Secondary power operations and the Brown-Peterson spectrum at the prime 2
aa r X i v : . [ m a t h . A T ] M a y Secondary power operations and theBrown–Peterson spectrum at the prime 2
Tyler Lawson ∗ May 4, 2018
Abstract
The dual Steenrod algebra has a canonical subalgebra isomorphic to the ho-mology of the Brown–Peterson spectrum. We will construct a secondary oper-ation in mod-2 homology and show that this canonical subalgebra is not closedunder it. This allows us to conclude that the 2-primary Brown–Peterson spec-trum does not admit the structure of an E n -algebra for any n ≥
12, answering aquestion of May in the negative.
The following appeared as Problem 1 in J.P. May’s “Problems in infinite loop spacetheory” [May75].
Problem 1.0.1.
For any prime p , does the p -local Brown–Peterson spectrum BP of [BP66]admit the structure of an E ∞ -algebra? Our goal in this paper is to address this question when p =
2. We will construct asecondary operation in the homology of E ∞ -algebras at the prime 2 and show, with ananalysis that begins with the calculations of Johnson–Noel [JN10], that the homology H ∗ BP cannot admit such a secondary operation. Thus, Problem 1.0.1 has a negative answer at the prime 2. Coherently commutative multiplication structures have a long history in homotopytheory, originating in the study of the cup product. The cup product in the coho-mology of a space X comes from the structure of a differential graded algebra onthe cochains C ∗ ( X ) , and while there are many variants on this algebra structure thatall give rise to the cup product there is no natural cochain-level cup product that isgraded-commutative. Instead, the cup product α ⌣ β and its reverse ± β ⌣ α are chainhomotopic by a natural operation α ⌣ β , called the cup-1 product. The cup-1 product ∗ The author was partially supported by NSF grant 1610408.
1s not graded-commutative either, but differs from its reverse by an operation calledthe cup-2 product, and these constructions extend out both to arbitrarily high “coher-ences” (giving cup- i products for all i ) and to operations accepting arbitrarily manyinputs (giving a more complicated set of operations of several variables discussed in[MS04]). The result is called an E ∞ -algebra structure and Steenrod’s reduced poweroperations in the cohomology of spaces are built from it [Ste62].Since then, these coherently commutative multiplications have been recognized inmany other areas: iterated loop spaces, monoidal structures on categories, structuresin mathematical physics related to string theory, and multiplications in cohomologytheories. By contrast with algebra, where commutativity is simply a property of a ring,coherent multiplications come in a hierarchy: there are E -algebra structures that cor-respond to associative products and there are E ∞ -algebra structures that correspondto commutative products, but there are also E n -algebra structures for 1 < n < ∞ thatinterpolate between these concepts.When we switch from ordinary cohomology to generalized cohomology theories,chain complexes become replaced with spectra . A ring spectrum R is a representingobject for a cohomology theory R ∗ (−) so that cohomology with coefficients in R natu-rally takes values in rings. This was refined to the concept of an E n -algebra structureon the spectrum R in [BMMS86, I.4], and these more refined algebras have come tooccupy a central role because E n -algebra structures produce concrete tools that arenot available to an ordinary ring spectrum [Man12, Lur17].• An E -algebra R can be given categories of left R -modules and right R -modules ,whose homotopy categories are triangulated categories. These enjoy severalforms of compatibility as R varies, extend to categories of bimodules, and haverelative smash products ∧ R with properties much like the tensor product.• The category of left modules over an E -algebra R is canonically equivalentto the category of right modules, and the smash product ∧ R makes the cate-gory of left R -modules into a monoidal category. The homotopy category ofleft R -modules has the structure of a (neither symmetric nor braided) tensortriangulated category.• The homotopy category of left modules over an E -algebra has the structure ofa braided monoidal category.• The homotopy category of left modules over an E -algebra has the structure ofa symmetric monoidal category.• The category of modules over an E ∞ -algebra R has homotopy-theoretic ver-sions of symmetric power operations, making it possible to discuss a relativeversion of the above: we can define E n R -algebras which satisfy all of the aboveproperties.• An E ∞ -algebra R has, for any principal Σ n -bundle P → B , natural geomet-ric power operations R ( X ) → R ( P × Σ n X n ) and R ( X ) → R ( B × X ) in R -cohomology that enhance the multiplicative structure. When P = Σ n theserecover the operation that sends a class to its n th power.2any examples of E ∞ -algebras exist. Commutative rings A produce E ∞ -algebras HA via the Eilenberg–Mac Lane construction; the spectra KO and KU , representing realand complex K -theory, have E ∞ -algebra structures whose origin is the tensor productof vector bundles; bordism spectra like MO , MSO , MU , and the like have E ∞ -algebrastructures whose origin is the product structure on manifolds; if Y is an infinite loopspace, then there is a spherical group algebra S [ Y ] = Σ ∞ + Y with an E ∞ -algebra struc-ture; and if R is an E ∞ -algebra and X is a space, there is a spectrum R X (playing the roleof “cochains on X with coefficients in R ”) with an E ∞ -algebra structure that combinesthe multiplication on R with the diagonal map X → X × X .Problem 1.0.1 dates back to the first systematic studies of E ∞ -algebras. Understand-ing why this result is so desirable requires knowing a little about what the Brown–Peterson spectrum is and how important it is in stable homotopy theory.The complex bordism spectrum MU has an E ∞ -algebra structure and it is cen-tral to Quillen’s relation between stable homotopy theory and formal group laws[Qui69], which initiated the subject of chromatic homotopy theory. However, whilealmost the entirety of chromatic theory is possible to phrase in terms of MU , the p -localization MU ( p ) decomposes into summands equivalent to this irreducible Brown–Peterson spectrum BP . The Brown–Peterson spectrum has simpler cohomology andhomotopy groups than MU and has canonical descriptions that are internal to the sta-ble homotopy category [Pri80]. The Brown–Peterson spectrum also exhibits the closeconnection between p -local stable homotopy theory and the theory of formal grouplaws, but with the added benefit that nearly every deep structural property of chro-matic homotopy theory or formal group law theory is made more concise and moreconceptually accessible through the eyes of BP -theory (see, for example, [Rav86] forextensive applications).The existence of an E ∞ -algebra structure on BP would be useful in several ways.• The Adams–Novikov spectral sequence is a method for calculating the set of ho-motopy classes of maps between two spectra X and Y and can be derived fromeither their MU -homology or BP -homology. The computational tools using MU -theory (such as the cobar complex) are well behaved with respect to the ge-ometric power operations discussed earlier, which appear in places such as theconstruction of manifolds of Kervaire invariant one [Bru01]. If BP had an E ∞ -algebra structure then computations of these geometric power operations using MU -theory could instead be related to simpler computations in BP -theory.• Such a structure would allow more concise constructions of many important ob-jects in chromatic theory, such as the Morava K -theories K ( n ) and the truncatedBrown–Peterson spectra BP h n i , as BP -algebras rather than as MU -algebras.• These algebra structures would mean that several computations with these ringspectra could be governed by the computations for BP -theory, such as compu-tations of topological Hochschild homology and topological cyclic homologythat are important to current work in algebraic K -theory [AR02]. These canalso be extended by relative computations in BP -modules, which are much sim-pler than the relative calculations in MU -modules.3 Perhaps most importantly, the Brown–Peterson spectrum is one of the mostprominent examples of an important homology theory where our knowledgeof geometric interpretations (e.g. via Baas–Sullivan theory [Baa73]) lags farbehind our algebraic knowledge. Many of the prominent examples of E ∞ -algebras, such as K -theory and bordism theory, originate in cohomology the-ories with geometric cycles or cocycles that have a product. The existence ofan E ∞ -algebra structure on BP would be a good indicator that a strong geomet-ric interpretation existed.This problem has generated a great deal of interesting research. The existence ofmultiplication structures in the homotopy category has a long history (for example,see the introduction of [Str99]). Several forms of obstruction theory have been devel-oped which showed that many spectra constructed by Baas–Sullivan theory admit E -algebra structures [Rob89, Laz01, BJ02, Ang08]. More sophisticated obstruction the-ory has appeared for E ∞ -algebras [Rob03, GH04], and Richter obtained lower boundson the amount of commutativity present in BP based on Robinson’s obstruction the-ory [Ric06]. Techniques such as localization and idempotent splitting were developedin [May01] to handle additive and multiplicative versions of the construction of BP .More recently Basterra–Mandell showed that BP is a split summand of MU ( p ) as an E -algebra [BM13], and so the homotopy category of BP -modules has a symmetricmonoidal structure; Chadwick–Mandell used idempotent splittings to show that thiscould be done with the Quillen idempotent as E -algebras [CM15]. Both Hu–Kriz–May [HKM01] and Baker [Bak14] gave iterative constructions by methods that killtorsion, producing two different types of closest possible torsion-free E ∞ -algebra to BP . An unpublished paper of Kriz attempted to prove that BP admits an E ∞ -algebrastructure, and Basterra developed the theory of topological André-Quillen (TAQ) co-homology based on his ideas—this theory allows the construction of E ∞ -algebras bysystematically lifting the k -invariants in the Postnikov tower from ordinary coho-mology to TAQ-cohomology [Kri95, Bas99]. Kriz’s original program foundered on atechnical detail, but TAQ has been central in a great deal of research since. However, the hope that Problem 1.0.1 has a positive solution perhaps originatedin a time of much greater optimism, and the intervening years have shown that theadditive and multiplicative structure of a spectrum are difficult to untangle from eachother. Indeed, there is something closer to a reciprocity relationship, where require-ments of the additive structure are rewarded with constraints on the multiplicativeand vice versa. In line with this, there have been several more recent calculationsshowing that desirable properties of a multiplication on BP cannot be realized. Hu–Kriz–May showed that there cannot be a map of E ∞ -algebras BP → MU ( p ) becauseit conflicts with calculation of Dyer–Lashof operations in their homology, despite thepresence of the Quillen idempotent which describes such a splitting additively andalgebraically [HKM01]. In the reverse direction, Johnson–Noel showed with hard cal-culations that the particular map of ring spectra MU ( p ) → BP employed to great effect From [May75]: “The point here is that the notion of an E ∞ ring spectrum seems not to be a purelyhomotopical one; good concrete geometric models are required, and no such model is known for BP .” The problem, insofar as the author understands, was establish certain elements in the Miller spectralsequence computing TAQ-cohomology needed to be shown to be permanent cycles, but the operationsused to establish this were insufficiently compatible with the differentials.
4n chromatic theory cannot be a map of E ∞ -algebras for p ≤
13 [JN10], based on apower operation criterion due to McClure [BMMS86, VIII.7.7, 7.8].The Hu–Kriz–May result seems more decisive mainly because it uses structurethat is forced. The mod- p homology groups H ∗ BP are identified as a canonical sub-algebra of the dual Steenrod algebra H ∗ H F p , and this means that the ring structureon H ∗ BP and operations coming from any E n -algebra structure (including the Dyer–Lashof operations mentioned above) are completely determined by those in the dualSteenrod algebra. It is straightforward to show that H ∗ BP is closed in H ∗ H F p un-der the Dyer–Lashof operations, and so we cannot exclude the possibility that BP isan E ∞ -algebra using a relation between these primary operations. This paper showsthat, at the prime 2, there does exist a contradiction for a more subtle reason: while H ∗ BP is closed under primary operations, it is not closed under secondary operations.This parallels Adams’ solution of the Hopf invariant one problem using secondarycohomology operations [Ada66]. The proof will critically rely on Johnson–Noel’s cal-culation of power operations in complex bordism. Theorem 1.1.1 (5.4.2, 5.4.5) . There exists a natural secondary operation in the mod- homology of E -algebras with the following properties. For R an E -algebra, thesecondary operation is defined on the subset of H R satisfying certain identities betweenDyer–Lashof operations in H k R for ≤ k ≤ , and the secondary operation takes valuesin a quotient of H R . This operation is preserved by maps R → R ′ of E -algebras.In the dual Steenrod algebra H ∗ H F (cid:27) F [ ξ , ξ , . . . ] , this operation is defined onthe element ξ ∈ H ( H F ) and, mod decomposables, unambiguously takes the value ξ ∈ H ( H F ) . With this theorem, we can exclude the existence of an E ∞ -algebra structure onthe 2-local Brown–Peterson spectrum and several related objects (e.g. the generalized BP h k i whose cohomology is discussed in [LN14, 4.3] and whose additive uniquenessis discussed in [AL17]). Theorem 1.1.2 (5.4.6, 5.5.4) . Suppose that R is a connective E -algebra with a ringhomomorphism π R → F such that the induced map on mod- homology H ∗ R → H ∗ H F is injective in degrees through . If ξ is in the image of H ( R ) , then theelement ξ is in the image of H R mod decomposables.In particular, the -local Brown–Peterson spectrum BP , the (generalized) truncatedBrown–Peterson spectra BP h k i for k ≥ , and their -adic completions do not admit thestructure of E n -algebras for any ≤ n ≤ ∞ . The secondary operation we will define is determined by a relation between Dyer–Lashof operations (the full relation is rather large, and is displayed in Proposition 5.4.1).For us, this relation is not obvious; it is not obvious that this particular relation is rele-vant; and it is not obvious that the resulting secondary operation is calculable. We didnot find this relation by trial and error—or, more accurately, we tried to find relevantsecondary operations by trial and error and failed. All of our preliminary attemptsresulted in combinations that were excluded by necessity of compatibility with the5teenrod operations. In this section we will indicate a little about how the main resultof this paper was found, as opposed to how it is written. The obstruction theory of Goerss–Hopkins [GH] takes as input a simplicial op-erad, an appropriate homology theory E ∗ , and an algebra A for this simplicial operadin E ∗ E -comodules. From this, it produces an obstruction theory to calculate the mod-uli space of algebras over the geometric realization of this operad whose E -homologyis A . Senger specialized this to the case where E is mod- p homology and the operadis a constant E ∞ -operad [Sen]. His work produced an obstruction theory whose in-put is an algebra A with Steenrod operations and Dyer–Lashof operations satisfyinginstability relations and Nishida relations, whose obstruction groups are Ext-groupsin this category, and which calculated the moduli space of E ∞ -algebras whose mod- p homology is A . He also developed several tools for reducing these calculations tomore tractable Ext-groups that could, in the case of BP or BP h n i , be calculated witha Koszul resolution [Pri70]. By construction, this obstruction theory remembers thatthe Nishida relations will exclude a number of possible obstructions. (The problemencountered in Kriz’s preprint [Kri95] could be viewed as the accidental exclusion oftoo many obstructions in this fashion.)In the case of the 2-primary Brown–Peterson spectrum, calculations with this ob-struction theory indicated two first potential nonzero obstruction classes. We can de-fine y , R n , and v m to be, respectively, Ext-Koszul dual to the generator ξ ∈ H ( BP ) , theDyer–Lashof operation Q n − , and the Milnor primitive Q m − in the Steenrod algebra.(The R n are closely related to unpublished work of Basterra–Mandell on operations inTAQ-cohomology.) Then, using this notation, the first possible obstruction classes are v R R y and v R R y . Under the yoga of secondary operations described by Adams[Ada66], the potential obstruction class v R R y would detect an obstruction from asecondary operation whose value involved ξ (detected by the Milnor primitive), com-bining relations that (at least) involved the Adem relations for Q Q and an identitysatisfied by Q ξ .Indeed, our main result is that this is the case. However, much of the progressin this paper traces its origin back to the actual calculation of these relations. Afterdetermining the needed identities in Proposition 5.4.1, we could identify most of therelations in H ∗ BP ⊂ H ∗ H F as already holding true in H ∗ MU , making it possible tobegin juggling this secondary operation through much simpler ones passing through H ∗ MU . In the original version of this paper, we expressed the following strong belief thatthe 2-primary Brown–Peterson spectrum was not unique in failing to admit an E ∞ -algebra structure. Conjecture 1.3.1.
For any odd prime p , the p -local Brown–Peterson spectrum does notadmit the structure of an E ∞ -algebra. Senger has already extended the methods of this paper to prove Conjecture 1.3.1, A more detailed explanation of this calculation is now in [Law17]. BP (and BP h n i for n ≥
4) do not admit the structure of E ( p + ) -algebrasat any prime p [Sen17].Our keystone computation in this paper is a Dyer–Lashof operation in a versionof the 2-primary dual Steenrod algebra for MU -modules. Problem 1.3.2.
Determine how the Dyer–Lashof operations act on the p -primary MU -dual Steenrod algebras π ∗ ( H F p ∧ MU H F p ) . Baker has shown in [Bak15] how to derive the Nishida relations, describing theinteraction between cohomology operations and Dyer–Lashof operations, from theDyer–Lashof operations in the ordinary dual Steenrod algebra. This suggests that asolution to the previous problem would give additional constraints on MU -algebrasby describing additional relations that have to hold in their mod- p homology relativeto MU . Problem 1.3.3.
Determine analogues of Nishida relations between the homology oper-ations on H MU ∗ R = π ∗ ( H F p ∧ MU R ) and the Dyer–Lashof operations. In particular, Remark 4.4.7 describes how the Dyer–Lashof operation that we havecalculated seem to place a cap on multiplicative structure for the map MU → BP at theprime 2—a stronger cap than the one we have shown for the amount of multiplicativestructure on BP . Problem 1.3.4.
Find constraints on the values of n for which the p -local Brown–Petersonspectrum can admit the structure of an E n MU -algebra. Again, in the time since we raised this question, Senger has shown that BP doesnot admit the structure of an E p + MU -algebra at any prime p [Sen17].The calculations of this paper deduce our unexpected Dyer–Lashof operation inthe MU -dual Steenrod algebra from a multiplicative Dyer–Lashof operation in theHopf ring for MU , and an induced operation in the homology of the space SL ( MU ) ⊂ GL ( MU ) of strict units. This is a first step towards determining the homology ofthe spectrum дl ( MU ) , about which very little is known, using the Miller spectralsequence [Mil78]. Problem 1.3.5.
Determine multiplicative Dyer–Lashof operations in the Hopf ring for MU and in the homology of GL ( MU ) . Determine homology groups of the unit spectrum дl ( MU ) and the Picard spectrum pic ( MU ) , as well as information about their homotopytypes. Remark 2.1.9 points out that our description of secondary operations and Todabrackets is not optimal. For example, it sometimes requires strict basepoints for map-ping spaces, strict unitality, and strict initial and terminal objects, all of which are notinvariant under unbased homotopy equivalences between objects and not invariantunder Dwyer–Kan equivalences between topological categories. However, the toolsshould apply in much wider generality; investigations in this direction have been car-ried out by Bhattacharya and Hank.
Problem 1.3.6.
Develop a homotopical framework for secondary operations. ∞ -category, extending Lurie’s version ofthe Dold–Kan correspondence [Lur17, 1.2.4].Our calculations with power operations in the Hopf ring make use of the H ∞ -algebra structure on MU , a concept from [BMMS86] that has been largely neglectedin the modern literature. It should be possible to describe a fully coherent version ofthis structure using the language of Picard spaces and Picard spectra [MS16]. Problem 1.3.7.
Give a systematic development of E d ∞ -algebras as homotopy coherentversions of H d ∞ -ring spectra, and show that the H d ∞ -structures on classical Thom spectraconstructed in [BMMS86, VIII.5.1] lift to E d ∞ -algebra structures. An E d ∞ -structure on an E ∞ -algebra R should be a lift of the map of spaces d Z ⊂ Z ⊂ Pic ( S ) → Pic ( R ) to a map of E ∞ -spaces, corresponding to a functor of symmetric monoidal ∞ -categories.In close analogy with the work of Ando–Blumberg–Gepner–Hopkins–Rezk [ABG + { dk } ֒ → Pic ( S ) representing S dk gives rise (via the E ∞ -space structure) to adiagram B Σ m → Pic ( S ) , whose homotopy colimit is a Thom spectrum on B Σ m repre-senting the extended power construction on S dk . An E d ∞ -structure on R should thenmake the resulting diagram B Σ m → Pic ( R ) factor through a constant diagram withvalue S dkm , allowing us to conclude that the smash product of R with the Thom spec-trum has an equivalence to R ∧( B Σ m ) + ∧ S dkm . We will begin by calculating a Dyer–Lashof operation in the MU -dual Steenrod alge-bra π ∗ ( H ∧ MU H ) , where H is the Eilenberg–Mac Lane spectrum H F . This has mapsin from the dual Steenrod algebra π ∗ ( H ∧ H ) and out to the homology of SU whichbecome a left exact sequence0 → Qπ ∗ ( H ∧ H ) → Qπ ∗ (cid:18) H ∧ MU H (cid:19) → QH ∗ SU on indecomposables. The Dyer–Lashof operations on the left are known by work ofSteinberger, and were employed by Tilson [Til16] to calculate operations in the middleterm. The Dyer–Lashof operations on the right are known by work of Kochman. Theoperation we will calculate is the first possible hidden extension and it turns out tobe nontrivial.To carry out this calculation we rely on calculations of unstable multiplicativeDyer–Lashof operations. This uses Ravenel–Wilson’s description of Hopf ring struc-ture on the homology of the spaces in the Ω -spectrum for MU [RW74] and a compar-ison between Dyer–Lashof operations and the tom Dieck–Quillen power operations8n MU -cohomology. The relevant portion of this extension is ultimately determinedby the calculation of Johnson–Noel discussed earlier [JN10]. We will make exten-sive use of the results of Bruner–May–McClure–Steinberger in doing this calculation[BMMS86].We will then give an alternative description of this operation in the MU -dualSteenrod algebra as a Dyer–Lashof operation applied to the result of a secondary oper-ation. This allows us to use juggling formulas for secondary operations to determinea more complicated secondary operation in the dual Steenrod algebra, showing that H ∗ BP is not closed under secondary operations. (We are fortunate in this regard thatmost of our calculations can be carried out mod decomposable elements.) In order towork with this we will describe a framework for secondary operations in Section 2based on Harper’s book [Har02], with our emphasis shifted from suspension and loopoperators to loops inside mapping spaces. The notation Map always denotes a space, or simplicial set, of maps. We will referto a diagram as homotopy commutative if it commutes in the homotopy category, and homotopy coherent if we have further chosen compatible homotopies and higher ho-motopies to recover a coherent diagram [Vog73, Lur09].We will adhere to the standard conventions for function composition and pathcomposition, even though they make no sense. Maps in a category are written usingarrows X → Y , and given f : X → Y and д : Y → Z there is a composite д f . Pathsin a space are written using arrows p ⇒ q , and given h : p ⇒ q and k : q ⇒ r thereis a path composite h · k .Throughout this paper, we will write H ∗ ( X ; R ) for the homology groups of X withcoefficients in R , and similarly for cohomology. If R is not specified, we view these asbeing taken with coefficients in a fixed finite field F p of prime order. Homology andcohomology groups of spaces are unreduced unless otherwise specified.When X is a spectrum, π n ( X ) always denotes the set of maps S n → X in the stablehomotopy category.We will let MU be the complex cobordism spectrum and F be the formal grouplaw of MU , writing it as F ( x , y ) = x + F y = Í a i , j x i y j with a i , j ∈ π ( i + j − ) MU . We are in the position that we require tools from both classical and modern frame-works.In Section 2, we will require highly structured categories of algebras, well-behavedadjunctions between them, relative smash products, and the like. To our knowledge,the only literature that accommodate our needs for E n -algebras is due to Elmendorff–Mandell [EM06], which works in the category of symmetric spectra of with the posi-tive stable model structure [HSS00, MMSS01]. We will use the term commutative ringspectrum for a commutative monoid in symmetric spectra, and the term E n -algebra for an algebra over a fixed E n -operad in simplicial sets—for this it is convenient touse the E ∞ -operad of Barratt–Eccles [BE74] with its filtration by E n -suboperads due9o Berger [Ber97]. In this framework, Elmendorf–Mandell show that each category of E n -algebras is a simplicial model category. For m ≤ n , the forgetful functors from E n -algebras to E m -algebras or to symmetric spectra are right Quillen functors, and thereis a Quillen equivalence between E ∞ -algebras and commutative ring spectra [EM06,1.3, 1.4].In Section 3 and beyond, where we are calculating with MU and the dual Steen-rod algebra, we require classical results: particularly results of Cohen–Lada–May[CLM76], May–Quinn–Ray [May77], Bruner–May–McClure–Steinberger [BMMS86],and Ravenel–Wilson [RW74]. All of these results rest on the interaction betweena (possibly highly structured) ring spectrum E and the spaces E n in an Ω -spectrumrepresenting it, an item not immediately available in the positive stable model struc-ture. Most of these references use more classical categories of spectra, such as thosefrom [LMSM86]. In particular, comparisons are easiest to draw to the S -modules of[EKMM97], and these all have homotopically equivalent notions of commutative ringspectra as shown in [MMSS01]. This gives us a path to show that operations and rela-tions between them that we construct in Section 2 can be related to our calculations.(We do not mean to assert that the constructions in Section 2 cannot be carried outwithin S -modules. To our knowledge, ours is the shortest path without the hard workinvolved in creating an equivalent of [EM06].) The author has benefited from discussions and perspectives provided by many peoplein the course of developing this paper. The author would particularly like to thank An-drew Baker, Clark Barwick, David Benson, Andrew Blumberg, Robert Bruner, John R.Harper, Fabien Hebestreit, Mike Hill, Paul Goerss, Weinan Lin, Michael Mandell, Pe-ter May, Haynes Miller, Ulrich Pennig, Charles Rezk, Andrew Senger, Neil Strickland,Markus Szymik, Sean Tilson, Craig Westerland, Dylan Wilson, and Steven Wilson fortheir assistance. The anonymous referee also provided a great deal of useful feedbackon this paper.
A secondary composite is the first basic type of obstruction encountered when liftinga homotopy commutative diagram to a homotopy coherent diagram.
Definition 2.0.1.
Let D be a category enriched in spaces. Suppose that we are giventhe following data:1. a sequence ( X , X , X , X ) of objects of D ,2. maps f ij : X i → X j for i < j , and3. paths h ijk : f jk f ij ⇒ f ik in Map D ( X i , X k ) for i < j < k .Then the associated secondary composite is the element of π ( Map D ( X , X ) , f ) rep-resented by the path composite ( h ) − · ( f h ) − · ( h f ) · h , f . f h − + f f f h − (cid:11) (cid:19) f f h K S f f f h f k s (In Remark 2.1.9 we will discuss a quasicategorical expression of this data.)In the following sections we will describe secondary composites which are com-parable with Massey products or Toda brackets; they rely on the existence of dis-tinguished “null” maps so that we can make sense of composites being trivial. Ourperspective is based on Harper’s book [Har02]. Throughout this section, let C be a category enriched in pointed spaces (or, with ap-propriate modifications, pointed simplicial sets) under ∧ , and write Map C ( x , y ) forthe mapping space between any pair of objects of C . We refer to the basepoint of thismapping space as the null map or ∗ ; null maps satisfy f ∗ = ∗ f = ∗ for any f . Definition 2.1.1.
Suppose we have maps X f −→ X д −→ X in C . A tethering of this composite is a homotopy class of nullhomotopy of д f : ahomotopy class of path h : д f ⇒ ∗ in Map C ( X , X ) (cf. [Har02, 4.1.2]). We will write д h ! f to indicate such a tethering, and д ! f to indicate that there is a chosentethering which is either implicit or not important to name. Remark . If a triple composite kд f is nullhomotopic, then a tethering kд h ! f isthe same data as a tethering k h ! д f . Definition 2.1.3.
Suppose we have maps X f −−→ X f −−→ X f −−→ X , and tetherings f h ! f h ! f . Then we define the element h f h ! f h ! f i ∈ π ( Map C ( X , X ) , ∗) to be the path composite ( f h ) − · h f obtained by gluing together the twonullhomotopies f f f ⇒ ∗ . This is the secondary composite, as in Definition 2.0.1,obtained by choosing f = f = f = ∗ and the trivial nullhomotopies h and h . Strictly speaking, the smash product on pointed spaces is nonassociative and so does not give rise to amonoidal category [MS, §1.7]. We really mean that we are working in an appropriate “convenient category,”such as compactly generated spaces. efinition 2.1.4. Suppose we have maps X f −−→ X f −−→ X f −−→ X . If we have chosen a tethering f h ! f and f f is nullhomotopic, we write h f h ! f , f i ⊂ π ( Map C ( X , X ) , ∗) for the set of all elements h f h ! f k ! f i as k ranges over possible tetherings, andrefer to h f h ! f , −i as the secondary operation determined by the tethering. The setof maps f such that f f is nullhomotopic is referred to as the domain of definition of this secondary operation, and the possibly multivalued nature of this function asthe indeterminacy of the secondary operation.The secondary operations h− , f ! f i are defined in the same way. Definition 2.1.5.
Suppose we have maps X f −−→ X f −−→ X f −−→ X such that the double composites f f and f f are nullhomotopic. We define thesubset h f , f , f i ⊂ π ( Map C ( X , X ) , ∗) , or bracket , to be the set of all secondary composites h f ! f ! f i . Proposition 2.1.6.
Changing the tethering and homotopy class of maps alters the valueof a secondary composite by multiplication by loops, as follows. If f is homotopic to f ′ ,we have h f h ! f k ! f i = h f ′ h ′ ! f k ! f i · ( u f ) for some u ∈ π Map C ( X , X ) that is determined by h , h ′ , and a homotopy between f and f ′ . Similarly, if f is homotopic to f ′ , we have h f h ! f k ! f i = ( f v ) · h f h ! f k ′ ! f ′ i for some v ∈ π Map C ( X , X ) .If we replace all three maps with homotopic maps and choose new tetherings, we have h f h ! f k ! f i = ( f v ) · h f ′ h ′ ! f ′ k ′ ! f ′ i · ( u f ) for some u ∈ π Map C ( X , X ) and v ∈ π Map C ( X , X ) . In particular, this describes completely the indeterminacy in secondary operationsand brackets, and shows that (up to this indeterminacy) a secondary operation or abracket is well-defined on homotopy classes of maps.12 roof.
We will prove the first identification, as the second is symmetric. Since f and f ′ are homotopic, there is a path j : f ⇒ f ′ in Map C ( X , X ) . The composition jk : ∆ × ∆ j × k −−−→ Map C ( X , X ) × Map C ( X , X ) → Map C ( X , X ) determines a homotopy from f k to ( j f f ) · ( f ′ k ) , making them equal in the fun-damental groupoid of Map C ( X , X ) .In this fundamental groupoid, we then have the following sequence of identities: h f h ! f k ! f i = ( f k ) − · ( h f ) = ( f ′ k ) − · ( j f f ) − · ( h f ) = ( f ′ k ) − · ( h ′ f ) · ( h ′ f ) − · ( j f f ) − · ( h f ) = h f ′ h ′ ! f k ! f i · [( j f · h ′ ) − · h ] f Letting u = ( j f · h ′ ) − · h ∈ π Map C ( X , X ) gives the desired result. (cid:3) Corollary 2.1.7.
A secondary operation h f h ! f , −i determines a well-defined map Φ on ker f ⊂ π Map C ( X , X ) whose values are right cosets: ker f Φ −→ ( f π Map C ( X , X )) / π Map C ( X , X ) . If two tetherings h , h ′ give rise to operations Φ , Φ ′ , then there exists an element u ∈ π Map C ( X , X ) such that Φ x = Φ ′ x · ( ux ) for all x ∈ ker f ⊂ π Map C ( X , X ) .Dual results hold for h− , f ! f i . Corollary 2.1.8.
Suppose we have maps X f −−→ X f −−→ X f −−→ X such that the double composites f f and f f are nullhomotopic. Then the bracket h f , f , f i depends only on the homotopy classes of f i , i + and is a well-defined doublecoset in ( f π Map C ( X , X )) / π Map C ( X , X ) . ( π Map C ( X , X ) f ) . Remark . A more flexible version of the above constructions should exist, wherebasepoints are replaced by some appropriate system of maps E i , j → Map C ( X i , X j ) from contractible spaces E i , j , together with appropriate lifts of the composition maps.For example, the category of diagrams of spaces E → X is a monoidal category un-der the pushout-product, and so we could ask for C to be enriched in this categorywith the constraint that the space E is always contractible. We might instead try tofind an appropriate analogue in terms of quasicategories satisfying certain basepoint13onditions: in the notation of [Lur09], the data to describe a secondary composite inDefinition 2.0.1 defines a map of enriched categories C [ ∂ ∆ ] → D , and the secondarycomposite is the obstruction to extending it to a map C [ ∆ ] → D (a homotopy coher-ent triple composite). Both of these constructions would apply more widely, but in-volve more bookkeeping and possibly require a more advanced technical framework.We have elected to use constructions in categories where this will not be necessaryin order to minimize the technical load.The definitions of secondary operations and brackets are preserved in an obviousway under functors between enriched categories. Proposition 2.1.10.
Suppose F : C → C ′ is an enriched functor between categoriesenriched in pointed spaces. Then any tethering д h ! f in C induces a tethering Fд Fh ! F f in C ′ . We have an equality F (h f h ! f k ! f i) = h F f Fh ! F f Fk ! F f i , and we have containments as follows: F (h f h ! f , f i) ⊂ h F f Fh ! F f , F f i F (h f , f k ! f i) ⊂ h F f , F f Fk ! F f i F (h f , f , f i) ⊂ h F f , F f , F f i There is a further extension in the case where we have an enriched adjunction . Anexample of such a result appears below.
Proposition 2.1.11.
Suppose that we have an enriched adjoints F : C → D and G : D → C , encoded by a natural based homeomorphism θ : Map C ( X , GY ) (cid:27) Map C ( FX , Y ) . Given maps X f −−→ X f −−→ X д −→ GY and tetherings д h ! f k ! f , the map θ induces an identity θ ∗ h д h ! f k ! f i = h θд θh ! F f Fk ! F f i . Corollary 2.1.12.
There are containments θ ∗ h д , f k ! f i ⊂ h θд , F f Fk ! F f i and θ ∗ h д , f , f i ⊂ h θд , F f , F f i . .2 Pointings and augmentations In this section we let D be a category enriched in spaces (now assumed to have nobasepoint). In this section we indicate a construction that replaces D with a categoryenriched in pointed spaces. Definition 2.2.1. An augmented object of D is an object X ∈ D equipped with a map Y → ∅ to an initial object of D . The space of maps between two augmented objectsis the subspace of ordinary maps that commute with the augmentations.A pointed object of D is an object Z ∈ D equipped with a map ∗ → Z from aterminal object of D . The space of maps between two pointed objects is the subspaceof ordinary maps that commute with the pointings. Definition 2.2.2.
Suppose D is a category enriched in spaces. We define D ± , thecategory of possibly pointed or augmented objects of D , to be the following categoryenriched in based spaces.An object of D ± is one of three types:1. an augmented object X → ∅ of D ,2. an ordinary object Y of D , or3. a pointed object ∗ → Z of D .The mapping spaces in D ± are given as follows.1. The space of maps between two augmented objects X → ∅ , X ′ → ∅ ′ is thespace of maps of augmented objects, with basepoint given by the composite X → ∅ → X ′ .2. The space of maps between two pointed objects ∗ → Z , ∗ ′ → Z ′ is the space ofmaps of pointed objects, with basepoint given by the composite Z → ∗ ′ → Z ′ .3. The space of maps between two ordinary objects Y , Y ′ is the based space Map D ( Y , Y ′ ) + ,whose disjoint basepoint is called the formal null map .4. The space of maps from an augmented object X → ∅ to an ordinary object Y isthe space of maps X → Y , with basepoint given by the map X → ∅ → Y .5. The space of maps from an ordinary object Y to a pointed object ∗ → Z is thespace of maps Y → Z , with basepoint given by the map Y → ∗ → Z .6. The space of maps from an augmented object X → ∅ to a pointed object ∗ → Z is the space of maps X → Z , with basepoint given by the canonical mapfactoring through either ∅ or ∗ in the commutative diagram X / / (cid:15) (cid:15) ∅ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ (cid:15) (cid:15) ∗ / / Z .
15. All other mapping spaces are one-point spaces—there are no non-basepointmaps from ordinary objects to augmented ones, or from pointed objects to or-dinary ones. We also refer to these as formal null maps .We have full subcategories of D ± spanned by fewer than all three of these types ofobjects: for example, we have the categories of augmented objects , pointed objects , possibly augmented objects , and possibly pointed objects of D . Proposition 2.2.3.
The category D ± is enriched in pointed spaces under ∧ .In D ± , if a composite X → Y → Z is nullhomotopic then X is augmented, Z ispointed, or one of the maps is a formal null map (in which case there is a canonicaltethering). This construction makes it possible to take a category D and sensibly talk aboutsecondary operations and brackets for a composite X → X → X → X in D if thefirst map is a map of augmented objects, if the last map is a map of pointed objects, orif the first object is augmented and the last object is pointed. (If the maps arise from D then a formal null map cannot appear.) Example . If C has homotopy pushouts and we have augmented objects X → X → ∅ , the bracket can be identified with an element in π Map C ( Σ X , X ) , repre-sented by the outside rectangle in the homotopy coherent diagram X / / (cid:15) (cid:15) X / / (cid:15) (cid:15) ∅ (cid:15) (cid:15) ∅ / / ⇒ X / / ⇒ X . The indeterminacy in the bracket is given by path concatenation with composites ofeither of the following forms: Σ X v / / X f / / X Σ X Σ f / / Σ X u / / X Dual results hold if we are given pointed objects ∗ → X → X , so that the bracketcan be identified with an element in π Map C ( X , Ω X ) . To avoid grief in these iden-tifications, especially with respect to a loop-suspension adjunction, it is important topay attention to the orientation of S as detailed at length in [Har02]. This is why wehave indicated directions for 2-cells.In the “mixed” case, there is little profound that we can say other than identifica-tion of a element in the bracket with the loop determined by a homotopy coherentdiagram X / / (cid:15) (cid:15) X / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∅ / / ⇒ X / / ⇒ X . .3 Juggling and Peterson–Stein formulas In this section we return to assuming that we have a category C enriched in basedspaces.There are several “juggling” formulas that describe the relationship between brack-ets and function composition. All of them are obtained by choosing representativenullhomotopies and composing them appropriately, as in the Peterson–Stein formu-las [PS59]. Lemma 2.3.1.
Suppose we have a sequence of objects ( X , . . . , X ) , together with maps f i , i + : X i → X i + and tetherings f ! f ! f ! f . Then there is an identity f h f ! f ! f i − = h f ! f ! f i f in π Map C ( X , X ) .Example . In the case where X → X → X are maps of pointed objects in acategory D , this Peterson–Stein relation expresses that both loops in Map D ± ( X , X ) are homotopic to the loop determined by the following homotopy coherent diagram: X / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ❆❆❆❆❆❆❆❆❆ ❆❆❆❆❆❆❆❆❆ X (cid:15) (cid:15) / / ⇒ X / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∗ ⇒ / / X / / ⇒ X Similarly, in the mixed case we will need to derive Peterson–Stein relations from dia-grams such as the following: X / / (cid:15) (cid:15) ∅ (cid:15) (cid:15) ❆❆❆❆❆❆❆❆❆ X (cid:15) (cid:15) / / ⇒ X / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) ∅ ⇒ / / X / / ⇒ X Lemma 2.3.3.
Each of the following juggling formulas holds whenever defined. f h f h ! f k ! f i = h f f f h ! f k ! f ih f f h ! f k ! f i = h f h ! f f f k ! f ih f hf ! f f k ! f i = h f h ! f k ! f f ih f h ! f kf ! f f i = h f h ! f k ! f i f
17s we range over possible choices of tethering, these lemmas expressing equalityof secondary composites become containment relations for secondary operations andbrackets.
Proposition 2.3.4.
Each of the following juggling formulas for secondary operationsholds whenever both sides are defined: h f ! f , f i f = f h f , f , f i − f h f h ! f , f i = h f f f h ! f , f ih f f h ! f , f i ⊂ h f h ! f f , f ih f hf ! f f , f i = h f h ! f , f f ih f h ! f , f f i ⊃ h f h ! f , f i f Dual results hold for secondary operations h− , f ! f i .Proof. We will give the argument for the first statement, as the others are similar butless complex. Given fixed tetherings f h ! f k ! f ℓ ! f , we find that the left-handside consists of elements of the following form: h f h ! f k ′ ! f i f = (cid:20) ( f v ) · h f h ! f k ! f i (cid:21) f = ( f v f ) · h f h ! f k ! f i f The right-hand side consists of elements of the following form: f h f k ′ ! f ℓ ′ ! f i − = f (cid:20) ( f w ) · h f h ! f k ! f i · ( u f ) (cid:21) − = f (cid:20) ( u − f ) · h f h ! f k ! f i − · ( f w − ) (cid:21) = ( f u − f ) · f h f h ! f k ! f i − · ( f f w − ) However, f f w − is always trivial because f f is nullhomotopic, and so the two setscoincide by Lemma 2.3.1. (cid:3) Proposition 2.3.5.
Each of the following juggling formulas for brackets holds wheneverboth sides are defined: h f , f , f i f = f h f , f , f i − f h f , f , f i ⊂ h f f , f , f ih f f , f , f i ⊂ h f , f f , f ih f , f f , f i ⊃ h f , f , f f ih f , f , f f i ⊃ h f , f , f i f
18e end with a remark on adjunctions. In the presence of an (enriched) adjunctionbetween categories C and D , we can describe relationships between secondary oper-ations. Recall that an enriched functor F : C → D with enriched left adjoint G deter-mines (and is determined by) an enriched category E with object set Ob (C) ∪ Ob (D) ,such that: Map E ( x , y ) = Map C ( x , y ) if x , y ∈ C Map D ( x , y ) if x , y ∈ D Map C ( x , Gy ) (cid:27) Map D ( Fx , y ) if x ∈ C , y ∈ D∅ otherwiseThis allows us to describe augmented and pointed objects in the presence of an ad-junction and define brackets even amongst objects in categories related by adjunc-tions. We could, if desired, rephrase several of our constructions in these terms, inparticular with respect to brackets that involve maps out of free objects. In prominent examples, some of the mapping spaces in C have natural “addition”structures. Definition 2.4.1.
An object Y ∈ C is an H-object if Map C (− , Y ) naturally takes valuesin H -spaces: it is equipped with a natural homotopy-unital binary operation + whoseunit is the basepoint. A map of H-objects is a map Y → Y ′ preserving this structure.An object X ∈ C is an co-H-object if Map C ( X , −) naturally takes values in H -spaces:it is equipped with a natural homotopy-unital binary operation + whose unit is thebasepoint. A map of co-H-objects is a map X → X ′ preserving this structure. Proposition 2.4.2.
Suppose X is a co-H-object in C and that we have maps f , f ′ : X → Y and д : Y → Z , together with tetherings д h ! f and д h ′ ! f ′ . Then the pointwiseproduct on paths in Map C ( X , −) gives a tethering д h + h ′ ! ( f + f ′ ) . Proposition 2.4.3.
Each of the following addition formulas holds whenever both sidesare defined and the source object is an co-H-object in C : h f h ! f k + k ′ ! ( f + f ′ )i = h f h ! f k ! f i + h f h ! f k ′ ! f ′ ih f h ! f , ( f + f ′ )i = h f h ! f , f i + h f h ! f , f ′ ih f , f , ( f + f ′ )i ⊂ h f , f , f i + h f , f , f ′ i Dual results hold for H -objects. Here the addition on paths is the pointwise H -space structure. The addition on π Map C ( X , X ) is, by the Eckmann–Hilton argument, equivalent to either path con-catenation or the pointwise H -space structure on paths, and makes this group abelian.19 roof. The first identity is expressed by the following interaction between path com-position and the pointwise H -space structure: [ f ( k + k ′ )] − · [ h ( f + f ′ )] = [( f k ) − + ( f k ′ ) − ] · [ h f + h f ′ ] = [( f k ) − · h f ] + [( f k ′ ) − · h f ′ ] Letting k and k ′ vary over possible tetherings, this then shows that h f h ! f , ( f + f ′ )i ⊃ h f h ! f , f i + h f h ! f , f ′ i . The indeterminacy on the left-hand side consists precisely of adding elements of theform f u , while on the right-hand side it consists of adding elements of the form f v + f v ′ = f ( v + v ′ ) . Because the indeterminacy group is the same, this containmentmust be an equality of cosets.Now letting h vary over possible tetherings (which produces a restricted set ofelements on the right-hand side), we obtain the third identity. (cid:3) Working in a model category often requires attention to objects that are not cofibrantor fibrant, and function spaces for such objects are poorly behaved. In this sectionwe will spell out adjustments to the construction of secondary operations which aremore convenient but equivalent to our standard construction.Let M be a model category. Associated to this data there is a hammock localization L H M [DK80a]. This is a simplicial category with a functor M → L H M , bijective onobjects, that turns weak equivalences into homotopy equivalences. In [DK80b] it isshown that L H M recovers the homotopy theory of M : it is invariant under Quillenequivalence, the homotopy category of L H M is localization of M with respect toweak equivalences, and if M is a simplicial model category there is a chain of weakequivalences between L H M and the simplicial category of cofibrant-fibrant objectsof M .With this in mind, for (possibly pointed or augmented) objects of M it makessense to calculate secondary composites and brackets in either M or L H M . Thereare natural maps π k Map M ( X , Y ) → π k Map M ( X cof , Y f ib ) (cid:27) π k Map L H M ( X , Y ) , where the first is an isomorphism if X is cofibrant and Y is fibrant. This natural mapis compatible with function composition.This means that a tethering, secondary composite, secondary operation, or bracketin M determines a compatible one in L H M . This use of L H M then allows us to dis-cuss brackets, and identities between them, for maps in the homotopy category of M without the inconvenience of using cofibrant or fibrant replacements to obtain mapsin M . When discussing secondary composites in M , we will regard this process asimplicit. 20 .6 Secondary power operations The study of secondary operations can now be specialized to homotopy operationsfor algebras over a fixed commutative ring spectrum A . Definition 2.6.1.
Given a commutative ring spectrum A , we let P E n A be the left adjointto the forgetful functor from E n A -algebras to spectra; if n = ∞ we simply write P A ,and if A = S then we will omit A from the notation.In particular, there is an isomorphism P E n A ( X ) (cid:27) Ü A ∧ (cid:18) E n ( k ) + ∧ Σ k X ∧ k (cid:19) , where the spaces E n ( k ) are the terms in our chosen E n -operad, and the set of homotopyclasses of maps of E n A -algebras P E n A (∨ S k i ) → C is naturally isomorphic to Î π k i C .The natural map X → ∗ becomes a natural augmentation P E n A ( X ) → A , and a pinchmap X → X ∨ X gives P E n A ( X ) the structure of a co-H-object. Definition 2.6.2. A homotopy operation on E n A -algebras is a natural transformationof functors Ö π k i (−) → π j (−) , represented by a homotopy class of map of E n A -algebras P E n A ( S j ) → P E n A (∨ S k i ) or equivalently an element of π j P E n A (∨ S k i ) (cid:27) π j ( A ∧ P E n (∨ S k k )) If this operation preserves the zero element, we view it as determined by a map ofaugmented objects via the canonical projection to A ; if it preserves addition, we viewit as determined by a map of co-H-objects.Similarly, if B is an E n A -algebra, a homotopy operation on E n A -algebras under B is a natural transformation in the homotopy category of E n A -algebras under B ,represented by a homotopy class of map B ∐ P E n A ( S j ) → B ∐ P E n A (∨ S k i ) . Here the coproduct ∐ takes place in the category of E n -algebras. If this operationpreserves the zero element, we view it as determined by a map of augmented objectsvia the canonical projection to B ; if it preserves addition, we view it as determined bya map of co-H-objects.Taking B = A shows that the first type of operations are a special case of thesecond, so there is no loss of generality in restricting our attention to operations inthe relative case. If n = ∞ , then conversely E ∞ A -algebras under B are equivalent to E ∞ B -algebras. 21 xample . For any b ∈ π k ( B ) and any n >
0, multiplication by b determines anadditive homotopy operation on E n A -algebras under B . Remark . As above, the Yoneda lemma allows homotopy operations to be ex-pressed as pre -composition with maps of free algebras. We usually write precomposi-tion on the right, but this is at odds with the standard convention of writing operators(such as the Dyer–Lashof operations) on the left. We could attempt to solve this inmany ways. One would be to work in an opposite category so that function applica-tion is on the right. One would be to notationally distinguish between maps betweenfree algebras (operations), maps from free algebras to ordinary algebras (homotopyelements), and maps between ordinary algebras (maps). One is to accept the state ofaffairs, and resist the urge to use the same names for a Dyer–Lashof operation Q n andthe map P H ( S j + n ) → P H ( S j ) that represents it. None of these solutions are good, butwe have adopted the third because (in all honesty) it has confused us the least.Relations between homotopy operations allow us to define secondary operationsin the following way. Definition 2.6.5.
Let A be a commutative ring spectrum and B an E n A -algebra. Sup-pose we have homotopy operations Q i : Î s π l i , s → π k i and R : Î i π k i → π j thatpreserve zero such that R ◦ ( Î i Q i ) =
0, realized by a homotopy coherent diagram B ∐ P E n A ( S j ) R / / (cid:15) (cid:15) B ∐ P E n A (∨ i S k i ) Q (cid:15) (cid:15) B / / ⇒ B ∐ P E n A (∨ i , s S l i , s ) of augmented E n A -algebras under B . We refer to R as a relation between the opera-tions Q i . The coherence produces a tethering homotopy h , and the secondary operation associated to this relation is h− , Q h ! R i . Proposition 2.6.6.
Given C any E n A -algebra under B , the domain of definition of thesecondary operation h− , Q h ! R i is the subset of Î π l i , s C of collections of elements x i , s ∈ π l i , s C such that Q i ( x i , s ) = for all i . These are represented by homotopy commutativediagrams B ∐ P E n A (∨ i S k i ) Q i (cid:15) (cid:15) / / B (cid:15) (cid:15) B ∐ P E n A (∨ i , s S l i , s ) x i , s / / C of E n A -algebras under B . The value of h− , Q h ! R i is a subset of π j + C , and the indeter-minacy consists of adding elements in the image of the suspended operation σR : Î π k i + C → π j + C . Proposition 2.6.7.
Maps f : C → D of E n A -algebras under B preserve secondaryoperations. roof. This is the statement that f (h x , Q ! R i) ⊂ h f ( x ) , Q ! R i , which is an application of the juggling formulas from Proposition 2.3.4. (cid:3) Remark . If n < m ≤ ∞ , then the forgetful functors from E m A -algebras under B to E n A -algebras under B also preserve secondary operations in the following sense.The forgetful functor U from E m A -algebras under B to E n -algebras under B has a leftadjoint F , giving rise to an enriched adjunction. Since adjoints are preserved undercomposition, it preserves free objects: F (cid:16) B ∐ P E n A ( X ) (cid:17) (cid:27) B ∐ P E m A ( X ) . In particular, any homotopy operation Q : B ∐ P E n A ( S j ) → B ∐ P E n A (∨ i S k i ) for E n A -algebras under B gives rise to a homotopy operation for E m A -algebras under B , defined by applying FQ or, equivalently, by applying U and then applying Q . ByCorollary 2.1.12, the enriched adjunction gives us canonical identifications h U − , Q h ! R i E n = h− , FQ Fh ! FR i E m showing that secondary operations are preserved by the forgetful functor.We can also define functional homotopy operations as the analogues of Steenrod’sfunctional cohomology operations. Definition 2.6.9.
Suppose A is a commutative ring spectrum and that we have maps B → C f −→ D of E n A -algebras, making f : C → D a map under B . Suppose that wehave a homotopy operation Q : Î s π l s → π k for E n A -algebras under B that preserveszero, realized by a commutative diagram B ∐ P E n A ( S k ) Q / / ( ( PPPPPPPPPPPPP B ∐ P E n A (∨ s S l s ) (cid:15) (cid:15) B . The functional homotopy operation associated to this relation is the bracket h f , − , Q i . Proposition 2.6.10.
For any maps of E n A -algebras B → C f −→ D , the domain ofdefinition of the functional operation h f , − , Q i is the subset of Î π l s C of collections ofelements x s ∈ π l s C such that f ( x s ) = and Q ( x s ) = . These are represented byhomotopy commutative diagrams B ∐ P E n A ( S k ) Q / / (cid:15) (cid:15) B ∐ P E n A (∨ s S l s ) x s (cid:15) (cid:15) / / B (cid:15) (cid:15) B / / C f / / D f E n A -algebras. The value of h f , − , Q i is a subset of π k + D , and the indeterminacyconsists of adding elements in the image of the suspended operation σQ : Î π l s + D → π k + D and elements in the image of f : π k + C → π k + D . We now specialize the previous discussion to the category of E n -algebras over themod-2 Eilenberg-Mac Lane spectrum H . As in Example 2.6.3, multiplication is oneclassical example of a homotopy operation. Other examples of homotopy operations,and relations between them, are furnished by power operations . Theorem 2.6.11 ([BMMS86, III.3]) . For any commutative H -algebra A , there are ho-motopy operations Q s : π k → π k + s for E n A -algebras when s < k + n − . These satisfy the following relations.1. The additivity relation : Q s ( x + y ) = Q s ( x ) + Q s ( y )
2. The instability relations : Q s x = x when | x | = s , Q s x = when | x | > s
3. The
Cartan formula : Q s ( xy ) = Í p + q = s Q p ( x ) Q q ( y )
4. The
Adem relations : If r > s , Q r Q s ( x ) = Í (cid:0) i − s − i − r (cid:1) Q r + s − i Q i For m ≤ n , the forgetful map from E n -algebras to E m -algebras preserves Dyer–Lashofoperations. Proposition 2.6.12.
For any commutative H -algebra A , all homotopy operations for E ∞ A -algebras C are composites of the following types:1. the constant operation associated to an element α ∈ π n A , which takes no argu-ments and whose value on C is the image of α under the map π ∗ A → π ∗ C ;2. the Dyer–Lashof operations Q s : π n ( C ) → π n + s ( C ) ;3. the binary addition operations π n ( C ) × π n ( C ) → π n ( C ) ;4. the binary multiplication operations π n ( C ) × π m ( C ) → π n + m ( C ) .Proof. The set of homotopy operations Î s π l s → π ∗ in this category is isomorphic to π ∗ ( A ∧ P (∨ s S l s )) (cid:27) π ∗ A ⊗ H ∗ P (∨ s S l s ) . Therefore, any homotopy operation is a sum of homotopy operations for H -algebrasmultiplied by constants from π ∗ A . However, in [BMMS86, IX.2.1] it is shown thatthe homology H ∗ P ( X ) is the free commutative algebra with Dyer–Lashof operations(subject to the additivity formula, instability relations, Cartan formula, and Ademrelations) on H ∗ X , and so the homotopy operations for H -algebras are generated byconstants, addition, multiplication, and the Dyer–Lashof operations Q s . (cid:3) The category of E n A -algbras under B has suspensions, and the suspension of theaugmented object B ∐ P E n A (∨ s S l s ) is B ∐ P E n A (∨ s S l s + ) roposition 2.6.13. The suspension operator σ , on homotopy operations for E n A -algebras under B , takes zero-preserving homotopy operations Î π l s → π k to homotopyoperations Î π l s + → π k + . Suspension preserves addition, composition, and multiplica-tion by scalars from B . Suspension also takes Q s to Q s and takes the binary multiplicationoperation π p × π q → π p + q to the trivial operation.Remark . For E n H -algebras, there is also a “top” operation ξ n − which, if C ex-tends to an E n + -algebra, agrees with to Q k + n − on classes in π k C . However, the topoperation satisfies less tractable versions of the identities enjoyed by the remainingoperations—most prominently, additivity requires correction by a new binary opera-tion called the Browder bracket [BMMS86, III.3.3]. For the following, we note that a tethering of a composite map of spectra X f −→ Y д −→ Z is equivalent to a homotopy class of extension from the mapping cone C f to Z , up toorientation for the interval component of the mapping cone. Proposition 2.7.1.
Suppose X , Y , and Z are spectra, X f −→ Y д −→ Z is nullhomotopic,and that α ∈ ker ( f ) ⊂ π n ( X ) is represented by a map S n → X . Given any extension Y → C f h −→ Z from the mapping cone representing a tethering, the secondary operation h д ! f , α i is (up to sign) the set h ( ∂ − α ) , where ∂ : π n + C f → π n X is the connectinghomomorphism in the long exact sequence of homotopy groups. Corollary 2.7.2.
Suppose that X ⋆ is a simplicial spectrum with geometric realization | X ⋆ | and that F is the homotopy fiber in the sequence F j −→ X d −→ X . Then the composite F d j −−→ X i −→ | X ⋆ | has a canonical tethering. If α ∈ π n ( F ) ⊂ π n X is in the kernel of d ,then in the geometric realization spectral sequence H p ( π q X ⋆ ) ⇒ π p + q | X ⋆ | the secondary operation h i ! d j , α i is represented (up to sign) by the element [ α ] ∈ H ( π n X ⋆ ) in the spectral sequence.Proof. The 1-skeleton of | X ⋆ | , by definition, has a canonical diagram X ∨ X / / d ∨ d (cid:15) (cid:15) X ∧[ , ] + (cid:15) (cid:15) X / / sk ( ) | X ⋆ | . This defines a homotopy between the maps id and id . The map d j has a canonicalnullhomotopy by definition, and composing these two homotopies gives a canonicaltethering i ( d j ) ⇒ i ( d j ) ⇒ ∗ i ( d j ) . In particular, there is a canonical map Cj → sk ( ) | X ⋆ | from the mapping coneof j to the 1-skeleton of the geometric realization; by more carefully understandingthe degeneracies, we can show that this map is a homotopy equivalence.By Proposition 2.7.3, in the resulting long exact sequence . . . π n + X i −→ π n + sk ( ) | X ⋆ | ∂ −→ π n F d j −−→ π n X i −→ . . . , any α ∈ π n ( F ) which maps to zero under d j has a bracket h i ! d j , α i in π n + sk ( n + ) | X ⋆ | ,represented by any lift of α ∈ π n F , with indeterminacy given by the image of i .The spectral sequence for the homotopy groups of the geometric realization | X ⋆ | is the spectral sequence associated to the following (unrolled) exact couple: ∗ / / π ∗ sk ( ) | X ⋆ | / / (cid:15) (cid:15) π ∗ sk ( ) | X ⋆ | / / (cid:15) (cid:15) . . . π ∗ sk ( ) c c ●●●●●●●●●● π ∗ sk ( ) / sk ( ) g g ◆◆◆◆◆◆◆◆◆◆◆ Identifying the 0-skeleton with X and the next layer with the suspension of F , weobtain our desired identification of the element in the E -term with α . (cid:3) We will now specialize to discuss how certain elements in a Künneth spectralsequence can be identified with the results of secondary operations.
Proposition 2.7.3.
Suppose f : R → S is a map of commutative ring spectra, and let i = ∧ f : S ∧ R → S ∧ S . Then, in the (pointed) category of augmented commutative S -algebras, there is a canonical tethering p t ! i for the composite S ∧ R i −→ S ∧ S p −→ S ∧ R S . Let x ∈ π n ( S ∧ R ) map to zero in π n ( S ∧ S ) , so that σx = h p t ! i , x i ⊂ π n + S ∧ R S isdefined. Then σx is detected by the image of x under π n ( S ∧ R ) → π n ( S ∧ R ∧ S ) in thetwo-sided bar construction spectral sequence H p π q ( S ∧ R ∧ ⋆ ∧ S ) ⇒ π p + q ( S ∧ R S ) . Proof.
The relative smash product receives a map from the end of the augmentedsimplicial bar construction S ∧ R ∧ S / / / / S ∧ S / / S ∧ R S , a diagram of commutative ring spectra. The face maps d j : S ∧ R → S ∧ R ∧ S → S ∧ S are the null map S ∧ R → S η L −−→ S ∧ S for j = S ∧ R i −→ S ∧ S for j = S ∧ R → S ∧ R S are homotopic, this provides a canonicaltethering in the category of S -algebras. 26 homotopy element x as described comes from a homotopy coherent diagram asfollows: S n / / (cid:15) (cid:15) ! ! ❈❈❈❈❈❈❈❈❈ f ib ( d ) / / (cid:15) (cid:15) ∗ (cid:15) (cid:15) P S S n x / / (cid:15) (cid:15) ⇒ S ∧ R / / i (cid:15) (cid:15) ⇒ S (cid:15) (cid:15) ∗ / / S / / ⇒ S ∧ S p / / ⇒ S ∧ R S The two lower right-hand squares define the bracket h p , i , x i in augmented commu-tative S -algebras, while the outside of the diagram is made up of two large (2-by-2and 2-by-1) rectangles that are the result of forgetting down to spectra. However, byCorollary 2.7.2 the outside square determines an element of π n + ( S ∧ R S ) which liftsto the desired element in the two-sided bar construction spectral sequence. (cid:3) Remark . The tethering plays an important role here. If we do not impose thatthe tethering p t ! i comes from a tethering in E ∞ -algebras, rather than spectra, thenthe indeterminacy for the bracket in spectra is too large to determine anything aboutbracket in E ∞ -algebras.If π ∗ ( S ∧ R ) is flat over π ∗ S , we can identify the E -term in the two-sided bar con-struction spectral sequence: E ∗∗ = Tor π ∗ ( S ∧ R )∗∗ ( π ∗ ( S ∧ S ) , π ∗ S ) ⇒ π ∗ (cid:18) S ∧ R S (cid:19) The element x gives rise to the corresponding element in Tor , n . In particular, we havethe following result when the target is the mod-2 Eilenberg–Mac Lane spectrum. Proposition 2.7.5.
Suppose R → H is a map of E ∞ -algebras and x ∈ H n R maps tozero in the dual Steenrod algebra H ∗ H . Then there is an element σx = h p t ! i , x i in the R -dual Steenrod algebra π ∗ ( H ∧ R H ) which is detected by the image of x in homologicalfiltration of the spectral sequence Tor H ∗ R ∗∗ ( H ∗ , H ∗ H ) ⇒ π ∗ (cid:18) H ∧ R H (cid:19) Proof.
In this case, we can rectify the map R → H to a weakly equivalent map betweencommutative ring spectra and apply Proposition 2.7.3. (cid:3) We now specialize this result to the case where R is the complex bordism spectrum. Proposition 2.7.6.
Let n be an integer which is not of the form k − for any k , so thatthe corresponding generator b n ∈ H n MU (cid:27) F [ b , b , . . . ] in mod- homology is theHurewicz image of the generator x n ∈ π n MU (cid:27) Z [ x , x , . . . ] . Then the diagram of E ∞ H -algebras P H ( S n ) b n −−→ H ∧ MU p −→ H ∧ H i −→ H ∧ MU H , determines a bracket, and h p , i , b n i ≡ σx n mod decomposables. roof. The map H ∗ MU → H ∗ H is isomorphic to a map of polynomial algebras F [ b , b , . . . ] → F [ ξ , ξ , . . . ] that sends b k − to ξ k and sends the other generators to zero [Rav86, 3.1.4]. In partic-ular, the Künneth spectral sequenceTor H ∗ MU ∗∗ ( H ∗ , H ∗ H ) ⇒ π ∗ (cid:18) H ∧ MU H (cid:19) (2.1)has as E -term an exterior algebra Λ [ ξ k ] ⊗ Λ [ σb n | k , k − ] . By comparison withthe Künneth spectral sequenceTor π ∗ MU ∗∗ ( H ∗ , H ∗ ) ⇒ π ∗ (cid:18) H ∧ MU H (cid:19) , which degenerates and has E -term Λ [ σx k ] of the same (graded) dimension, we findthat spectral sequence (2.1) degenerates and that σb n is congruent to σx n mod decom-posables for n not of the form 2 k −
1. We can then apply Proposition 2.7.5 to identify σx n as a secondary operation. (cid:3) In this section we will recall some of the work of Ravenel–Wilson on Hopf rings[RW74].Let E be a spectrum with a homotopy commutative multiplication and let { E n } n ∈ Z be an associated Ω -spectrum. Then for any ring R the homology groups H ∗ ( E ⋆ , R ) have the structure of a Hopf ring: they have a coproduct ∆ , an additive product ◦ satisfying associativity, commutativity, unitality, anddistributivity laws that make them into a graded ring object in coalgebras [RW74,1.12]. The constants c ∈ E n = π E n give rise to elements [ c ] ∈ H ( E n ; R ) under theHurewicz map. Definition 3.1.1.
Suppose E has a complex orientation x ∈ e E ( CP ∞ ) realized by abased map b : CP ∞ → E , and let β i ∈ H i ( CP ∞ ; R ) be dual to the generator t i ∈ H ∗ ( CP ∞ ; R ) (cid:27) R [ t ] . We define the classes b i ∈ H i ( E ; R ) to be the images of β i under f . Theorem 3.1.2 ([RW74, 4.6, 4.15, 4.20]) . Let { MU n } be an Ω -spectrum associated tocomplex cobordism. For any ring R and any n ∈ Z , H ∗ ( MU n ; R ) is, as an algebra under , the tensor product of the group algebra Z [ π − n MU ] with a polynomial algebra over R .The even-degree indecomposables Q e H ∗ ( MU ⋆ ; R ) under the -product form a com-mutative graded ring under ◦ , with relations as follows. If we define a formal power Ravenel–Wilson write x ∗ y for the additive product and x ◦ y for the multiplicative product, whileCohen–Lada–May [CLM76] write xy for the additive product and x y for the multiplicative product. eries b ( s ) = Í b i s i and write x + F y = Í a i , j x i y j for the formal group law of MU ∗ , thenwe have the Ravenel–Wilson relations b ( s + t ) = Õ [ a i , j ] ◦ b ( s ) ◦ i ◦ b ( t ) ◦ j . (3.1) The ring Q e H ∗ ( MU ⋆ ; R ) is a quotient of the graded ring R [ b i ] ⊗ MU − ⋆ by a regular sequence, determined by the Ravenel–Wilson relations. Both Q e H ∗ ( MU ⋆ ; R ) and H ∗ ( MU ⋆ ; R ) are free over R . Corollary 3.1.3.
For all n and all primes p , we have commutative diagrams of thefollowing form: H ∗ ( MU n ; Q ) (cid:15) (cid:15) H ∗ ( MU n ; Z ) ? o o / / / / (cid:15) (cid:15) H ∗ ( MU n ) (cid:15) (cid:15) H ∗− n ( MU ; Q ) H ∗− n ( MU ; Z ) ? o o / / H ∗− n ( MU ) In the following, for spaces X and Y we will find it convenient to identify Hom ( H ∗ X , H ∗ Y ) with the isomorphic completed tensor product H ∗ ( Y ) ˆ ⊗ H ∗ ( X ) . Here H ∗ ( Y ) is discrete, while H ∗ ( X ) inherits an inverse limit structure dual to thefiltration of H ∗ ( X ) by finite-dimensional subspaces.The invariant below, in a slightly different form, appears as the “total unstableoperation” in [Goe99, 10.2] and is credited to Strickland. Definition 3.2.1.
Let E be a multiplicative generalized cohomology theory repre-sented by an Ω -spectrum { E n } n ∈ Z . The unstable homology invariant for E -cohomologyis the collection of natural transformations of sets Λ : E n ( X ) = [ X , E n ] → Hom ( H ∗ X , H ∗ E n ) (cid:27) H ∗ E n ˆ ⊗ H ∗ ( X ) . Remark . For any α , the element Λ ( α ) ∈ Hom ( H ∗ X , H ∗ E n ) is a coalgebra mapthat respects the Steenrod operations. This restriction will not be necessary for us totake into account here.The groups H ∗ E n ˆ ⊗ H ∗ ( X ) have products ◦ , each individually induced by thecorresponding product in the Hopf ring and the cup product in H ∗ ( X ) . Using these,we can determine how Λ interacts with the ring structure in E -cohomology. Proposition 3.2.3.
The unstable homology invariant Λ satisfies the following formulas: Λ ( x + y ) = Λ ( x ) Λ ( y ) Λ ( xy ) = Λ ( x ) ◦ Λ ( y ) Λ ([ c ]) = [ c ] ⊗ ore specifically, for an element z ∈ H k ( X ) with coproduct ∆ z = Í z ′ ⊗ z ′′ , we have theidentities Λ ( x + y )( z ) = Õ ( Λ x )( z ′ ) ( Λ y )( z ′′ ) , Λ ( xy )( z ) = Õ ( Λ x )( z ′ ) ◦ ( Λ y )( z ′′ ) . For z ∈ H k ( X ) with augmentation ϵ ( z ) ∈ H k (∗) and c ∈ E n , we have Λ ([ c ])( z ) = ϵ ( z )[ c ] . Proof.
Given elements x , y ∈ E n ( X ) , represented by maps X → E n , the sum is repre-sented by the composite X ∆ / / X × X ( x , y ) / / E n × E n / / E n . Similarly, a product is represented by a composite X ∆ / / X × X ( x , y ) / / E p × E q ◦ / / E p + q , and a constant c ∈ E n by a composite X → ∗ c −→ E n . The desired identities follow by applying H ∗ . (cid:3) Remark . In particular, for X = CP ∞ with mod- p graded cohomology ring F p [ t ] ,we can view the unstable homology invariant as a map E n ( CP ∞ ) → H ∗ ( E n )[[ t ]] . When E is complex oriented, the orientation class x ∈ E ( CP ∞ ) is taken to the powerseries Λ ( x ) = Õ i ≥ b i t i ∈ H ∗ ( E )[[ t ]] (Definition 3.1.1) denoted by b ( t ) in [RW74]. In these terms, Ravenel–Wilson’s iden-tity b ( s + t ) = b ( s ) + [ F ] b ( t ) = i , j [ a i , j ] ◦ b ( s ) ◦ i ◦ b ( t ) ◦ j is proved by first applying Λ to the identity m ∗ ( t ) = Í a i , j s i t j in E ( CP ∞ × CP ∞ ) andthen using naturality of Λ to write Λ m ∗ ( t ) = m ∗ b ( t ) = b ( s + t ) .While we will not require it, it can be clarifying to examine a “reduced” versionof this invariant, especially in cases where X has a basepoint. We begin by observingthat Λ ( α ) − [ ] takes values in reduced homology for any α ∈ E ∗ ( X ) .30 efinition 3.2.5. Let E be a multiplicative generalized cohomology theory repre-sented by an Ω -spectrum { E n } n ∈ Z . The reduced unstable homology invariant for E -cohomology is the natural transformation of sets λ : E n ( X ) = [ X , E n ] → Hom ( H ∗ X , e H ∗ E n ) (cid:27) e H ∗ ( E n ) ˆ ⊗ H ∗ ( X ) given by λ ( α ) = Λ ( α ) − [ ] . The identities for the operator Λ translate into ones for λ which are particularlytransparent when taken mod decomposables for Proposition 3.2.6.
The reduced unstable homology invariant λ satisfies the followingformulas: λ ( x + y ) = λ ( x ) + λ ( y ) + λ ( x ) λ ( y ) λ ( xy ) = λ ( x ) ◦ λ ( y ) λ ([ c ]) = [ c ] − [ ] The composite map E ⋆ ( X ) λ −→ e H ∗ ( E ⋆ ) ˆ ⊗ H ∗ ( X ) → ( Q e H ∗ ( E ⋆ )) ˆ ⊗ H ∗ ( X ) which reduces mod -decomposables is a natural homomorphism of graded E ⋆ -algebras. Finally, we consider the case of reduced cohomology.
Proposition 3.2.7.
Suppose α ∈ e E n ( X ) corresponds to a based map X → E n . Then thereduced unstable invariant λ ( α ) naturally takes values in e H ∗ E n ˆ ⊗ e H ∗ ( X ) .Proof. There is a restriction map e H ∗ E n ˆ ⊗ H ∗ ( X ) → e H ∗ E n ⊗ F p induced by the inclu-sion of the basepoint ∗ → X . An element α ∈ E n ( X ) which restricts to an element c ∈ E n (∗) at the basepoint is sent to the element λ ( α ) = Λ ( α ) − [ ] which restricts to [ c ] − [ ] ∈ e H ∗ E n . If the map is based, then c = λ ( α ) lifts to the tensor withreduced cohomology. (cid:3) For a ring spectrum E , the space SL ( E ) ⊂ Ω ∞ E of strict units is the path componentof the multiplicative unit 1 ∈ π ( E ) . This construction is functorial in E . If we define e E ⊂ E to be the path component of 0, then there is a homotopy equivalence e E → SL ( E ) given by applying [ ] (−) . In particular, there are canonical isomorphisms π k ( SL ( E )) (cid:27) π k ( E ) for k > H k ( SL ( E )) (cid:27) H k ( e E ) . When E is an E ∞ -algebra,the space of units inherits a corresponding structure. Theorem 3.3.1 ([May77, IV.1.8]) . For E an E ∞ -algebra, the space SL ( E ) has a naturalstructure of an infinite loop space such that the map Σ ∞ + SL ( E ) → E is a natural map of E ∞ -algebras. roposition 3.3.2. Suppose E is an E ∞ -algebra, HR is an Eilenberg-Mac Lane spectrumfor a commutative ring R , and E → HR is a map of E ∞ -algebras. Then there is a naturalsuspension map σ : SL ( E ) → Ω SL ( HR ∧ E HR ) , of infinite loop spaces realizing, for k > , the natural map π k E → Tor E ∗ , k ( R , R ) in theKünneth spectral sequence Tor E ∗ ∗∗ ( R , R ) ⇒ π ∗ (cid:16) HR ∧ E HR (cid:17) of [EKMM97, IV.4.1].Proof. Since SL only depends on connective covers, without loss of generality wecan assume that E is connective. We consider the commutative diagram E / / (cid:15) (cid:15) HR (cid:15) (cid:15) HR / / HR ∧ E HR . We then apply SL to this diagram. The space SL ( HR ) is contractible, so the commu-tative diagram of infinite loop spaces SL E / / (cid:15) (cid:15) SL ( HR ) (cid:15) (cid:15) SL ( HR ) / / SL ( HR ∧ E HR ) determines (up to contractible indeterminacy) two nullhomotopies of the diagonalmap as infinite loop space maps. Gluing these nullhomotopies together gives a mapof infinite loop spaces SL ( E ) → Ω SL (cid:16) HR ∧ E HR (cid:17) . To show compatibility with the Künneth spectral sequence, we begin by recallingits construction. Setting HR = M , we iteratively find fiber sequences M i + → F i → M i of E -modules which are exact on homotopy groups, where F i ≃ ∨ α Σ n α E is afree graded E -module, and smash over E with HR ; the resulting long exact sequencesassemble into an exact couple that calculates π ∗ ( HR ∧ E HR ) with E -term the desiredTor-groups. In particular, we may choose the unit map E → HR as one of the factorsin the map F → M , which gives us a map Σ ∞ + SL ( E ) → E → F .Let β : S k → SL ( E ) represent an element in π k SL ( E ) for k >
0, and consider the32iagram Σ ∞ SL ( E ) / / (cid:15) (cid:15) (cid:127) (cid:127) Σ ∞ + SL ( E ) / / (cid:15) (cid:15) Σ ∞ + SL ( HR ) ≃ S (cid:15) (cid:15) M / / (cid:15) (cid:15) F / / (cid:15) (cid:15) HR (cid:15) (cid:15) Ω ( HR ∧ E HR ) / / HR ∧ E M / / HR / / HR ∧ E HR , whose rows are fiber sequences and where the dotted arrow is the map induced bythe map σ . The composite map S k → Σ ∞ SL ( E ) → M represents the element β ∈ ker ( π k E → π k HR ) , and lifts to a map S k → F . The image in π k ( HR ∧ E F ) is the element corresponding to β in Tor E ∗ , k ( R , R ) . However, this also coincides withthe suspension of β under the dotted arrow that uses the two nullhomotopies of Σ ∞ SL ( E ) → HR . (cid:3) Corollary 3.3.3.
For a ring R , there are suspension maps σ : e H ∗ ( SL ( E ) ; R ) → H ∗ + ( BSL E ; R ) → π ∗ + (cid:16) HR ∧ E HR (cid:17) . These are natural in maps E → HR of E ∞ -algebras, and on the Hurewicz image of π ∗ BSL ( E ) these are given by the suspension map. When R = F , this map commuteswith the Dyer–Lashof operations.Proof. The map SL ( E ) → Ω SL ( HR ∧ E HR ) is adjoint to a map BSL ( E ) → SL ( HR ∧ E HR ) of infinite loop spaces. We begin with the map of E ∞ -algebras Σ ∞ + BSL ( E ) → Σ ∞ + SL ( HR ∧ E HR ) → HR ∧ E HR . The adjunction between E ∞ -algebras and E ∞ HR -algebras (using the left unit HR → HR ∧ E HR ) then produces a natural map HR ∧ BSL ( E ) + → HR ∧ E HR of E ∞ HR -algebras realizing our desired map. In particular, if R = F this map of H -algebras commutes with the Dyer–Lashof operations. (cid:3) Corollary 3.3.4.
In the commutative diagram e H ∗ ( SL ( MU ) ; Q ) (cid:15) (cid:15) e H ∗ ( SL ( MU ) ; Z ) ? o o / / / / (cid:15) (cid:15) e H ∗ ( SL ( MU )) (cid:15) (cid:15) π ∗ + ( H Q ∧ MU H Q ) π ∗ + ( H Z ∧ MU H Z ) ? o o / / π ∗ + ( H ∧ MU H ) , where the vertical maps are suspensions, the left-hand horizontal arrows are injectiveand the right-hand top horizontal arrow is surjective. In particular, the suspension mapin mod- homology is determined by the rational suspension map. In addition, the right-hand vertical map preserves the Dyer–Lashof operations. roof. The injectivity and surjectivity of the top rows was shown in Corollary 3.1.3.The injectivity of the bottom-left map follows because the comparison map of Kün-neth spectral sequences Tor π ∗ MU ∗∗ ( Z , Z ) → Tor π ∗ MU ∗∗ ( Q , Q ) becomes an inclusion of exterior algebras Λ [ σx i ] → Q ⊗ Λ [ σx i ] , and both spectral se-quences degenerate at the E -term. Therefore, the map π ∗ ( H Z ∧ MU H Z ) → π ∗ ( H Q ∧ MU H Q ) is injective. (cid:3) We can now examine the properties of the rational suspension map by using therational Hopf ring.
Proposition 3.3.5.
In the rational Hopf ring, the suspension map e H ∗ ( SL ( MU ) ; Q ) → H ∗ + ( H Q ∧ MU H Q ) , in terms of the Ravenel–Wilson basis, is a composite e H ∗ ( SL ( MU ) ; Q ) ։ Q ◦ Q e H ∗ ( MU ; Q )/( b , b , . . . ) → π ∗ + ( H Q ∧ MU H Q ) that kills -decomposables, ◦ -decomposables, and b i for i > , and sends any of theremaining basis elements [ α ] ◦ b ◦ s to the suspension class σα in the Künneth spectralsequence from Proposition 2.7.5.Proof. There is a commutative diagram of E ∞ rings over H Q : Σ ∞ + SL ( MU ) / / (cid:15) (cid:15) H Q ∧ SL ( MU ) + (cid:15) (cid:15) MU / / H Q ∧ MU Applying the natural map e H ∗ ( SL (−) ; Q ) → H Q ∧ (−) H Q , we find that the suspensionmap e H ∗ ( SL ( MU ) ; Q ) → π ∗ + ( H Q ∧ MU H Q ) (cid:27) π ∗ + ( H Q ∧ H Q ∧ MU H Q ) can be computed as the composite e H ∗ ( SL ( MU ) ; Q ) → H ∗ ( MU ; Q ) → H ∗ + ( H Q ∧ MU H Q ) . The first map, under the isomorphism [− ] (−) : e H ∗ ( SL ( MU )) (cid:27) e H ∗ ( g MU ) , sends ◦ -products to products, and takes the elements b i for i > ◦ -decomposable elements b i ≡ [ a i ] ◦ b ◦ i due to the Ravenel–Wilson relation(3.1). The second is the suspension map σ , which carries ◦ -decomposables to zero.The element [ α ] ◦ b ◦ s is the Hurewicz image of α which, by definition, is carried tothe suspension σα . (cid:3) Taking this together with Corollary 3.3.4, we find the following.34 orollary 3.3.6.
The suspension map e H ∗ ( SL ( MU )) → π ∗ + (cid:18) H ∧ MU H (cid:19) on mod- homology, in terms of the Ravenel–Wilson basis, is a composite e H ∗ ( SL ( MU )) ։ Q ◦ Q H ∗ ( MU )/( b , b , . . . ) → π ∗ + (cid:18) H ∧ MU H (cid:19) that kills -decomposables, ◦ -decomposables, and b i for i > , and sends any of theremaining elements [ α ] ◦ b ◦ s in the Ravenel–Wilson basis to the suspension class σα from the Künneth spectral sequence. Proposition 3.3.7.
The suspension map σ on mod- homology commutes with Dyer–Lashof operations.Proof. This map is the composite e H ∗ ( SL ( MU )) → H ∗ + ( BSL ( MU )) → π ∗ + (cid:18) H ∧ MU H (cid:19) . The Dyer–Lashof operations on the homology of infinite loop spaces are stable, andhence preserved by the first map; the compatibility of the second map is Corollary 3.3.3. (cid:3)
In this section we will recall the work from [BMMS86] on power operations in co-homology theories, and specifically results on H ∞ -algebra structures from [BMMS86,VIII].For an E ∞ (and hence H ∞ ) ring spectrum E , the E -cohomology of a (based) space X has natural power operations as follows. Fix m > D m for the extendedpower functor given by D m ( Y ) = ( Y ∧ m ) h Σ m . Representing an element α ∈ E ( X ) as a map α : Σ ∞ X → E , we form the commutativediagram Σ ∞ D m X D m α / / P m ( α ) $ $ ■■■■■■■■■■ D m E (cid:15) (cid:15) Σ ∞ X ∧( B Σ m ) + ∆ ♦♦♦♦♦♦♦♦♦♦♦ P m ( α ) / / E , where the right-hand map is induced by the multiplicative structure on E . In particular,this produces natural power operations: P m : e E ( X ) → e E ( D m X ) P m : e E ( X ) → e E ( X ∧( B Σ m ) + ) X with X + we obtaincompatible unbased versions: P m : E ( X ) → E (( X m ) h Σ m ) P m : E ( X ) → E ( X × B Σ m ) Outside degree 0, we cannot draw conclusions which are as strong in general.Given an element α ∈ e E n ( X ) represented by a map Σ ∞ X → E ∧ S n , we can onlydefine part of the desired diagram: Σ ∞ D m X D m α / / P m ( α ) & & D m ( E ∧ S n ) ? (cid:15) (cid:15) Σ ∞ X ∧( B Σ m ) + ♦♦♦♦♦♦♦♦♦♦♦ P m ( α ) / / E ∧ S nm With extra structure on E we can complete this diagram when n is a multiple of somefixed constant d : this is the case where E is H d ∞ -algebra [BMMS86, I.4]. An H d ∞ -algebrais an algebra equipped with explicit extra structure maps D m ( E ∧ S dn ) → E ∧ S dmn ,multiplicative and compatible across n and m . These allow us to obtain power opera-tions: P m : e E dk ( X ) → e E dmk ( D m X ) P m : e E dk ( X ) → e E dmk ( X ∧( B Σ m ) + ) These are multiplicative, and replacing X with X + gives compatible unbased versions: P m : E dk ( X ) → E dmk (( X m ) h Σ m ) P m : E dk ( X ) → E dmk ( X × B Σ m ) . Cohomology is representable, so we may apply the Yoneda lemma. Restricting tothe case where m is a chosen prime p and d =
2, we get the following.
Theorem 4.1.1. If E is an H ∞ -algebra, there are natural based and unbased poweroperations for n ∈ Z : P : e E n ( X ) → e E pn ( X ∧( B Σ p ) + ) P : E n ( X ) → E pn ( X × B Σ p ) These are universally represented by maps of based spaces E n ∧( B Σ p ) + → E pn , andsatisfy P ( x ) P ( y ) = P ( xy ) . For instance, the complex bordism spectrum MU is an H ∞ -algebra [BMMS86,VIII.5.1], giving us power operations on even-degree classes previously studied bytom Dieck and Quillen [tD68, Qui71] that extend the power operations in degree zero.The spectrum MU , which is complex oriented and has canonical Thom classes forcomplex vector bundles, also has the special property that these operations are com-patible with the Thom isomorphism, as described by Quillen.36 roposition 4.1.2 ([Qui71]) . For any complex vector bundle ξ → X of dimension k ,write t ( ξ ) ∈ MU k ( Th ( ξ )) for the canonical Thom class of ξ and e ( ξ ) ∈ MU k ( X + ) forthe Euler class.The based operation P m preserves Thom classes: we have P m ( t ( ξ )) = t ( D m ξ ) , where D m ξ is the extended power bundle over ( X m ) h Σ m . Restricting along the diagonal,we have P m t ( ξ ) = e ( ξ ⊠ ρ ) t ( ξ ) where ρ is the bundle on B Σ m induced by the reduced permutation representation of Σ m and ξ ⊠ ρ is the exterior tensor bundle on X × B Σ m . In particular, the Thom isomorphismfits into a commutative diagram MU n ( X ) P m / / t ( ξ ) (cid:15) (cid:15) MU mn ( X × B Σ m ) e ( ξ ⊠ ρ ) t ( ξ ) (cid:15) (cid:15) g MU ( n + k ) ( Th ( ξ )) P m / / g MU m ( n + k ) ( Th ( ξ ) ∧( B Σ m ) + ) . The cohomology of symmetric groups is closely related to formal group law theory[Qui71], and in particular the effect of the power operation P on the canonical firstChern class x ∈ g MU ( CP ∞ ) was determined by Ando [And95]. Theorem 4.1.3.
The inclusion C p ֒ → Σ p induces inclusions: MU ∗ ( B Σ p ) ֒ → MU ∗ [[ α ]]/[ p ] F ( α ) MU ∗ ( CP ∞ × B Σ p ) ֒ → MU ∗ [[ x , α ]]/[ p ] F ( α ) In these coordinates, the power operation P satisfies P ( x ) = Î p − i = ( x + F [ i ] F ( α )) . The power operations P : MU k → MU pk ( B Σ p ) are in principle determined bythese results, naturality, and multiplicativity, and are closely related to the Lubinisogeny in the theory of formal group laws. However, it has been difficult to obtainclosed-form expressions for these power operations. We will require the followingcomputation of Johnson–Noel, using the fact that the generator x of the complexcobordism ring in dimension 4 is CP . Theorem 4.1.4 ([JN10, 6.3]) . The polynomial generator x of the complex bordism ring MU ∗ (cid:27) Z [ x , x , x , . . . ] , appearing in π ( MU ) , has image P ( x ) ≡ α ( v + v ) + α ( v + v ) in BP ∗ [[ α ]]/([ ] F ( α ) , α ) , where P is the -primary power operation. In particular, P ( x ) ≡ x α mod decomposables and higher order terms in α .Remark . The powers of α appearing in the above result differ by a shift fromthose in [JN10] because their identity occurs after multiplication by a power of anEuler class. 37he main result of this paper hinges on this theorem. In Appendix A we willshow that Johnson–Noel’s method can be adapted to one that works in torsion-freequotients of the Lazard ring. This tweak allows us to give an abbreviated version oftheir proof at the prime 2, ignoring decomposables, that is easier to carry out withoutcomputer assistance. We recall the computation of the cohomology of the symmetric group Σ p : H ∗ ( B Σ p ) (cid:27) ( F [ u ] if p = , F p [ u ] ⊗ Λ [ v ] if p > . Here u has degree 1 if p =
2, while u has degree 2 p − v has degree 2 p − p isodd. Definition 4.2.1. If E is an H ∞ -algebra, the homology power operation Q : H ∗ ( E n ) → H ∗ ( E pn ) ˆ ⊗ H ∗ ( B Σ p ) is adjoint to the map H ∗ P : H ∗ ( E n ) ⊗ H ∗ ( B Σ p ) → H ∗ ( E pn ) induced by the map E n ∧( B Σ p ) + → E pn of based spaces from Theorem 4.1.1.The multiplicativity of the natural power operation P has the following conse-quence. Proposition 4.2.2.
The operation Q satisfies Q( x ) ◦ Q( y ) = Q( x ◦ y ) and Q([ ]) = [ ] . Proposition 4.2.3.
Suppose E is an H ∞ -algebra. Then for all n ∈ Z we have a commu-tative diagram of sets e E n ( X ) Λ (cid:15) (cid:15) P / / e E pn ( X ∧( B Σ p ) + ) Λ (cid:15) (cid:15) H ∗ ( E n ) ˆ ⊗ H ∗ ( X ) Q ⊗ / / H ∗ ( E pn ) ˆ ⊗ H ∗ ( B Σ p ) ˆ ⊗ H ∗ ( X ) that is natural in X . The horizontal maps preserve products and the bottom map is amap of abelian groups.Proof. The power operation P sends an element represented by a map α : X → E n to the composite P ( α ) : X ∧( B Σ p ) + α ∧ −−−−→ E n ∧( B Σ p ) + P −→ E pn . The value of Λ ( P ( α )) is the effect on homology, which is the composite H ∗ ( X ) ⊗ H ∗ ( B Σ p ) H ∗ α ⊗ −−−−−→ H ∗ ( E n ) ⊗ H ∗ ( B Σ p ) H ∗ P −−−→ H ∗ ( E pn ) . Taking adjoints recovers the statement about completed tensor products. (cid:3) emark . The map BC p → CP ∞ induces a map e E ( CP ∞ ) → e E ( BC p ) that takesthe orientation class x to the generator α described in Theorem 4.1.3, and the map H ∗ ( CP ∞ ) → H ∗ ( BC p ) is the ring map that sends t to u if p is 2 or to a generator w = u /( p − ) in degree 2 if p is odd. By naturality of Λ , we find that Λ ( α ) is equal to b ( u ) if p = b ( w ) if p is odd.For the remainder of this section we will focus on the prime 2. We first recallthe following calculation, which is dual to the identity used to define the Steenrodoperations in [Ste62, VII.3.2, VII.6.1]. Lemma 4.2.5.
For a space X with second extended power D ( X ) , the composite diagonalmap H ∗ ( X ) ⊗ H ∗ ( B Σ ) → H ∗ ( X × B Σ ) → H ∗ ( D ( X )) on mod- homology is given by v ⊗ β n Õ j ≥ Q j + n ( P j v ) . Here β n is dual to u n and P j is the homology operation dual to Sq j . As a result, the Dyer–Lashof operations can be recovered from this diagonal mapinto the extended power.
Theorem 4.2.6.
Consider the homology operations Q : H ∗ ( MU n ) → H ∗ ( MU n ) ˆ ⊗ H ∗ ( B Σ ) from Definition 4.2.1. Then there are multiplicative Dyer–Lashof operations b Q s : H ∗ ( MU n ) → H ∗ ( MU n ) , extending the Dyer–Lashof operations in degree zero of [CLM76, II.1] (coming from themultiplicative E ∞ -space structure) to Dyer–Lashof operations in even degrees. Thesesatisfy the Cartan formula b Q s ( x ◦ y ) = Õ p + q = s b Q p ( x ) ◦ b Q q ( y ) and are related to Q by the identity Q( x ) = Õ n , j b Q j + n ( P j x ) u n . In particular, if all nontrivial Steenrod operations vanish on x then Q( x ) = Í b Q n ( x ) u n .This property is invariant under the product ◦ .39 .3 Power operations in the Hopf ring We can now begin to use the results of the previous sections to calculate multiplica-tive Dyer–Lashof operations in the Hopf ring for MU (the additive ones having beendetermined by Turner [Tur93]). First we will find the effect on the class b ∈ H ( MU ) of Definition 3.1.1, because ◦ -multiplication by b represents suspension. Proposition 4.3.1 (cf. [Pri75]) . Let b k ∈ H k ( MU ) denote the fundamental classes ofDefinition 3.1.1. Then the -primary multiplicative Dyer–Lashof operations satisfy b Q n b = b ◦ b n for all n ≥ .Proof. For a general prime p , we consider the commutative diagram MU ( BU ( )) P / / t ( γ ) (cid:15) (cid:15) MU ( BU ( ) × B Σ p ) e ( γ ⊠ ρ ) t ( γ ) (cid:15) (cid:15) g MU ( MU ( )) P / / Λ (cid:15) (cid:15) g MU p ( MU ( ) ∧( B Σ p ) + ) Λ (cid:15) (cid:15) H ∗ ( MU ) ⊗ H ∗ ( MU ( )) Q ⊗ / / H ∗ ( MU p ) ⊗ H ∗ ( B Σ p ) ⊗ H ∗ ( MU ( )) , where the top square expressing compatibility of P with the Thom isomorphism isfrom Proposition 4.1.2. Because x is the Thom class of the canonical bundle on BU ( ) , Λ ( t ( γ )) = b ( s ) . The image of the unit 1 ∈ MU ( BU ( )) along the left-to-bottomcomposite is then (Q ⊗ )( Λ ( x )) = (Q ⊗ )( b ( s )) = Õ Q( b k ) s k . On the other hand, the image along the top-right composite is Λ x p − Ö k = ( x + F [ k ] F α ) ! = b ( s ) ◦ ( b ( s ) + [ F ] b ( u )) ◦ · · · ◦ ( b ( s ) + [ F ] [ p − ] [ F ] b ( u )) , using the expression for the Euler class of the exterior tensor bundle γ ⊠ ρ on BU ( ) × B Σ p .Taking the coefficient of s , which involves only the linear term of b ( s ) and theconstant coefficients (in terms of s ) of the factors b ( s ) + [ F ] [ k ] [ F ] b ( u ) , we find that Q( b ) = b ◦ b ( u ) ◦ [ ] [ F ] b ( u ) ◦ · · · ◦ [ p − ] [ F ] b ( u ) . When we specialize to p = Õ j ≥ b Q j ( b ) u j = b ◦ b ( u ) = Õ n ≥ ( b ◦ b n ) u n as desired. (cid:3) roposition 4.3.2. Suppose that y ∈ π n MU and that, in the coordinates of Theo-rem 4.1.3, we have P ( y ) = ∞ Õ i = c i α i for some elements c i ∈ π n + i MU . Then Q([ y ]) = ∞ i = [ c i ] ◦ b ( u ) ◦ i Proof.
Taking X = ∗ in Proposition 4.2.3 identifying [ y ] with Λ ( y ) , we find Q([ y ]) = Λ ( P ( y )) = Λ (cid:16)Õ c i α i (cid:17) = ∞ i = [ c i ] ◦ b ( u ) ◦ i by Proposition 3.2.3 and Remark 4.2.4. (cid:3) Corollary 4.3.3.
Mod -decomposables and the ◦ -ideal generated by b , b , . . . , theHurewicz image [ x ] ◦ b ◦ n ∈ H n ( MU ) of x ∈ π n ( MU ) satisfies Q([ x ] ◦ b ◦ n ) ≡ ∞ Õ i = [ c i ] ◦ ( b ) ◦( i + n ) u ( i + n ) . In particular, b Q k ([ x ] ◦ b ◦ n ) ≡ [ c k − n ] ◦ b ◦( k + n ) in this quotient.Proof. The first part follows from the multiplication formula
Q([ x ] ◦ b ◦ n ) = Q([ x ]) ◦Q( b ) ◦ n . The second part follows from Theorem 4.2.6 and the fact that the operations P j vanish on [ x ] ◦ ( b ) ◦ n for j > (cid:3) MU -dual Steenrod algebra We will now apply the previous technology to compute a multiplicative Dyer–Lashofoperation in H ∗ SL ( MU ) . In order to do so, we need some preliminary results abouthow the additive product interacts with multiplicative Dyer–Lashof operations. Proposition 4.4.1. At p = , the multiplicative and additive Dyer–Lashof operationsin the Hopf ring of an E ∞ -algebra satisfy the following identities.1. When x and y are in the positive-degree homology of the path component of zero,we have b Q s ( x y ) ≡ Q s ( x ◦ y ) mod -decomposables.2. When y is in the positive-degree homology of the path component of zero, we have b Q s ([ ] y ) ≡ [ ] Q s ( y ) + [ ] b Q s ( y ) mod -decomposables. . For any positive-degree element x there exist elements z i for < i < | x | such that Q s ( x ) = Q s [ ] ◦ x + Õ Q s + i [ ] ◦ z i . In particular, Q s ( x ) is ◦ -decomposable for any x and any s > .Proof. The mixed Cartan formula [CLM76, II.2.5] takes the following form. If x and y are elements with coproducts given by ∆ x = Í x ′ ⊗ x ′′ and ∆ y = Í y ′ ⊗ y ′′ , then b Q s ( x y ) = Õ p + q + r = s Õ b Q p ( x ′ ) Q q ( x ′′ ◦ y ′ ) b Q r ( y ′′ ) . In the case of the first identity, the only time this is not decomposable under b Q p ( x ′ ) and b Q r ( y ′′ ) are of degree zero; this occurs when p = r = [ ] ⊗ x and y ⊗[ ] of the coproduct.In the case of the second identity, the only nonzero terms in the mixed Cartanformula occur when p = y ′ = [ ] or y ′′ = [ ] .The third identity is proven by induction on the degree of x , using the formula Q s ([ ]) ◦ x = Õ Q s + i ([ ] ◦ P i x ) from [CLM76, II.1.6]. (cid:3) Corollary 4.4.2.
When x is in the positive-degree homology of the path component ofzero, we have b Q s ([ ] x ) ≡ b Q s ( x ) mod -decomposables and ◦ -decomposables, and hence Q([ ] x ) ≡ Q( x ) . Proof.
We have b Q s ([ ] x ) − b Q s ( x ) ≡ [ ] Q s ( x ) + [ ] b Q s ( x ) − [ ] b Q s ( x ) = ([ ] − [ ]) ( Q s ( x ) + b Q s ( x )) + Q s ( x )≡ ◦ -decomposable. (cid:3) Proposition 4.4.3.
Suppose that x ∈ π n MU and that, in the coordinates of Theo-rem 4.1.3, we have P ( x ) = ∞ Õ i = c i α i for some elements c i ∈ π n + i MU . Then mod -decomposables, ◦ -decomposables, andthe ideal generated by b , b , . . . , the Hurewicz image [ ] ([ x ] ◦ b ◦ n ) of x ∈ π n SL MU satisfies Q([ ] ([ x ] ◦ b ◦ n )) ≡ ∞ Õ i = [ c i ] ◦ ( b ) ◦( i + n ) u ( i + n ) . In particular, b Q k ([ ] ([ x ] ◦ b ◦ n )) ≡ [ c k − n ] ◦ b ◦( k + n ) in this quotient. emark . We are working mod ◦ -decomposables in H ∗ ( MU ) , and not in the entireHopf ring, and so the right-hand side is not necessarily ◦ -decomposable unless c k − n is. Proof.
By Corollary 4.4.2, we have
Q([ ] ([ x ] ◦ b ◦ n )) ≡ Q([ x ] ◦ b ◦ n ) , and by Corollary 4.3.3 this is congruent to ∞ Õ i = [ c i ] ◦ ( b ) i + n u ( i + n ) . In particular, taking coefficients of both sides gives us that b Q k ([ ] ([ x ] ◦ b ◦ n )) ≡ [ c k − n ] ◦ b ◦( k + n ) in this quotient. (cid:3) Remark . The expression for P ( x ) as a series in α is not unique due to the fact thatit takes place in a quotient ring, and it is not immediately clear that the identity in thisproposition is independent of this choice. However, any indeterminacy is a multipleof the identity [ ] F ( α ) =
0, whose image in the Hopf ring under the total unstableinvariant translates into an identity in terms of the Ravenel–Wilson relations.We can now apply the results of Johnson–Noel from Theorem 4.1.4, as well asCorollary 3.3.6 and Proposition 3.3.7.
Corollary 4.4.6.
The Dyer–Lashof operations in H ∗ SL ( MU ) satisfy b Q ([ ] ([ x ] ◦ b ◦ )) ≡ [ ] ([ x ] ◦ b ◦ ) mod -decomposables, ◦ -decomposables, and the ideal ( b , b , . . . ) .The Dyer–Lashof operations in π ∗ ( H ∧ MU H ) satisfy Q ( σx ) = σx . Remark . We can take a brief pause to sketch why no map MU → BP can begiven the structure of a map of E -algebras at the prime 2, extending [JN10]. If itcould, then we can obtain a map of E -algebras H ∧ MU H → H ∧ BP H , on homotopygiven by a map of exterior algebras Λ [ σx i ] → Λ [ σx i − ] . However, this map wouldbe zero on the element σx and nonzero on the element σx = Q ( σx ) . (Here weuse that Q , on a class in degree 5, is realized by an operation for E -algebras—seeRemark 2.6.14.) This argument has been expanded in [Sen17].43 Calculations with MU , H Z / , and BP In order to begin with more specific computations of secondary operations, we willuse the following convenient definitions.
Definition 5.0.1.
For a symbol a and an integer k , we define P E n H ( a k ) to be the free E n H -algebra P E n H ( S k ) , writing a k ∈ π k P E n H ( a k ) for the generator represented by theunit map S k → P E n H ( S k ) .Similarly, we use the coproduct in E n H -algebras to define P E n H ( a k , b k , . . . ) = P E n H ( a k ) ∐ . . . P E n H ( b k ) ∐ · · · (cid:27) P E n H (∨ S k i ) for a sequence ( a k , b k , . . . ) . If a generator has a known, fixed, degree, we will leaveoff the subscript. Definition 5.0.2.
Let D be the category of E ∞ H -algebras under P H ( x ) , where x hasdegree 2, and D n the category of E n H -algebras under P E n H ( x ) .Let C = D ± and C n = (D n ) ± as in Definition 2.2.1.There are forgetful functors between these categories, using the compatible maps P E n H S → P E m H S that are adjoint to the units S → P E m H S . The generator of H MU (cid:27) Z / P H ( x ) → H ∧ MU up to equivalence, lifting it to an object of C . MU The 2-primary power operations in H ∗ MU are known by work of Kochman [Koc73],but the following closed-form formula is due to Priddy. Theorem 5.1.1 ([Pri75]) . The Dyer–Lashof operations in H ∗ MU (cid:27) H ∗ BU are deter-mined by the following identity: Õ Q j b k = ∞ Õ n = k k Õ u = (cid:18) n − k + u − u (cid:19) b n + u b k − u ! ∞ Õ n = b n ! − Here b = by convention. In particular, we have Õ Q j b = ∞ Õ n = ( b n b + ( n − ) b n + ) ! ∞ Õ n = b n ! − . This allows the following direct computation. (Compare [Pri75, 2.5], which carriesout this computation for MO ). 44 roposition 5.1.2. We have the following Dyer–Lashof operations in H ∗ MU : Q b = b Q b = b + b b + b Q b = b Q b = b + b b + b b + b b + b b + b b + b Q b = b + b b + b Q b = b + b b + b b + b b Q b = b b + b b + b b b + b b In particular, the following identities hold: = Q b + b = Q b + ( Q b ) Q b = Q b + b Q b = Q b + b Q b H The power operations in the dual Steenrod algebra are known by work of Steinberger.
Theorem 5.2.1 ([BMMS86, III.2.2, III.2.4]) . The -primary Dyer–Lashof operations inthe dual Steenrod algebra satisfy the following identities: + ξ + Q ξ + Q ξ + Q ξ + · · · = ( + ξ + ξ + . . . ) − Q s ξ i = ( Q s + i − ξ if s ≡ , − i , otherwise. Q i ξ i = ξ i + This, again, allowed direct computation.
Proposition 5.2.2 ([BMMS86, III.5]) . We have the following Dyer–Lashof operationsin the -primary dual Steenrod algebra: Q ξ = ξ Q ξ = ξ Q ξ = ξ ξ Q ξ = ξ Q ξ = ξ n particular, the Cartan formula implies that the following identities hold: = Q ξ + ξ = Q ξ + ξ Q ξ = Q ξ + ( Q ξ ) Remark . While the identity Q ξ = ξ is valid, the results of this paper onlyrequire us to know the easier statement that Q ξ ≡ ξ mod decomposable elements. MU → H Z / Recall that the category C is the category of E ∞ H -algebras under P H ( x ) , where x hasdegree 2. Theorem 5.3.1.
Consider the maps P H ( x , z ) Q −→ P H ( x , y ) f −→ H ∧ MU p −→ H ∧ H in the category C , where Q sends z to Q y + x Q y and f sends ( x , y ) to ( b , b ) .Then a functional homotopy operation h p , f , Q i is defined in P H ( x ) -algebras and satisfies h p , f , Q i ≡ ξ mod decomposables.Proof. The identities Q b + b Q b = p ( b ) = E ∞ P H ( x ) -algebras over H : P H ( x , z ) Q (cid:15) (cid:15) / / P H ( x ) (cid:15) (cid:15) % % ▲▲▲▲▲▲▲▲▲▲▲ P H ( x , y ) f / / (cid:15) (cid:15) H ∧ MU / / p (cid:15) (cid:15) H (cid:15) (cid:15) P H ( x ) / / H ∧ H i / / H ∧ MU H In particular, Q is a map of augmented objects and H ∧ MU H is a pointed object, en-suring that the secondary operation is defined. As a result, we can define h p , f , Q i andapply the Peterson–Stein formula of Proposition 2.3.5 to find that there is an identity h i , p , f i Q = i h p , f , Q i . (Note that there is no inversion in this Peterson–Stein formula because the targetgroup is a vector space over F .)The bracket h i , p , f i takes y , which maps under f to the Hurewicz image b of x ∈ π MU , to the suspension class σb up to indeterminacy by Proposition 2.7.5.46he operation Q sends this to Q ( σb ) + x Q ( σb ) = Q ( σb ) because x acts by 0on H ∧ MU H . Then Corollary 4.4.6 implies that Q ( σb ) ≡ σx mod decomposables,and the proof of Proposition 2.7.6 shoes that σx ≡ i ( ξ ) mod decomposables. Thuswe find that i h p , f , Q i = h i , p , f i Q ≡ i ( ξ ) mod decomposables.The indeterminacy in the functional homotopy operation h p , f , Q i consists of el-ements in the image of p and elements in the image of σQ , which are of the form Q ( y ′ ) + ξ Q ( y ′ ) . However, there are no indecomposables in the image of p andno indecomposables in the dual Steenrod algebra in degree 5, and so the indetermi-nacy consists completely of decomposable elements. The map i is an isomorphism onhomotopy in degree 15 mod decomposables, and hence h p , f , Q i ≡ ξ mod decompos-ables. (cid:3) Proposition 5.4.1.
Suppose that R is an E H -algebra and x ∈ π ( R ) . Define thefollowing classes: y = Q xy = Q xy = Q xy = Q xy = Q x + x y = Q x + x Q xy = Q x + ( Q x ) Then there is an identity = Q y + Q y + Q y + x ( Q y ) + y ( Q x ) + y Q Q x + y Q Q x + ( Q y )( Q x ) + ( Q y )( Q x ) + y ( Q Q x + Q Q x + x Q Q x ) Proof.
The following table breaks this down term-by-term, substituting in the values47f the y i . Q y = Q Q x + Q ( x Q x ) Q y = Q Q x + Q (( Q x ) ) Q y = Q Q xx ( Q y ) = x ( Q Q x ) + x Q Q x + x ( Q x ) Q Q xy ( Q x ) = ( Q x ) ( Q x ) y Q Q x = ( Q x ) Q Q xy Q Q x = ( Q x ) Q Q x + x Q Q x ( Q y )( Q x ) = ( Q Q x )( Q x ) ( Q y )( Q x ) = ( Q Q x )( Q x ) y ( Q Q x ) = ( Q x ) ( Q Q x ) y ( Q Q x ) = ( Q x ) Q Q xy ( x Q Q x ) = ( Q x ) ( x Q Q x ) The reader who is interested in ensuring that these cancel is encouraged to do so withthe aid of a pen. To assist this, we list the following needed identities deduced fromthe Cartan formula, Adem relations, and instability relations where appropriate. Q Q x = Q Q x + Q Q xQ ( x Q x ) = x Q Q x + ( Q x ) Q Q x + ( Q x ) Q Q x + ( Q x ) Q Q x + ( Q x ) Q Q x + ( Q x ) ( Q x ) Q (( Q x ) ) = x Q Q x = x Q Q x ( Q x ) Q Q x = ( Q x ) Q Q x + ( Q x ) Q Q x ( Q x ) Q Q x = ( Q x ) Q Q x + ( Q x ) Q Q x ( Q x ) Q Q x = ( Q x ) Q Q x To apply Q r to an element in degree s , as well as make use of the Adem relations,Cartan formula, and instability relations, we require the presence of an E n -algebra for n ≥ r − s +
2. The greatest value of n required from the equations above is when we take Q y , and in particular use additivity for Q , which requires an E -algebra. (cid:3) We can use this relation to build secondary operations.
Proposition 5.4.2.
Suppose n ≥ and let R be an object of C n , corresponding to an E n H -algebra with an element x ∈ π ( R ) , such that the classes y i of Proposition 5.4.1vanish. Then there is a secondary operation on x given by h x , Q h ! R i ∈ π R . Theindeterminacy in this secondary operation consists of elements of the form Q y ′ + Q y ′ + Q y ′ nd decomposables. This secondary operation is preserved by the forgetful functors C m → C n for m > n .Proof. For any n ≥
12, Proposition 5.4.1 describes a relation between homotopy oper-ations, in the form of a homotopy commutative diagram of E n H -algebras P E n H ( z ) ϵ (cid:15) (cid:15) R / / P E n H ( x , y , y , y , y , y , y , y ) Q (cid:15) (cid:15) H / / P E n H ( x ) , adjoint to a commutative diagram of E n -algebras under P E n H ( x ) of the form P E n H ( x , z ) ϵ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ R / / P E n H ( x , y , y , y , y , y , y , y ) Q (cid:15) (cid:15) P E n H ( x ) . Here the maps Q and R are defined by the equations of Proposition 5.4.1. The map R is a map of augmented objects, the domain by the map ϵ sending z to 0 and therange by the map sending all y i to zero. In particular, the homotopy commutativityof the above diagrams shows that there exists a tethering Q h ! R in the category C n .The indeterminacy in this secondary operation consists of elements in the imageof the suspended operation σR : P E n H ( x , z ′ ) → P E n H ( x , y ′ , y ′ , y ′ , y ′ , y ′ , y ′ , y ′ ) . Proposition 2.6.13 implies that σR is given by z ′ Q y ′ + Q y ′ + Q y ′ + x ( Q y ′ ) + ( Q y ′ )( Q x ) + ( Q y ′ )( Q x ) , since the other terms involve binary products that map to zero. However, the termsother than Q y ′ + Q y ′ + Q y ′ always take decomposable values. (cid:3) Proposition 5.4.3.
In the -primary dual Steenrod algebra H ∗ H (cid:27) F [ ξ , ξ , . . . ] , viewed as the homotopy of the H -algebra H ∧ H , the bracket h ξ , Q , R i is defined, andthe indeterminacy is zero mod decomposables.Proof. The Cartan formula for Dyer–Lashof operations immediately implies that Q k + ( ξ ) = k . The remaining identities0 = Q ξ + ξ = Q ξ + ξ Q ξ = Q ξ + ( Q ξ ) ξ Q = h ξ , Q , R i is defined.We now consider the indeterminacy. The indeterminacy is generated by addingthe results of degree-29 homotopy operations applied to ξ , Dyer–Lashof operationsapplied to elements in degrees 11, 13, and 14, and decomposables. Proposition 2.6.12showed that all nonconstant homotopy operations are generated by multiplication, ad-dition, and the operations Q n , all of which preserve decomposables. The dual Steenrodalgebra contains no indecomposables in degrees 11, 13, and 14, and so any operationapplied to such an element is decomposable. (cid:3) The operations Q and R , while complex, can be related to simpler operations usingthe following diagram. Proposition 5.4.4.
Consider the maps µ : P H ( x , y , y , y , y , y , y , y ) → P H ( x , y ) ν : P H ( x , z ) → P H ( x , z ) α : P H ( x , z ) → P H ( x , z ) β : P H ( x , z ) → P H ( x , y ) of augmented objects, defined by the identities µ ( y i ) = for i , µ ( y ) = Q y ν ( z ) = Q z α ( z ) = ( z ) β ( z ) = Q xQ y Then there is an identity µR = Qν + βα and a homotopy commutative diagram in C ofthe form P H ( x , y i ) µ (cid:15) (cid:15) Q / / P H ( x ) b (cid:15) (cid:15) ξ % % ❏❏❏❏❏❏❏❏❏❏ P H ( x , y ) f / / H ∧ MU p / / H ∧ H , where f and Q are from Theorem 5.3.1.Proof. It is classical that the map p : H MU → H H takes b to ξ , making the right-hand triangle commute.To verify that the map µ makes the square diagram commute in the homotopy50ategory, we need to know that the Dyer–Lashof operations on b satisfy0 = Q b = Q b = Q b = Q b = Q b + b = Q b + ( Q b ) Q b = Q b + b Q b The odd operations vanish automatically because H ∗ MU is concentrated in even de-grees, and the remaining three identities were proven in Proposition 5.1.2.Finally we need to verify the identity Qν = βα + µR . Using the definition of µ andthe formula from Proposition 5.4.1 for R , we find that µ ( R ( z )) = Q Q y + x Q Q y In the Adem relation Q Q = Q Q + Q Q + Q Q , the last two terms auto-matically vanish on classes in degree four. Therefore, we can continue to simplify,finding µ ( R ( z )) = Q Q y + x Q Q y = Q Q y + Q ( x Q y ) + ( Q x ) ( Q y ) = Q ( Q y + x Q y ) + ( Q xQ y ) = Q ( Q ( z )) + β ( z ) = Qν ( z ) + βα ( z ) , as desired. (cid:3) Corollary 5.4.5.
In the dual Steenrod algebra, any element in the bracket h ξ , Q , R i iscongruent to ξ mod decomposables.Proof. We first observe that three types of elements in degree 31 are decomposable inthe dual Steenrod algebra.• The first are elements in the image of p : H ∗ MU → H ∗ H : the only indecompos-able element in the image of p is 1 ∈ H H .• The second are elements in the image of σR , which (as in Proposition 5.4.2)consists of multiples of Dyer–Lashof operations applied to elements in degrees11, 13, and 14. Degrees 11, 13 and 14 contain no indecomposables, and so theCartan formula for Dyer–Lashof operations implies that any elements in theimage of σR are decomposable. 51 The third are elements in the image of σ ( Qν ) or σ ( βα ) , both of which are multi-ples of Dyer–Lashof operations applied to classes in degree 5. Degree 5 containsno indecomposables, and thus similarly the images of these elements are inde-composable.Multiple applications of Proposition 2.3.5 and Proposition 2.4.3 give us the follow-ing string of identities. h ξ , Q , R i = h pb , Q , R i⊂ h p , b Q , R i = h p , f µ , R i⊃ h p , f , µR i = h p , f , Qν + βα i⊂ h p , f , Qν i + h p , f , βα i⊃ h p , f , Q i ν + h p , f , β i α . We note that in all of these brackets, the indeterminacy is contained in the three typesmentioned above: the image of p , the image of σR , and the images of σ ( Qν ) or σ ( βα ) . Itsuffices to check at the local maxima for indeterminacy in this chain of containments:the brackets h p , b Q , R i and h p , f , Qν i + h p , f , βα i . Therefore, if we work mod decom-posables we get unambiguous values and these containments become equalities. Wefind h ξ , Q , R i ≡ h p , f , Q i ν + h p , f , β i α . By Theorem 5.3.1, we have h p , f , Q i( νz ) = Q (h p , f , Q i( z )) ≡ Q ξ ≡ ξ mod decomposables. On the other hand, h p , f , β i α ( z ) = (h p , f , β i( z )) which is automatically decomposable. Therefore, every element in h ξ , Q , R i is con-gruent to ξ mod decomposables. (cid:3) Theorem 5.4.6.
Suppose that n ≥ and R is an E n ring spectrum with a map д : R → H and an element x ∈ H ( R ) such that д ( x ) = ξ in H H . If the element x makes theclasses y i of Proposition 5.4.1 zero, then the map H R → H H has ξ in its image moddecomposables.In particular, if H ∗ R → H ∗ H is injective through degree 13, this result holds.Proof. Under these conditions, H ∧ R → H ∧ H is a map of E n H -algebras, and (upto equivalence) the map P E n H ( x ) → H ∧ H lifts to a map P E n H ( x ) → H ∧ R . Thus, H ∧ R → H ∧ H can be lifted to a map in C n which, on homotopy groups, gives themap д : H ∗ R → H ∗ H .Then the secondary operation h x , Q , R i is defined and the map д carries h x , Q , R i into a subset of h ξ , Q , R i , all of whose elements are congruent to ξ mod decompos-ables. (cid:3) .5 Application to the Brown–Peterson spectrum Using Theorem 5.4.6, we can now exclude the existence of E n -algebra structures onspectra related to the Brown–Peterson spectrum. We first recall the homology of theBrown–Peterson spectrum, dual to the cohomology described in [BP66]. Proposition 5.5.1.
The Brown–Peterson spectrum BP is connective, with π BP (cid:27) Z ( ) .The map BP → H F induces an inclusion H ∗ BP ֒ → H ∗ H F whose image is the subalge-bra F [ ξ , ξ , . . . ] ⊂ F [ ξ , ξ , . . . ] of the dual Steenrod algebra. The image in positive degrees consists entirely of decom-posables. Similarly, we have truncated Brown–Peterson spectra BP h k i and their generalizedversions. Proposition 5.5.2 ([LN14, 4.3]) . Any generalized truncated Brown–Peterson spectrum BP h k i is connective, with π BP h k i = Z ( ) . The map BP h k i → H F induces an inclusion H ∗ BP ֒ → H ∗ H F whose image is the subalgebra F [ ξ , ξ , . . . ξ k + , ξ k + , ξ k + , . . . ] ⊂ F [ ξ , ξ , . . . ] of the dual Steenrod algebra. The image in positive degrees consists entirely of decom-posables until dimension k + − . In particular, the element ξ is not in the image mod decomposables for k ≥ unique non-trivial map of spectra BP → H F , and similarly for BP h k i . (At odd primes, this mapis unique up to scalar.) As E n -algebras have Postnikov towers, there is the followingconsequence. Corollary 5.5.3. If BP or BP h k i admits the structure of an E n -algebra, then the uniquenontrivial map to H F lifts to a map of E n -algebras. We can now apply Theorem 5.4.6.
Theorem 5.5.4.
The -local Brown–Peterson spectrum BP , the (generalized) truncatedBrown–Peterson spectra BP h k i for k ≥ , and their -adic completions do not admit thestructure of E n -algebras for any ≤ n ≤ ∞ .Remark . The above results can also be applied to appropriate truncations in thePostnikov tower for BP . A Power operations in the Lazard ring
In this section we will extend Johnson–Noel’s proof of Theorem 4.1.4 to a proof thatworks in torsion-free quotients of the Lazard ring. The following calculations arespecialized to the prime 2. 53he power operation P of Section 4.1 takes the form of a natural transformation P : MU n ( X ) → MU n ( X × B Σ ) ֒ → MU n ( X )[[ α ]]/[ ] F ( α ) . Writing the 2-series as [ ] F ( α ) = α · h i F ( α ) , we have the following properties.• The identity P ( uv ) = P ( u ) P ( v ) holds.• The identity P ( u ) ≡ u holds mod α .• The identity P ( u + v ) ≡ P ( u ) + P ( v ) holds mod h i F ( α ) . In particular, P becomesa ring homomorphism in this quotient.• On the canonical orientation x ∈ g MU ( CP ∞ ) , we have P ( x ) = x ( x + F α ) .Let L = MU ∗ be the Lazard ring, and define д ( x , α ) = x ( x + F α ) in the power seriesring L [[ x , α ]] . Applying the identities for P to the spaces ( CP ∞ ) n and the natural mapsbetween them, we deduce the following. Proposition A.0.1.
The map P induces a ring homomorphism Ψ : L → L [[ α ]]/h i F ( α ) and the power series д ( x , α ) defines an isogeny F → Ψ ∗ F : д ( x , α ) + Ψ ∗ F д ( y , α ) ≡ д ( x + F y , α ) . (A.1)The rings L and L [[ α ]]/h i F ( α ) are torsion-free, and so the formal group laws F and Ψ ∗ F have logarithms: ℓ F ( x ) = Õ CP n − x n n ℓ Ψ ∗ F ( x , α ) = Õ Ψ ( CP n − ) x n n By choosing any lifts of Ψ ( CP n ) to L [[ α ]] , we can view these formulas as definingpower series ( ℓ F ) ′ ( x ) ∈ L [[ x ]] and ( ℓ Ψ ∗ F ) ′ ( x , α ) ∈ L [[ x , α ]] .Taking derivatives of (A.1) with respect to y and evaluating at y =
0, we find д ′ ( , α )( ℓ Ψ ∗ F ) ′ ( д ( x , α ) , α ) ≡ д ′ ( x , α )( ℓ F ) ′ ( x ) in L [[ x , α ]]/h i F ( α ) , and thus д ′ ( x , α )( ℓ Ψ ∗ F ) ′ ( д ( x , α ) , α ) = α · ( ℓ F ) ′ ( x , α ) + h ( x , α ) · h i F ( α ) (A.2)for some power series h ( x , α ) ∈ L [[ x , α ]] .We now substitute x = αy and observe that д ( αy , α ) = α k ( y , α ) , k ( y , α ) has the form y + O ( y ) . Hence, there a compositioninverse: a series k − ( y , α ) of the same form such that k ( k − ( y , α ) , α ) = y .Substituting x = αy in to (A.2), we obtain an identity αk ′ ( y , α )( ℓ Ψ ∗ F ) ′ ( α k ( y , α ) , α ) = α · ( ℓ F ) ′ ( αy ) + h ( αy , α ) · h i F ( α ) which can be simplified to the statement ( ℓ Ψ ∗ F ) ′ ( α y , α ) = ( ℓ F ) ′ ( αk − ( y , α ))( k − ) ′ ( y , α ) + ˜ h ( y , α ) · h i F ( α ) for some series ˜ h ( y , α ) ∈ L [[ y , α ]] . If we write f n ( α ) for the coefficient of y n in ( ℓ F ) ′ ( αk − ( y , α ))( k − ) ′ ( y , α ) , we then find that Ψ ( CP n ) α n = f n ( α ) + ˜ h n ( α ) · h i F ( α ) for some series ˜ h n ( α ) . If f : L → S is any ring homomorphism, there is a degree-2 n polynomial h n ( α ) ∈ S [ α ] such that f n ( α ) − h n ( α ) · h i F ( α ) ≡ f ( CP n ) in S [[ α ]]/( α n + ) . If 2 is not a zero divisor in the ring S , this determines h n ( α ) uniquelyand so it can be calculated in S . We deduce that f ( P ( CP n )) ≡ α − n ( f n ( α ) − h n ( α ) · h i F ( α )) in S [[ α ]]/[ ] F ( α ) .In particular, we may take S = Z [ v ]/( v ) , which has logarithm x + v x . We canthen expand out the definitions in this ring. x + F y = x + y + v ( x + y − ( x + y ) )h i F ( α ) = − v α ( ℓ F ) ′ ( x ) = + v x д ( x , α ) = αx + x + v ( α + x − ( α + x ) ) = αx + ( − v α ) x − v α x + O ( x ) k ( y , α ) = y + ( − v α ) y − v α y + O ( y ) k − ( y , α ) = y + ( v α − ) y + ( − v α ) y + O ( y )( ℓ F ) ′ ( αk − ( y , α )) = + O ( y )( k − ) ′ ( y , α ) = + ( v α − ) y + ( − v α ) y + O ( y ) f ( α ) = − v α h ( α ) = f ( P ( CP )) = v α ≡ v α . Here the last congruence follows because, in the ring S [ α ]/[ ] F ( α ) ,2 αv ≡ v α ≡ v =
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