G] is abelian (which is always the case for motivic spectra), then α, β, γ is a coset of the subgroup where α is an extension of α to cone(β) and γ is a coextension of γ with respect to β [26,Prop. 1.7]. Any coextension of γ with respect to β is a lift of γ along cone(β) → E. In the stable case of motivic spectra, a coextension of γ with respect to β is the same as a lift of γ along cone(β) → E [11,Prop. 5.2]. Two straightforward extensions of statements from [26] will be stated explicitly for reference purposes. By adjointness, an equivalent description in terms of maps to loop spaces can be given. From the definition it follows quite immediately that any functor induced by a pointed simplicial left Quillen functor on homotopy categories preserves Toda brackets in the sense of providing an inclusion ( α, β, γ ) ⊂ (α), (β), (γ ) whenever the Toda bracket α, β, γ is defined. In particular, if the target Toda bracket (α), (β), (γ ) does not contain the zero element, neither does the source Toda bracket α, β, γ . This applies in particular to base change functors, stabilization, and to complex and étale realization.