TRANSCHROMATIC EXTENSIONS IN MOTIVIC BORDISM

. We show a number of Toda brackets in the homotopy of the mo-tivic bordism spectrum MGL and of the Real bordism spectrum MU R . These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n , the bracket itself will be of chromatic height p n ` 1 q . Using these, we deduce a family of exotic multiplications in the π p˚ , ˚q MGL -module structure of the motivic Morava K -theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π ‹ MU R -module structure on the Real Morava K -theories.


Introduction
Complex bordism has played a fundamental role in stable homotopy since the 1960s.Work of Quillen connected complex bordism to formal groups, and this gives rise to the chromatic approach to stable homotopy theory.Building on Atiyah's Real K-theory [2], which can be viewed as Galois descent in families, Fujii and Landweber defined Real bordism [7,17].This theory plays an analogous role in C 2 -equivariant homotopy theory that ordinary complex bordism does classically, and a detailed exploration of it was carried out by Hu-Kriz [13].
The Real bordism spectrum has proven central to understanding classical chromatic phenomena.By the Goerss-Hopkins-Miller theorem, the Lubin-Tate spectra E n are acted upon by the Morava stabilizer group, and in particular, they can be viewed as genuine G-equivariant spectra for any finite subgroup G. Work of Hahn and the third author proved that at the prime 2, there is a Real orientation of all of the Lubin-Tate spectra, and for a finite subgroup G of the Morava stabilizer group that contains C 2 , this extends to a G-equivariant map where M U ppGqq is the norm of M U R [8], introduced by Hill-Hopkins-Ravenel in the solution to the Kervaire invariant one problem [10].This has turned questions about computations with the Lubin-Tate theories into questions about computations with the norms of M U R and its quotients (see, for example [3]).
In motivic homotopy over R, there is a beautifully parallel story.The role of complex bordism is played by the spectrum M GL.Just as in classical and equivariant homotopy theory, M GL is a fundamental object of study in motivic homotopy, providing not only a chromatic filtration but also a way to understand the motivic homotopy sheaves of the sphere spectrum via Voevodsky's slice filtration.
Voevodsky's slice filtration is an analogue of the Postnikov tower, where instead of killing all maps of spheres of a particular degree, we kill off all maps out of The first author was supported by NSF Grant DMS-1906227.The second author was supported by NSF Grant DMS-1811189.sufficiently many smash powers of P 1 .Applied to the algebraic K-theory spectrum KGL (an M GL-module spectrum), this yields the motivic cohomology to algebraic K-theory spectral sequence considered by Friedlander-Suslin, Voevodsky, and others (See [6,27]).Hopkins and Morel generalized this, showing that the slices of M GL are suspensions of the spectrum representing motivic cohomology, and Hoyois provided a careful treatment and generalization of this result [12].Work of Levine further connected the slice filtration to M GL, showing that the slice filtration for the sphere can be built out of the slice filtrations of the Adams-Novikov resolution based on M GL [18], and for the latter, the Hopkins-Hoyois-Morel result describes all of the starting pieces.
The spectrum M GL is a commutative ring spectrum, so in addition to a commutative multiplication on the bigraded homotopy groups, we have higher operations and products like Toda brackets.In this paper, we produce a family of Toda brackets in the homotopy groups of M GL which describe unexpected trans-chromatic phenomena.Recall that there is a canonical map π 2˚M U Ñ π p2˚,˚q M GL classifying the canonical group law for the motivic orientation (see [4] or [24]), and hence we have associated to the chromatic classes v n P π 2 n`1 ´2M U the motivic classes vn P π p2 n`1 ´2,2 n ´1q M GL.Recall also that we have a canonical element ρ P π p´1,´1q S 0 corresponding to the unit ´1 P R ˆ, by Morel's computation of the motivic zero stem [22].Finally, let I be the kernel of the map Theorem.For all n ě 0 and k ě 0, in the motivic homotopy of M GL over R, we have If we also consider the shifts of vn to other weights, using some of the other motivic lifts vn pbq of the chromatic classes (the definitions of which we recall below), then we have even longer transchromatic connections.
Theorem.For all n, j ě 0 and k ě 0, in the motivic homotopy of M GL over R, These transchromatic shifts imply a surprising number of hidden extensions in very naturally occurring quotients like the motivic Morava K-theories, the definitions of which we recall below in Section 3. Put in a pithy way, killing vn without also killing vn`1 does not fully kill vn : Theorem.For all n ě 1, for all 0 ď k ď n and for all b ě 0, in the homotopy of motivic Morava K-theory K GL pnq, there are nontrivial multiplications by vk pbq.
Note that even though the brackets had ambiguity caused by higher Adams filtration, these statements do not.
There is a natural functor from motivic spectra over R to C 2 -equivariant spectra, extending the functor "take complex points of a variety defined over R", and this takes M GL to the Real bordism spectrum M U R [14].Moreover, as studied by Hu-Kriz, Hill, and Heard, this connects the motivic slice filtration to Dugger's C 2 -equivariant slice filtration [14], [11], [9], [5].This functor takes ρ to the Euler class a σ P π C2 ´σ M U R and the copy of the Lazard ring in the homotopy of M GL to the copy of the Lazard ring described by Araki in the homotopy of M U R [1].In particular, the motivic classes vn are sent to the equivariant classes vn P π C2 p2 n ´1qρ2 M U R , where ρ 2 " 1 `σ is the regular representation of C 2 .The classes vn pbq are sent to the classes vn u 2 n b 2σ , using the notation of [10].We therefore deduce the equivariant versions of these transchromatic phenomena.We spell these out for those more familiar with the equivariant literature.
Corollary.For all n ě 0, in the ROpC 2 q-graded homotopy of M U R , we have an inclusions modulo I 2 , for I the kernel of π ‹ M U Ñ π ‹ HF 2 : In general, the indeterminacy of these brackets may be larger in the C 2 -equivariant context.We do not address this here.Using the Mackey functor structure on homotopy, we can actually identify the interesting element in the bracket as the unique non-zero element in the kernel of the restriction map.
There are also similar extensions in Hu-Kriz's Real Morava K-theories [13].
Theorem.For all n ě 1 and for all 0 ď k ď n and b ě 0, in the ROpC 2 qgraded homotopy of the Real Morava K-theory spectrum K R pnq, there are nontrivial multiplications by vk pbq.
For n " 1, this recovers the classical observation that multiplication by 2 is not identically zero in KO{2.For larger n and k, these seem unknown.
Acknowledgements.We thank Dan Isaksen for some very helpful conversations related to Massey products and to steering us towards a much more direct proof of our results.We thank Nitu Kitchloo and Steve Wilson for interesting discussions of Real K-theory and Paul Arne Østvaer for his help with the motivic slice spectral sequence for M GL and k GL pnq.We thank Irina Bobkova, Hans-Werner Henn and Viet Cuong Pham who made an observation which inspired our study of these brackets.We thank Mike Hopkins for several helpful conversations.We also thank the anonymous referee for their comments and suggestions.
2. Non-trivial brackets in π ˚,˚M GL 2.1.The homotopy of M GL.Throughout, we work with motivic spectra over R. We implicitly complete all spectra 2. Just as classically, we have a splitting of M GL into various suspensions of a motivic ring spectrum BP GL, and since this is smaller, we will mainly work with it.Then bigraded homotopy ring of BP GL has been completely determined [11], [16].We very quickly recall the answer here, using the description from [11].
As an algebra, it is generated by the classes vn pbq " for all n ě 0 and b ě 0, together with the class ρ P π p´1,´1q BP GL.
Here, the brackets indicate how the classes arise in the ρ-Bockstein spectral sequence.We will also normally use vn for vn p0q.
There are relations which reflect the underlying products with τ : if n ě m, then vm pbq ¨v n pcq " vm pb `2n´m cq ¨v n .
We also have relations involving ρ: for all n and b, The homotopy is actually largely concentrated in a particular bidegree sector.With the exception of the subalgebra generated by ρ, the first (i.e."topological") dimension is positive.
Proposition 2.1.Outside of the subalgebra spanned by ρ, all elements have nonnegative first coordinate.Moreover, the only generators with first coordinate zero are ρ 2 n`1 ´2 vn pbq, with b arbitrary.The products of any of these is zero.
Proof.Of the listed algebra generators, only ρ has a negative first coordinate, and the bidegree of ρ j vn pbq is Since the only non-zero values correspond to 0 ď j ď 2 n`1 ´2, we see that the first coordinate is always non-negative.This gives the first part.
For the second part, note that it is the last ρ power that gives a zero first coordinate.Since ρ times this is zero, we deduce that the product of any of these elements with first coordinate zero is zero.
Just as classically, the classes vn pbq depended on a choice of coordinate for the underlying formal group.However, we have a surprising invariance after multiplication by certain powers of ρ.Lemma 2.2.For any n and b, if p is an element in π ˚,˚B P GL such that p " vn pbq mod I 2 , then for all k ě p2 t n 2 u`1 ´1q, Proof.While there are many monomials in the vi paq in topological degree p2 n`1 2q, the only monomials of degree greater than one which survive even a single ρmultiplication are the monomials in the vi paq with i ă n.The order of the ρ-torsion on a monomial is governed by the smallest subscript present, so if we multiply by some ρ-power larger than any smallest subscript in a monomial in this degree, we annihilate any element of I 2 in this degree.Our choice of k accomplishes this.

2.2.
Toda brackets in the homotopy of M GL.The relations ρ 2 n`1 ´1 vn pkq " 0 show that we can form many Toda brackets in the homotopy of M GL and BP GL.
Proposition 2.3.Let n, m, b, and c be non-negative and let r, s, and t be nonnegative integers such that s `r ě 2 n`1 ´1 and s `t ě 2 m`1 ´1.
Proof.Kraines showed that we have a kind of symmetry invariance for higher brackets [15,Theorem 8], up to a sign, for Massey products.The proof goes through without change for Today brackets.
Remark 2.4.Whenever r `s, s `t ě p2 n`1 ´1q, we can also form the brackets xρ r , ρ s vn , ρ t y.For degree reasons, these are zero with zero indeterminacy.
One of the most useful parts of these brackets is that the indeterminacy is easily controlled.Because of Proposition 2.3, we may without loss of generality pick an ordering on the heights of chromatic classes.
Theorem 2.5.Fix non-negative numbers m ě n, b, and c.If r, s, and t are non-negative integers such that r `s ě p2 m`1 ´1q and s `t ě p2 n`1 ´1q, then the indeterminacy of xρ r vm pbq, ρ s , ρ t vn pcqy is nonzero in one case: when t " 0, r `s " 2 m`1 ´1, and m ą 0. In this case, the indeterminacy is Proof.The indeterminacy of the bracket xρ r vm pbq, ρ s , ρ t vn pcqy is the subgroup ρ r vm pbqπ pxt,ytq BP GL `ρt vn pcqπ pxr,yrq BP GL, where px t , y t q " |ρ s`t vn pcq| `p1, 0q " `2n`1 ´1 ´s ´t, 2 n ´1 ´s ´t ´2n`1 c ȋs the degree of the choices of null-homotopy of ρ s`t vn pcq, and similarly for px r , y r q.By our assumptions on r, s, and t, the first coordinate is always non-positive.By Proposition 2.1 if r `s ą p2 m`1 ´1q and s `t ą p2 n`1 ´1q, then we have π pxr,yrq BP GL " π pxt,ytq BP GL " 0, and hence, we have no indeterminacy.We need only consider the cases that pr`sq " p2 m`1 ´1q or ps `tq " p2 n`1 ´1q.
There is an obvious symmetry here in m and n, so it suffices to understand the case r `s " p2 m`1 ´1q.In this case, px r , y r q " `0, ´2m ´2m`1 b ˘.
Proposition 2.1 shows that the only classes with zero first coordinate are the classes ρ 2 k`1 ´2 vk paq, which is in bidegree We are therefore looking for all pairs pk, aq such that 1 ´2k p1 `2aq " ´2m p1 `2bq.
If m ą 0, then we must have k " 0 and a " 2 m´1 p1 `2bq, and this corresponds to the class v0 `p1 `2bq2 m´1 ˘.
This generates a Z 2 , and hence the contribution to the indeterminacy is the subgroup generated by v0 `p1 `2bq2 m´1 ˘¨ρ t vn pcq.
If t ą 0, then this is automatically zero (since ρv 0 paq " 0), so if r `s " p2 m`1 ´1q and t ą 0, then we have no contribution to the indeterminacy.On the other hand, if t " 0, then we have a contribution: Note that if t " 0, then the condition that s `t ě p2 n`1 ´1q implies that s ě p2 n`1 ´1q.This with the condition that r `s " p2 m`1 ´1q in particular shows the condition m ě n.Now let m " 0. We find k " `ν2 p1 `bq `1˘, for ν 2 the 2-adic valuation, and Note also that k ě 1, so the contribution to the indeterminacy is Again, if t ą 0, then this is automatically zero, since ρ 2 k`1 ´1 vk paq " 0. If t " 0, then the conditions r `s " 2 0`1 ´1 " 1 and s `t " p2 n`1 ´1q imply that in fact, n " 0 as well.Since k ě 1, p2 k`1 ´2q ě 1, and hence Thus in all cases, the contribution to indeterminacy here is zero.
This vanishing on indeterminacy actually lets us tightly connect the various brackets for a fixed m, n, b, and c via juggling formulae for Toda brackets.Proposition 2.6.Fix non-negative numbers m ě n, b, and c, and non-negative integers r, s, and t such that pr `sq ě p2 m`1 ´1q and ps `tq ě p2 n`1 ´1q.
Proof.Both of these are analogues of so-called "juggling" formulae for Massey products.These go through without change for Toda brackets.For the first, Kraines showed that we have an inclusion axb, c, dy Ď p´1q q xab, c, dy, where q is the degree of a [15, Theorem 6].Theorem 2.5 shows that the "larger" bracket never has any indeterminacy, giving the result, since ρ is simple 2-torsion in M GL.
We have a similar juggle for moving internal products through brackets, using the matric Massey juggling due to May [20, Corollary 3.4]: xab, c, dy Ď ˘xa, bc, dy.
By the first part of the proposition, the bracket xρ r`1 vm pbq, ρ s , ρ t vn pcqy is in the image of multiplication by ρ and hence the signs do not matter.The conditions on r, s, and t also guarantee that neither bracket in xρ r`1 vm pbq, ρ s , ρ t vn pcqy Ď xρ r vm pbq, ρ s`1 , ρ t vn pcqy has any indeterminacy.The last equality follows by symmetry.

2.3.
Identifying brackets via the Adams spectral sequence.We can describe what elements arise in these Toda brackets using the Adams spectral sequence.Just as classically, the homology of BP GL is cotensored up along a quotient Hopf algebroid.Voevodsky computed the dual Steenrod algebra over R, showing that as a Hopf algebroid over M 2 " F 2 rρ, τ s, the motivic homology of a point [28], we have The element ρ is primitive, the left unit on τ is the obvious inclusion, and the right unit is τ `ρτ 0 .The coproducts on the ξ i and the τ i are the classical ones [29].Let Ep8q " M 2 rτ 0 , τ 1 , . . .s{pτ 2 i ´ρτ i`1 q be the quotient of the motivic dual Steenrod algebra by the ideal generated by the ξ i s.This is a Hopf algebroid under the motivic dual Steenrod algebra, and the generators τ i are now primitive.In particular, all of the interesting behavior in Ext is determined by the left and right units on τ (since ρ, being in the Hurewicz image, is necessarily primitive).
Just as classically, this Hopf algebroid is related to the homology of BP GL, via the cotensor product.Recall that if Γ is a Hopf algebroid, M is a right Γ-comodule with coproduct ψ M , and N a left Γ-comodule with coproduct ψ N , then the cotensor product of M and N is defined by an equalizer diagram [25]).We have an isomorphism of A ‹ -comodule algebras Note that there are no ρ-torsion elements in either M 2 or Ep8q, so we can divide uniquely by ρ various ρ-divisible elements.
The elements vn pbq are detected by where d is the cobar differential, and the class ρ by itself in Ext 0,p´1,´1q .The spectral sequence collapses with no exotic extensions.
The Hopf algebroid `M2 , Ep8q ˘is computationally very simple: we have a primitive polynomial generator ρ and a second, non-primitive element τ .The ring Ep8q is not polynomial on τ 0 " η L pτ q ´ηR pτ q ρ .
Instead, we have a kind of "ρ-divided power algebra".The real power of Theorem 2.8 is that all of the generators of Ext 1 can be realized as ρ-fractional multiples of the usual cobar differential on powers of τ .That will allow us to easily compute Massey products in Ext.
It is helpful in what follows to blur the chromatic heights of the elements, focusing instead on the powers of τ and their differentials.Notation 2.9.For each k ě 1, let m k " 2 ν2pkq`1 ´1 for ν 2 pkq the 2-adic valuation of k and let vpkq " η L pτ q k ´ηR pτ q k ρ m k .
Note that vn pbq is detected by the element v`p 1 `2bq2 n ˘.
It is not hard to show that ρ m k is the largest power of ρ which divides the cobar differential on τ k .Theorem 2.10.Let k and be non-negative, and let r, s, and t be natural numbers such that s `r ě m k and s `t ě m .
Then in Ext, we have an inclusion ρ m k` ´mk ´m `r`s`t vpk ` q P @ ρ r vpkq, ρ s , ρ t vp q D .
Proof.There are preferred null-homotopies of vpkqρ r`s and vp qρ s`t : The bracket in question then contains ρ s`t`r´m vpkqη R pτ q `ρr`s`t´m k τ k vp q.
Unpacking the ρ-fractions giving vpkq and vp q and recalling that the cobar differential is a bimodule derivation, we see that this particular element is ρ m k` ´mk ´m `r`s`t vpk ` q.
Remark 2.11.These same relations will hold for any similar Hopf algebroid with a primitive element playing the role of ρ and the differential on a class analogous to τ involving only ρ divisibility.For example, we see the same brackets involving the α-family in the Miller-Ravenel-Wilson Chromatic Spectral Sequence approach to understanding the classical Adams-Novikov E 2 -term [21].
Remark 2.12.When n " 0, Theorem 2.13 gives the formula ρη P x2, ρ, 2y, since ρv 1 is the ordinary, topological η.Since 2ρ " 0 in the homotopy of M GL over R, this is recovering a universal classical formula in an A 8 -ring spectrum that forming the balanced bracket with 2 gives η multiplication [26].
2.4.Convergence.Moss's Convergence Theorem allows us to lift these Massey products in the Adams spectral sequence to Toda brackets in homotopy [23].Normally in applying Moss's Convergence Theorem, we have to worry about so-called "crossing differentials".In our case, the Adams spectral sequence collapsed at E 2 with no differentials.We deduce the following.
Theorem 2.13.Let b and c be nonegative numbers.Let r, s, and t be such that r `s, s `t ě m n " p2 n`1 ´1q.
Let j and d be such that we have Then, modulo elements of higher Adams filtration, we have an inclusion ρ p2 n`j`2 ´2n`2 `r`s`t`1q vn`j`1 pdq P @ ρ r vn pbq, ρ s , ρ t vn pcq D .
As stated, the theorem has markedly limited appeal, since there are in general a great many classes in higher Adams filtration.Using Proposition 2.6 and Lemma 2.2, we can remove some ambiguity, since the Adams filtration here coincides with the monomial filtration in induced by I. Then for any k ě p2 t n 2 u`1 ´1q, we have ρ 2 n`k`2 ´2n`2 `k`1 vn`j`1 pdq " xv n pbq, ρ 2 n`1 ´1`k , vn pcqy.

Application to Morava K-theories
The transchromatic brackets give us some surprising consequences for the action of π p˚,˚q M GL on various quotients.Since M GL is a commutative monoid, we have a good category of M GL-modules.For a single element x i , we define the quotient by x i via the cofiber sequence and for a family of elements x 1 , x 2 . .., we form For each n ě 0, let k GL pnq " BP GL{pv 0 , . . ., vn´1 , vn`1 , . . .q be the nth connective motivic Morava K-theory.
As an F 2 rρ, vn , τ 2 n`1 s-module, the only non-trivial differentials are generated by where 0 ď a ď 2 n ´1.
Away from the subalgebra F 2 rρs, the localization map In M GLrv ´1 n s, the class τ 2 n`1 is a permanent cycle and multiplication by this is well-defined in K GL pnq.By injectivity of the localization, we may therefore view everything as being a module over Z 2 rρ, vn , τ 2 n`1 s.
Corollary 3.4.As a module over Z 2 rρ, vn , τ 2 n`1 s, the homotopy of k GL pnq is generated by the classes τ a ι for 0 ď a ď p2 n ´1q.
Although the spectral sequence collapses here, we have a surprising number of non-trivial multiplications by the vk -generators that we killed to form k GL pnq, including non-trivial multiplications by 2. These are all detected by our brackets, using the sparseness of the spectral sequence (and hence sparseness of the bigraded homotopy groups).
We have a simple consequence of the module structure and the presence of τ 2 n`1 : we need only check a small number of hidden extensions.Corollary 3.5.We need only determine hidden extensions of the form vk pbqτ a ι for 0 ď b ď p2 n´k ´1q.
Proof.Since there is a class τ 2 n`1 , we have vk pb `2n´k q " vk pbqτ 2 n`1 .Remark 3.6.While going through our analysis of extensions, the reader is encouraged to consult Figure 1 which shows the slice associated graded for the bigraded homotopy groups of k GL p3q.The grading is the usual motivic bigrading.The circled classes are the generators as a module over Z 2 rρ, vn , τ The slice E 8 term for k GL p3q, together with extensions.See Remark 3.6 for the notation.
Lemma 3.7.Let a and b be non-negative integers and 0 ď k ă n.Then there is at most one non-zero class that could be vk pbqτ a ι.If a `2k p1 `2bq ă 2 n or 2 n`1 ´1 ă a `2k p1 `2bq, then we have vk pbqτ a ι " 0.Moreover, the ideal I 2 n acts trivially.Proof.The E 8 -page of the slice spectral sequence looks like a quilt.Applying the degree sheer px 1 , y 1 q " px, y ´xq, the homotopy groups are built out of rectangles of size p2 n`1 ´1q by p2 n q, with a single non-zero class in each degree.The generator of each rectangle is at the top right corner, given by τ 2 n`1 m v n ι for m ě 0 and ě 0.
Since vn has sheered bidegree `p2 n`1 ´2q, ´p2 n ´1q ˘, we see that the rectangle for α overlaps with the one for vn α in exactly one corner: the bottom right for α and the top left for vn α.Multiplication by τ 2 n`1 adds in a rectangle with generator in degree |α| `p0, ´2n`1 q.This does not intersect either the original rectangle or any of its vn -multiples.In particular, this proves the first claim, since almost all degrees have a single non-zero class in them, and the degrees with overlap have a single class in higher filtration, where any hidden extensions must land.
For the second and third claims, recall that the degree of vk pbqτ a ι is Since the classes τ a ι are at the rightmost edge of their rectangle, these extensions must show up in the rectangle from vn ι.The top edge of the rectangle for vn ι is given by the classes ρ m vn ι.The only class on that edge with first coordinate p2 k`1 ´2q is ρ 2 n`1 ´2k`1 vn ι, which is in bidegree Similarly, the bottom edge of the rectangle on vn ι is given by the classes ρ m vn τ 2 n ´1ι.
Proof.These follow surprisingly quickly from Theorem 2.13.We note that since multiplication by ρ is injective on both the source and target of any extension involving vk pbq for k ą 0, there is no ambiguity caused by the indeterminacy.There was no indeterminacy anyway for brackets involving v0 pbq.Since the ideal I 2 n acts trivially, we also have no ambiguity due to higher Adams filtration elements.For 0 ď b ď p2 n´k´1 ´1q, Theorem 2.13 shows that we have a bracket (where again, we ignore indeterminacy since it does not contribute).This gives us Lemma 3.7 shows that for a ă 2 k p1 `2bq, we have vk p2 n´k´1 ´1 ´bqτ a ι " 0, so we can shuffle the bracket, giving: ρ 2 n`1 ´2k`1 vn τ a ι " vn pbq @ ρ 2 k`1 ´1, vk p2 n´k´1 ´1 ´bq, τ a ι D .
A degree check shows that the only possible value for the bracket is @ ρ 2 k`1 ´1, vk p2 n´k´1 ´1 ´bq, τ a ι D " τ 2 n ´2k p1`2bq`a ι.
We are therefore at the leftmost edge of the "quilt rectangle" generated by vn ι.All of the elements with first coordinate zero here are annihilated by ρ, so we reach a contradiction: 0 ‰ ρ 2 n`1 ´2k`1 `1 vn p1qτ a ι " vk pcqρ @ ρ 2 k`1 ´1, vk pbq, τ a ι D " 0.
As always, there is a unique possibility for the value of vk pbqτ a ι, and checking degrees, we see it must be the listed one.
Specializing to the chromatic classes vk , we have a series of non-trivial extensions.
Since v0 detects multiplication by 2 in M GL-modules, we have non-trivial additive extensions.This resolves the question described in [16,Remark 9.8].
Remark 3.10.These extensions show just how far the quotient is from being a ring.One way to parse Theorem 3.8 is that although we cone-off vk for 0 ď k ď n ´1, and hence we seem to kill all of the classes vk pbq, we actually see non-trivial multiplications by all of the generators of π ˚,˚B P GL{pv n`1 , . . .q.
Even though the indeterminacy for brackets may grow in the passage from motivic homotopy over R to C 2 -equivariant homotopy, the extensions we found are visible just in the homotopy.They will in particular go through without change.We indicate the result using the usual equivariant names.Corollary 3.11.In the homotopy groups of K R pnq, we have exotic π ‹ M U R multiplications for 0 ď k ď pn ´1q: ‚ for 0 ď b ď p2 n´k´1 ´1q, and 0 ď a ď `2k p1 `2bq ´1˘, vk pbqu 2 n ´2k p1`2bq`a σ ι " a 2 n`1 ´2k`1 σ vn u a σ ι, ‚ and for 2 n´k´1 ď b ď p2 n´k ´1q, and 0 ď a ď `2n`1 ´1 ´2k p1 `2bq ˘, vk pbqu a σ ι " a 2 n`1 ´2k`1 σ vn u a`2 k p1`2bq´2 n σ ι.

Corollary 2 . 14 .
Let b and c be nonnegative numbers, and let j and d be such that 1 `b `c " p1 `2dq2 j .
The dotted lines indicate multiplication by ρ. ‚ The dashed blue lines indicate exotic multiplication by v2 , and the solid red lines indicate exotic multiplication by v1 .‚The bullets ‚ and solid black squares both indicate copies of F 2 ; bullets complexify to zero while black squares complexify to the corresponding generator.‚ The open circles are copies of Z{4 which complexify to Z{2.To avoid clutter, we do not draw the exotic multiplications by vk pbq for b ą 0 and k ă 3.