Multiplicative structure on Real Johnson-Wilson theory
aa r X i v : . [ m a t h . A T ] A ug MULTIPLICATIVE STRUCTURE ON REAL JOHNSON-WILSON THEORY
NITU KITCHLOO, VITALY LORMAN, AND W. STEPHEN WILSON
1. I
NTRODUCTION
At the prime 2, Johnson-Wilson theory E ( n ) [JW73] is a complex-oriented cohomologytheory which has a C -equivariant refinement, E ( n ) as a genuine C -equivariant spec-trum, where the action of C stems from complex conjugation. This was first constructedin [HK01], and Real Johnson-Wilson theory ER ( n ) is defined to be the C -fixed points of E ( n ) . The underlying nonequivariant spectrum of E ( n ) is Johnson-Wilson theory E ( n ) ,and it is a homotopy associative, commutative, and unital ring spectrum. The goal of thisnote is to investigate whether the same properties hold for E ( n ) and ER ( n ) .Interest in this problem comes from the fact that ER ( n ) is quickly becoming a useful andcomputable cohomology theory. For n = 1 and 2, it reproduces familiar cohomologytheories, ER (1) = KO (2) and ER (2) = T M F (3) (the latter after suitable completion,see [HM17]). The ER ( n ) -cohomology of a large (and growing) collection of spaces hasbeen computed: real projective spaces and their products [KW08a, KW08b, Ban13] (for n = 2 ), complex projective spaces [Lor16, KLW17], BO and some of its connective covers[KW14,KLW17], and half of all Eilenberg MacLane spaces [KLW16,KLW17]. Furthermore,these computations have applications. In [KW08a, KW08b], the first and third authorsused computations in ER (2) -cohomology to prove new nonimmersion results for realprojective spaces.The existence of a multiplicative structure on Real Johnson-Wilson theory has been sug-gested in a comment in [HK01] (Comment 5 following the proof of Theorem 2.28) whichclaims that E ( n ) ⋆ ( E ( n ) ∧ E ( n )) may be calculated and from this it may be shown that E ( n ) is a (homotopy) associative, commutative, and unital ring spectrum. The results inthis note were born in the attempt to verify the above claim. Unfortunately, we were un-successful in doing so. However, we show that E ( n ) represents an MU -algebra which ishomotopy unital, associative, and commutative up to phantom maps . By a phantom map,we mean a map f : X −→ Y which has trivial restriction to any finite CW complex map-ping into X . In addition, we show that the E ( n ) -cohomology of an equivariant topologicalspace is canonically a commutative ring. Date : August 31, 2017.The first author is supported in part by the NSF through grant DMS 1307875. heorem 1.1. E ( n ) is a homotopy commutative, homotopy associative, unital MU -algebra up tophantom maps. In other words, there exist unit and multiplication maps: MU −→ E ( n ) , ˆ µ : E ( n ) ∧ E ( n ) −→ E ( n ) , such that all the obstructions to ˆ µ being a homotopy associative and homotopy commutative MU -algebra structure are phantom maps. Differently said, all the corresponding structure diagramscommute up to phantom maps. Furthermore, the forgetful map: ρ : E ( n ) ( E ( n ) ∧ E ( n )) −→ E ( n ) ( E ( n ) ∧ E ( n )) , maps ˆ µ to the canonical product µ on the non-equivariant Johnson-Wilson spectrum E ( n ) . Theorem 1.1 tells us that the Real Johnson-Wilson theory is valued in commutative ringswhen applied to finite
CW complexes. Our second result extends this to the category ofall spaces.
Theorem 1.2.
With any choice of multiplication ˆ µ as above, the spectrum E ( n ) represents a multi-plicative cohomology theory on the category of C -spaces valued in (bigraded) commutative rings.There are natural transformations of ring-valued cohomology theories MU ⋆ ( − ) −→ E ( n ) ⋆ ( − ) and E ( n ) ⋆ ( − ) −→ E ( n ) ∗ ( − ) . The results of this document justify the assumption of commutativity in the computationsof the ER ( n ) -cohomology of topological spaces made by the authors in previous work.We conclude by revisiting a result of the first and third authors concerning the M R ( n ) -orientation of MO[2 n +1 ] to correct an error in the proof of [KW14, Theorem 1.4].In the course of proving Theorem 1.2 we show that the infinite loop space underly-ing ER ( n ) , ER ( n ) , is a homotopy commutative, associative, and unital H -ring space(Lemma 5.1). Applying the Bousfield-Kuhn functor shows that the K ( n ) -localization, L K ( n ) ER ( n ) , is in fact a homotopy commutative, associative and unital ring spectrum(not just up to phantom maps). The authors are grateful to Tyler Lawson for pointing thisout.Hahn and Shi [HS17] have recently proved that Johnson-Wilson theory E ( n ) admits an A ∞ ring structure for which the C -action is given by an A ∞ involution (but not necessar-ily an A ∞ ring automorphism). The question of (homotopy) commutativity is not readilyaddressed by their techniques. They show that these issues resolve on K ( n ) -localizationgiving rise to an E ∞ ring spectrum structure on L K ( n ) ER ( n ) .The present paper shows a (homotopy) commutative and associative ring structure up tophantom maps, but to the authors’ present knowledge, ER(n) is not (yet) known to be ahomotopy commutative and associative ring spectrum.The authors would like to thank Neil Strickland for helpful discussion related to thisdocument and the referee for diligent and valuable comments. . B ACKGROUND
In this section, we recall a few background definitions and theorems from [KW07b] and[KLW16] that we use in subsequent sections.A genuine C -equivariant spectrum E is a family of C -spaces E a + bα , indexed over ele-ments a + bα ∈ RO ( C ) where α denotes the sign representation, together with a compat-ible system of equivariant homeomorphisms E a − r +( b − s ) α ≃ / / Ω r + sα E a + bα where the right hand side denotes the space of pointed maps (endowed with the conjuga-tion action) from the one point compactification of the representation r + sα . The readermay refer to [HK01] for more details on C -spectra. We will denote by ER the homotopyfixed point spectrum of the C -action on E and by E the underlying nonequivariant spec-trum given by forgetting the C -action. An example of a C - spectrum of interest to us is MU , whose underlying nonequivariant spectrum is complex cobordism, MU , studied firstby Landweber [Lan68], Fujii [Fuj76], Araki and Murayama [AM78], and more recently byHu-Kriz [HK01]. The action of C is induced by the complex conjugation action on thepre-spectrum representing MU in the usual way (see, e.g. [HK01]).The p = 2 Johnson-Wilson theory E ( n ) lifts to a C -spectrum, E ( n ) , defined as an MU -module by coning off certain equivariant lifts of the Araki generators v i for i > n , andthen inverting the lift of v n . We shall call these equivariant lifts by the same names, v i .The Real Johnson-Wilson theories, ER ( n ) , are defined as the fixed points of E ( n ) .Working with cohomological grading, let Y be a C -space and let E ∗ (1+ α ) ( Y ) denote thesubgroup of diagonal elements in the equivariant E -cohomology of Y i.e. E ∗ (1+ α ) ( Y ) := π Maps C ( Y, E ∗ (1+ α ) ) . Consider the group homomorphism given by forgetting the C -action: ρ : E ∗ (1+ α ) ( Y ) −→ E ∗ ( Y ) . Notice that the image of ρ belongs to the graded sub-group of elements in even degree.We recall the following definitions from [KW07b]. Definition 2.1. A C -space Y is said to have the weak projective property with respect toa C -spectrum E if the map ρ : E ∗ (1+ α ) ( Y ) −→ E ∗ ( Y ) , is an isomorphism of graded abelian groups.Note that in the case that the C -spectrum E is an MU -module spectrum, the map ρ inthe definition above is a map of MU ∗ (1+ α ) ∼ = M U ∗ -modules by virtue of the fact that the MU -module structure on E forgets to an M U -module structure on E . e will make extensive use of spaces with the weak projective property. In order torecognize such spaces, we need some auxiliary definitions. Definition 2.2.
A pointed C -space X is said to be projective if(1) H ∗ ( X ; Z ) is of finite type.(2) X is homeomorphic to W I ( C P ∞ ) ∧ k I for some weakly increasing sequence of inte-gers k I , with the C action given by complex conjugation.By a C -equivariant H -space, we shall mean an H -space whose multiplication map is C -equivariant. Definition 2.3. A C -equivariant H -space Y is said to have the projective property if thereexists a projective space X , along with a pointed C -equivariant map f : X −→ Y , suchthat H ∗ ( Y ; Z / is generated as an algebra by the image of f .The following theorem, proved in [KLW16, Theorem 2.6], establishes that spaces with theprojective property have the weak projective property with respect to certain MU -modulespectra (in particular, E ( n ) ). Theorem 2.4.
Let Y be a C -equivariant H -space with the projective property. Let E denoteany complete MU -module spectrum with underlying spectrum E , satisfying the property that theforgetful map: ρ ∗ : E ∗ (1+ α ) −→ E ∗ , is an isomorphism. Then the space Y has the weak-projectiveproperty with respect to E . In other words, the following map is an isomorphism of MU ∗ (1+ α ) -modules: ρ : E ∗ (1+ α ) ( Y ) −→ E ∗ ( Y ) . Remark 2.5.
The smash product of a finite collection of spaces with the projective prop-erty is an example of a space that has the weak projective property with respect to any E as in Theorem 2.4, but not the projective property (since it is not an H -space). This followsfrom writing Y ∧ Y = ( Y × Y ) / ( Y ∨ Y ) and the Five Lemma.There are many examples of spaces with the projective property. The ones of interest inthis document will be E ( n ) and its products. Lemma 2.6. E ( n ) × j × MU × si (2 n − α ) has the weak projective property with respect to E ( n ) forall i, j, s ≥ .Proof. By [KW13, Theorem 1-4], the spaces E ( n ) and MU ( i (2 n − α ) have the projectiveproperty. Thus E ( n ) × j × MU × si (2 n − α ) is a restricted product of a family of spaces withthe projective property (i.e. it is the colimit of finite products of spaces with projectiveproperty). A finite product of spaces with projective property evidently has the weakprojective property, and since ρ : E ( n ) ∗ (1+ α ) ( − ) −→ E ( n ) ∗ ( − ) is an isomorphism at eachstage, it follows that it is an isomorphism in the limit. (cid:3) . A STABLE MULTIPLICATIVE STRUCTURE
We begin with the observation that the spaces E ( n ) are | v n | = (2 n − α ) -periodic.That is, the adjoint of the multiplication-by- v n map on the spectrum E ( n ) induces, on the0-space level, an equivalence E ( n ) ≃ / / Ω | v n | E ( n ) We now express the spectrum E ( n ) , and its products as colimits of shifted suspensionspectra E ( n ) = colim m Σ − m (2 n − α ) E ( n ) , E ( n ) ∧ k = colim m Σ − mk (2 n − α ) E ( n ) ∧ k , where the maps are given by successive v ∧ kn multiplications Σ − mk | v n | E ( n ) ∧ k −→ Σ − ( m +1) k | v n | E ( n ) ∧ k . Applying E ( n ) cohomology, Milnor’s lim -sequence gives us −→ lim E ( n ) − (Σ − mk | v n | E ( n ) ∧ k ) −→ E ( n ) ( E ( n ) ∧ k ) −→ lim E ( n ) ( E ( n ) ∧ k ) −→ . Using the | v n | -periodicity of E ( n ) noted above to identify Σ −| v n | E ( n ) with E ( n ) , we havethe sequence −→ lim E ( n ) − ( E ( n ) ∧ k ) −→ E ( n ) ( E ( n ) ∧ k ) −→ lim E ( n ) ( E ( n ) ∧ k ) −→ . We may now invoke the weak projective property of the spaces E ( n ) ∧ k (see Remark 2.5)and identify the last term with lim E ( n ) ( E ( n ) ∧ k ) . The space E ( n ) has evenly graded co-homology (since it has the projective property) and homotopy; it follows from the Atiyah-Hirzebruch spectral sequence that the E ( n ) -cohomology of E ( n ) is evenly graded aswell (as are its k -fold smash products). Thus, in the analogous non-equivariant Milnorsequence −→ lim E ( n ) − ( E ( n ) ∧ k ) −→ E ( n ) ( E ( n ) ∧ k ) −→ lim E ( n ) ( E ( n ) ∧ k ) −→ the lim term vanishes. Consequently, we identify lim E ( n ) ( E ( n ) ∧ k ) with E ( n ) ( E ( n ) ) and obtain our short exact sequence of interest(3.1) −→ lim E ( n ) − ( E ( n ) ∧ k ) −→ E ( n ) ( E ( n ) ∧ k ) −→ E ( n ) ( E ( n ) ∧ k ) −→ , with the last map being ρ .By writing E ( n ) ∧ k as a colimit of v kn -multiplication maps as above and MU ∧ s as a colimitof suspension spectra MU ∧ s = colim m Σ ∞− ms | v n | MU ∧ sm | v n | he above proof readily extends to show the existence of a short exact sequence −→ lim E ( n ) − ( E ( n ) ∧ k ∧ MU ∧ sm | v n | ) −→ E ( n ) ( E ( n ) ∧ k ∧ MU ∧ s ) (3.2) −→ E ( n ) ( E ( n ) ∧ k ∧ M U ∧ s ) −→ . where we have used Lemma 2.6 above to identify the right hand term as before.We are now ready to construct the MU -algebra structure on E ( n ) that will be shown to behomotopy commutative and homotopy associative up to phantom maps. Definition 3.3.
Define ˆ µ to be any element in E ( n ) ( E ( n ) ∧ ) that lifts the canonical ringstructure of E ( n ) ( E ( n ) ∧ ) along ρ . Define the unit map MU −→ E ( n ) to be thecanonical map expressing E ( n ) as a localized quotient of MU .As a formal consequence of the short exact sequence constructed above, we obtain Theo-rem 1.1 from the introduction: Theorem 1.1.
The class ˆ µ defines a homotopy commutative, homotopy associative, and unital MU -algebra structure on E ( n ) up to phantom maps.Proof. By construction, ˆ µ maps to the canonical ring structure on E ( n ) under the forgetfulmap, ρ . It follows that any obstruction to the homotopy associativity, commutativity, orunitality of ˆ µ , viewed as a class in E ( n ) ( E ( n ) ∧ k ∧ MU ∧ s ) , maps to zero under ρ . We claimthat the only elements of the kernel of ρ are phantoms.To see this, recall the short exact sequence (3.2) above: → lim E ( n ) − ( E ( n ) ∧ k ∧ MU ∧ sm | v n | ) → E ( n ) ( E ( n ) ∧ k ∧ MU ∧ s ) → E ( n ) ( E ( n ) ∧ k ∧ M U ∧ s ) → where the right hand side was identified vialim E ( n ) ( E ( n ) ∧ k ∧ MU ∧ sm | v n | ) ∼ = lim E ( n ) ( E ( n ) ∧ k ∧ M U ∧ sm | v n | ) ∼ = E ( n ) ( E ( n ) ∧ k ∧ M U ∧ s ) . Notice that by exactness, an element in the kernel of ρ is in the image of the lim -term in E ( n ) ( E ( n ) ∧ k ∧ MU ∧ s ) and so restricts trivially to all the terms E ( n ) ( E ( n ) ∧ k ∧ MU ∧ sm | v n | ) in the inverse system. Any map from a finite CW complex into E ( n ) ∧ k ∧ MU ∧ s mustfactor through a finite stage of the colimit, that is, through E ( n ) ∧ k ∧ MU ∧ sm | v n | for some m .It follows that any element in the image of the lim term is zero upon restriction to anyfinite CW-complex. In other words, the kernel of ρ consists entirely of phantoms. (cid:3) Remark 3.4.
One may attempt to compute the group of phantom maps lim E ( n ) − ( E ( n ) ∧ k ) explicitly by identifying it with the vector space E ( n ) − ( E ( n ) ∧ k ) ⊗ Z / (suitably extendedby a Z / -agebra). This is an open problem, but it appears to the authors that this vectorspace is trivial for n = 1 , but may fail to be so for n > . Hence we at present have no gen-eral way of ensuring that the ring structure we have constructed is rigid up to homotopyfor the spectra E ( n ) , n > . . M ULTIPLICATIVE STRUCTURE ON MU (2) [ v − n ] We would like to show that the multiplication ˆ µ constructed above naturally inducesa commutative algebra structure on the E ( n ) -cohomology of any (not necessarily finite-dimensional) space. An essential ingredient in our construction will be the multiplicationon MU (2) [ v − n ] . We pause to describe it in this section. The ingredient we need is thefollowing proposition, which appears as Proposition 9.15 in [HHR16] and is proved in[HH14]. Proposition 4.1. [HH14, Corollary 4.11] Let R be a G -equivariant commutative ring with D ∈ π G⋆ ( R ) . If D has the property that for every H ⊂ G , N GH i ∗ H D divides a power of D , then thespectrum D − R has a unique commutative algebra structure such that the map R −→ D − R is amap of commutative rings. We begin by constructing a C -equivariant associative and commutative ring (in the highlystructured sense) that lifts E ( n ) . We begin with the fact that MU has this structure (see[HHR16] or [HK01]). We localize at p = 2 . The spectrum MU (2) is a C -equivariant com-mutative ring, as shown in [HH14]. By [HK01], the forgetful map ρ : π ∗ (1+ α ) ( MU (2) ) −→ π ∗ ( M U (2) ) is an isomorphism. The classes v i (Araki, Hazewinkel, or others) in π ∗ ( M U (2) ) may nowbe lifted via ρ − to equivariant classes, ρ − ( v i ) . While in the rest of the manuscript, weabuse notation by denoting ρ − ( v i ) by v i , in the following lemma, we will distinguish be-tween the nonequivariant v i and the equivariant ρ − ( v i ) . Note that for i ≤ n and the Araki v i , the images of these lifts ρ − ( v i ) in the coefficients of E ( n ) are exactly the equivariant v i we have been working with throughout.Our next step is to invert ρ − ( v n ) . Lemma 4.2.
The spectrum ( ρ − ( v n )) − MU (2) is a C -equivariant commutative ring.Proof. We apply Proposition 4.1 above (quoted from [HHR16]). The map i ∗ H is exactly ρ , and so ρ ( ρ − ( v n )) = v n ∈ π n − ( M U (2) ) . We need to show that N C { e } ( v n ) divides apower of ρ − ( v n ) . In fact, we claim that N C { e } ( v n ) = − [ ρ − ( v n )] . To see this, we apply theisomorphism ρ to both sides. Let c denote the action of the generator of C . Using thefact that c ( v n ) = − v n , the double coset formula (see e.g. Proposition 10.9(v) in [Sch] or[May96]) reduces in our case to ρ ◦ N C { e } ( v n ) = v n · c ( v n ) = − v n , which completes the proof. (cid:3) Remark 4.3.
Let us again denote the spectrum ( ρ − ( v n )) − MU (2) by MU [ v − n ] . This spec-trum serves as a commutative proxy for E ( n ) . Indeed, essentially all prior results of theauthors that hold for E ( n ) extend verbatim to this spectrum. . U NSTABLE PROPERTIES OF THE MULTIPLICATIVE STRUCTURE
We now address the question of the (unstable) multiplicative structure on E ( n ) . We beginwith the following lemma: Lemma 5.1.
There is a unique (homotopy) commutative, associative, and unital equivariant H-ring structure on the infinite loop space of E ( n ) that lifts any fixed H-ring structure on E ( n ) : ˆ µ : E ( n ) × E ( n ) −→ E ( n ) Proof.
Recall that Lemma 2.6 shows that E ( n ) × j has the weak projective property. As E ( n ) is a (homotopy) associative and commutative ring spectrum, we may define themap ˆ µ : E ( n ) × E ( n ) −→ E ( n ) as the preimage of the multiplication on E ( n ) alongthe isomorphism E ( n ) ( E ( n ) × E ( n ) ) ρ ∼ = / / E ( n ) ( E ( n ) × E ( n ) ) The unity, commutativity, and associativity of ˆ µ are similarly verified by applying theisomorphism ρ , as the desired relations all hold in E ( n ) -cohomology. Likewise, given achoice of H -ring structure on E ( n ) , the uniqueness of the equivariant lift follows fromthe fact that ρ is an isomorphism. (cid:3) Remark 5.2.
The unstable multiplication ˆ µ we constructed in the previous lemma andthe stable multiplication ˆ µ we constructed in Definition 3.3 are compatible in the sensethat ˆ µ restricts to ˆ µ on the zero space of E ( n ) . To see this, apply the isomorphism ρ andnote that this claim is true nonequivariantly by construction.We now prove our second main result, Theorem 1.2. Proof. (of Theorem 1.2) Let X be a space and consider f ∈ E ( n ) V ( X ) and g ∈ E ( n ) W ( Y ) .We define the product f g ∈ E ( n ) V + W ( X ) as follows. First, note that since E ( n ) is an MU [ v − n ] -module, we may multiply f and g by classes in the coefficients MU [ v − n ] ⋆ . Let k and l be the minimal integers such that v kn f ∈ E ( n ) V ′ ( X ) , v ln g ∈ E ( n ) W ′ ( X ) with V ′ , W ′ ≤ (by this we mean that when we express each representation as a combina-tion of irreducibles, each coefficient should be nonpositive). These classes are representedby maps X v kn f / / E ( n ) V ′ = Ω − V ′ E ( n ) , X v ln g / / E ( n ) W ′ = Ω − W ′ E ( n ) We adjoin the loops over to form classes Σ − V ′ X v kn f / / E ( n ) , Σ − W ′ X v ln g / / E ( n ) ote that since − V ′ , − W ′ ≥ , these are positive suspensions and so the sources of thesemaps are spaces . We may thus smash them together, precompose with the diagonal on X and postcompose with the multiplication on the zero space (which factors through thesmash product) constructed in Lemma 5.1: Σ − V ′ − W ′ X ∆ / / Σ − V ′ X ∧ Σ − W ′ X v kn f ∧ v ln g / / E ( n ) ∧ E ( n ) µ / / E ( n ) This produces a class in E ( n ) V ′ + W ′ ( X ) . Finally, we multiply by v − k − ln to define the product f · g ∈ E ( n ) V + W ( X ) .The unit of this multiplication comes from the unit on MU [ v − n ] . The unity, associativity,and commutativity of this multiplication follow from the corresponding properties of E ( n ) .We have shown that for any space X , E ( n ) ⋆ ( X ) is a graded associative and commuta-tive ring. It remains to show that this is compatible with the multiplication on MU [ v − n ] -cohomology. If we carry out the above construction to define multiplication on MU [ v − n ] ⋆ ( X ) ,it is evident that this agrees with the multiplication coming from the ring spectrum struc-ture on MU [ v − n ] . To see that this multiplication agrees with the one defined on E ( n ) -cohomology, it suffices to check this fact on zero spaces. To see that this diagram MU × MU µ MU / / (cid:15) (cid:15) MU (cid:15) (cid:15) E ( n ) × E ( n ) µ E ( n ) / / E ( n ) commutes, we may use the fact that MU and MU × MU are spaces with weak projec-tive properties to map the diagram isomorphically along ρ where its commutativity isapparent. (cid:3)
6. T HE MR ( n ) ORIENTATION FOR MO [2 n +1 ] REVISITED
Let n > and let MO[2 n +1 ] denote the Thom spectrum for the virtual bundle over BO given by multiplication by n +1 seen as a self-map of BO . In other words, MO[2 n +1 ] is thespectrum that represents real vector bundles ξ endowed with an isomorphism ξ → n +1 ζ for some bundle ζ .In [KW14], the first and third authors showed that MO[2 n +1 ] admits an orientation withrespect to ER ( n ) . However, that proof requires the homotopy commutativity of E ( n ) which is unclear in light of this document. Consequently, in this section, we reproduce theargument in [KW14] to show that MO[2 n +1 ] admits a canonical orientation with respectto the commutative ring spectrum MR( n ) defined as the homotopy fixed points of thespectrum MU [ v − n ] , where it is understood that MU is 2-local. This orientation descendsto an orientation with respect to ER ( n ) , generalizing the ˆ A -genus for real K-theory. We lso take this opportunity to fix an error in the proof of this result given in [KW14], seeRemark 6.7. Theorem 6.1.
The spectrum
MO[2 n +1 ] supports a canonical orientation u n +1 with respect to MR( n ) . Furthermore, given a real vector bundle of the form n +1 ζ , this orientation is uniquelydetermined by the property that the image of u n +1 in MU[ v − n ] is given by u n +1 (2 n +1 ζ ) = µ (2 n ζ ⊗ C ) ∪ ψ ( ζ ⊗ C ) n − , where µ is the usual Thom class from usual complex orientation, and ψ ( ζ ⊗ C ) is the series in MU[ v − n ] ∗ ( BU ) generated from line bundles by: ψ ( x ) = [ − MU ( x ) − x . First, we lay some groundwork.Consider the Real orientation of MU given by a Z / -equivariant map:(6.2) µ : MU (1) −→ Σ (1+ α ) MU . where MU (1) ≃ CP ∞ denotes the C -space in the usual prespectrum defining MU . Weview M U ∗ ( M U (1)) as a rank one free module on µ over M U ∗ ( BU (1)) . We need thefollowing fact regarding the C -action on µ . Lemma 6.3.
Let x denote the first Chern class in M U ( BU (1)) and let ψ ( x ) denote the seriesdefined in Theorem 6.1 above. Then c ( µ ) = µ [ − MU ( x ) − x = µ ψ ( x ) . Proof.
The action of complex conjugation on Σ α MU can be identified with − c (the c from MU and the − from the orientation reversing action on the two sphere). On the otherhand, complex conjugation on MU(1) is induced by the (complex anti-linear) self mapof the universal line bundle γ over BU(1) that sends a vector to its complex conjugate.This map can be seen as a (complex linear) isomorphism from γ to γ , where γ is theopposite complex structure on the real bundle underlying γ . Since γ is isomorphic tothe dual bundle γ ∗ , we see that the action of complex conjugation on MU(1) sends theThom class µ ∈ MU (MU(1)) to the class [ − MU ( µ ) . MU(1) and
BU(1) are homotopyequivalent and the Thom isomorphism is an MU ∗ (BU(1)) module map so that µ = µ x ,where x = c ( γ ) .From this we have µ k = µ x k − , so any power series P a i µ i +11 can be rewriten as µ P a i x i .Hence [ − MU ( µ ) = µ [ − MU ( x ) x . ncorporating this observation into the Z / equivariance of µ from Equation (6.2) bycomputing on the left and the right, this translates to the equality: c ( µ ) = µ [ − MU ( x ) − x = µ ψ ( x ) . (cid:3) Outline of proof: (of Theorem 6.1) Before beginning the proof of Theorem 6.1 in earnest,we give a brief overview. We will make significant use of the Bockstein spectral sequenceconstructed from the fibration in [KW07a, Theorem 1.6]. Though the results in [KW07a]are stated for ER ( n ) , since the class v n has been inverted, they will apply verbatim to M R ( n ) . In particular, the proof of [KW07a, Theorem 1.6] shows that there is a Bock-stein spectral sequence E r (MO[4]) , starting with the MU[ v − n ] -cohomology of the spectrum MO[4] and converging to the
M R ( n ) -cohomology of MO[4] . The proof of Theorem 6.1 issomewhat technical and assumes familiarity with the properties of this spectral sequence,which may be found summarized in [KW14, Theorem 2.1]. In particular, note that thespectral sequence collapses at the E n +1 page. The final part of the proof will make useof the structure of the spectral sequence for a point and for BO . These are described inSection 3 and 5 of [KW14], respectively.The orientation u n +1 will be constructed inductively, starting with a class u ∈ MU[ v − n ] ∗ (MO[4]) .Once the class u is constructed, we will inductively define u k +1 in terms of u k (see Equa-tion 6.4 below) and begin our analysis of the Bockstein spectral sequence differentials.We will show that for < k < n + 1 that the class u k survives to the E k − page andthat E k − (MO[2 k ]) is a rank one free module over the E k − (BO) (the E k − page of thespectral sequence for BO ) on the distinguished generator u k . Continuing in this way, wewill conclude that u n +1 survives to the last stage E n +1 − where there is one last possibledifferential left in the spectral sequence. We will show that this differential on u n +1 mustbe zero, and thus M R ( n ) ∗ (MO[2 n +1 ]) is a rank one free module over M R ( n ) ∗ ( BO ) , whichwill establish the theorem. Proof. (of Theorem 6.1) We start by defining a class u . Let ζ denote the universal real vec-tor bundle over BO , ζ C the universal complex vector bundle over BU , and ζ C its conjugate.Consider the composite BU ∆ / / BU × BU / / BU classifying the bundle ζ C ⊕ ζ C over BU . Precomposing with the complexification map BO −→ BU , we have that ζ C ⊕ ζ C pulls back to ζ . Taking Thom spectra and mappinginto MU[ v − n ] yields the composite MO[4] −→ MU ∧ MU −→ MU[ v − n ] here the second map is induced by the twisted multiplication map (the counit of thenorm-forgetful adjunction) m ◦ ( id ∧ c ) : MU ∧ MU −→ MU . This is an
MU[ v − n ] -Thom class for ζ , and we define it to be u (4 ζ ) . We have u (4 ζ ) = µ (( ζ ⊗ C ) ⊕ ( ζ ⊗ C )) = µ ( ζ ⊗ C ) µ ( ζ ⊗ C ) Reasoning from line bundles using Lemma 6.3 and applying the splitting principle showsthat µ ( ζ ⊗ C ) = µ ( ζ ⊗ C ) ψ ( ζ ⊗ C ) It follows that u (4 ζ ) = µ (2 ζ ⊗ C ) ∪ ψ ( ζ ⊗ C )) . and that u is c -invariant.We now begin our investigation of the Bockstein spectral sequence. Since u is c -invariant,it follows that d ( u ) = 0 in the Bockstein spectral sequence converging to MR( n ) ∗ (MO[4]) .We also have d ( u ) = 0 for degree reasons, so that d r ( u ) = 0 for r < will beginour induction. If some Thom class u r (to be defined inductively in Equation 6.4 below)survives to E r +1 , notice that E r +1 for the Thom space is a rank one free module overE r +1 (BO) on the Thom class. We will show for < k < n + 1 that E k − (MO[2 k ]) is a rankone free module over E k − (BO) on a distinguished generator u k .By induction, assume this is true for k > . Writing k +1 ζ as k ζ ) , we have a factoriza-tion: [2 k +1 ] = ([2 k ] + [2 k ]) ◦ ∆ : BO −→ BO × BO −→ BO , which induces a map of Thom spectra: τ = m ◦ ∆ : MO[2 k +1 ] −→ MO[2 k ] ∧ MO[2 k ] −→ MO[2 k ] . This induces a map of spectral sequences τ ∗ : E k − (MO[2 k ]) −→ E k − (MO[2 k +1 ]) . Define(6.4) u k +1 := τ ∗ ( u k ) . Observe that since c ( u k +1 ) = u k +1 by naturality and the fact that c ( u k ) = u k by induction(the observation that u is c -invariant above started the induction). By the properties ofthe Bockstein spectral sequence, we have: d k − ( u k +1 ) = ∆ ∗ µ ∗ d k − ( u k ) = ∆ ∗ d k − µ ∗ ( u k ) = ∆ ∗ ( d k − ( u k ) ∧ u k + u k ∧ d k − ( u k )) . But notice that ∆ : MO[2 k +1 ] −→ MO[2 k ] ∧ MO[2 k ] is invariant under the swap mapon MO[2 k ] ∧ MO[2 k ] . Therefore ∆ ∗ ( d k − ( u k ) ∧ u k ) = ∆ ∗ ( u k ∧ d k − ( u k )) . It follows that d k − ( u k +1 ) is a multiple of and must consequently be zero since E k − (MO[2 k +1 ]) isa Z / -module for external degrees greater than zero . It follows that u k +1 survives toE k (MO[2 k +1 ]) .We now need to show that d k + r ( u k +1 ) = 0 by induction on r , for ≤ r < k − . Herewe use a degree argument that relies on the structure of the Bockstein spectral sequenceas described in Section 5 of [KW14] and we import our notation from there, in particular Proposition 6.5 below shows that d k − ( u k ) is in fact non-trivial for k ≤ n he ‘hatted’ classes below. We write the Bockstein spectral sequence for BO as a tensorproduct of a ring of permanent cycles and the Bockstein spectral sequence for a point, asin [KW14]: E r (BO) = ˆMU[ˆ v − n ] ∗ (BO) ⊗ ˆMU[ˆ v − n ] ∗ E r ( pt ) By construction, all classes in ˆMU[ v − n ] ∗ (BO) have internal degree divisible by n +2 . Onthe other hand, the differentials longer than d k − have external degree larger than k − ,and hence represent elements divisible by x k − .Using the structure of the spectral sequence for a point, the domain of these differentialsis generated by the classes y, ˆ v i,l := ˆ v i v l i +1 n and v ± k n for i > k − , where y is the permanentcycle representing the nilpotent class x . All of these classes have internal degree divisibleby k +1 . Therefore, for dimensional reasons, there can be no differentials in this spectralsequence that land in internal degree between k and k +1 , until we reach the differential d k +1 − .Continuing in this way, we notice that u n +1 survives until the last stage E n +1 − (MO[2 n +1 ]) .Now consider d n +1 − ( u n +1 ) in degree n +1 − λ . The image of this differential landsinside a subquotient of the group MU[ v − n ] ∗ (MO[2 n +1 ]) , which we henceforth identify (us-ing the Thom isomorphism) with the group u n +1 MU[ v − n ] ∗ (BO) . Furthermore, classesthat have survived past E n − must belong to the Z / v , . . . , ˆ v n − , v ± n n ] -submodule of u n +1 MU[ v − n ] ∗ (BO) generated by u n +1 ˆMU[ v − n ](BO) , modulo previous differentials. Thisallows us to express d n +1 − ( u n +1 ) as: d n +1 − ( u n +1 ) = v n n u n +1 w, where w ∈ MU[ v − n ] ∗ (BO) h ˆ v , . . . , ˆ v n − i , for some permanent cycle w . Furthermore, we know that d n +1 − ( u n +1 ) = 0 . Applyingthis to the above expression and using the derivation property, we see that: v n +1 n w = v n +1 − n n ˆ v n w. Replacing w with v − n n ˆ w for some (unique) element ˆ w ∈ MU[ v − n ] (1 − n )(1 − λ ) (BO) , we ob-tain the relation ˆ w = ˆ v n ˆ w . This implies that ˆ w (1 − ˆ v − n ˆ w ) = 0 . Since (1 − ˆ v − n ˆ w ) is a unit,we see that ˆ w = 0 . In other words, u n +1 survives the differential d n +1 − . The proof ofTheorem 6.1 is complete on observing that the spectral sequence collapses at E n +1 (by[KW14, Theorem 2.1(iv)]. (cid:3) The following proposition shows that we cannot expect to do better:
Proposition 6.5.
There exists a vector bundle ζ such that n ζ is not ER ( n ) -orientable. In par-ticular, the differential d k − in the Bockstein spectral sequence converging to MR( n ) ∗ (MO[2 k ]) isnontrivial on the generator u k for k ≤ n .Proof. Note that if d k − was trivial on u k for some k ≤ n , the proof of the above theoremwould show that MO[2 m ] was MR( n ) -orientable for some m ≤ n . Let us demonstrate acontradiction under that hypothesis by showing the existence of a vector bundle ζ such hat n ζ is not ER ( n ) -orientable. Let S α denote the one point compactification of the signrepresentation of Z / . Consider the virtual vector bundle ζ over B Z / with Thom spacegiven by: T h ( ζ ) = E Z / + ∧ Z / S ( α − . If n ζ were to admit an MR( n ) -orientation, then one would obtain a map representing theThom class µ : µ : E Z / + ∧ Z / S n ( α − −→ M R ( n ) . Postcomposing with the inclusion of fixed points map
M R ( n ) −→ MU [ v − n ] (where weview M R ( n ) as a C -spectrum with trivial action) and precomposing with the map to theorbits, we have the composite E Z / + ∧ S n ( α − −→ E Z / + ∧ Z / S n ( α − −→ M R ( n ) −→ MU [ v − n ] . which is a C -equivariant map (with trivial actions on the middle two terms). Takingadjoints, we obtain a map S n ( α − −→ F ( E Z / + , MU [ v − n ]) ≃ MU [ v − n ] , where we have used the fact that MU [ v − n ] is cofree. Since it comes from the Thom class,this map must represent a unit in π Z / n ( α − MU [ v − n ] . However, the computation of thebigraded homotopy of MU [ v − n ] given in [HK01] shows that there is no such class. Hencewe obtain a contradiction to the existence of µ . (cid:3) Remark 6.6.
As mentioned earlier, Theorem 6.1 descends to an ER ( n ) -orientation for MO[2 n +1 ] . However, this is by no means optimal. For example, for n = 1 we knowthat ER (1) is 2-localized real K -theory. Hence a bundle ξ is ER (1) -orientable if and onlyif it is Spin . This is equivalent to w ( ξ ) = w ( ξ ) = 0 . This holds for bundles of theform ξ = 4 ζ , but clearly there are Spin bundles that are not divisible by . Similarly,for ER (2) , the results of [KS04] suggest that a bundle ξ is ER (2) -orientable if and onlyif w ( ξ ) = w ( ξ ) = w ( ξ ) = 0 , which is clearly true for bundles of the form ξ = 8 ζ . Itis a compelling question to find a nice answer in general for when a bundle is ER ( n ) -orientable, or even to show that an answer to this question may be given in closed form. Remark 6.7.
The above Theorem 6.1 corrects an error given in the proof of [KW14, The-orem 1.4]. The induction process for the construction of u n +1 in [KW14] began with aclass in the Bockstein spectral sequence converging to ER ( n ) ∗ (MO[2]) . Unfortunately,that class turns out to not be conjugation invariant as required. Our current argumentstarts with a manifestly invariant class in the Bockstein spectral sequence converging to ER ( n ) ∗ (MO[4]) and use that to generate the other permanent cycles. The rest of the argu-ment is essentially the same as in [KW14].R EFERENCES [AM78] Sh ˆor ˆo Araki and Mitutaka Murayama. τ -cohomology theories. Japan. J. Math. (N.S.) , 4(2):363–416,1978.[Ban13] Romie Banerjee. On the ER (2) -cohomology of some odd-dimensional projective spaces. TopologyAppl. , 160(12):1395–1405, 2013. Fuj76] Michikazu Fujii. Cobordism theory with reality.
Math. J. Okayama Univ. , 18(2):171–188, 1975/76.[HH14] M. A. Hill and M. J. Hopkins. Equivariant multiplicative closure. In
Algebraic topology: applicationsand new directions , volume 620 of
Contemp. Math. , pages 183–199. Amer. Math. Soc., Providence,RI, 2014.[HHR16] M. A. Hill, M. J. Hopkins, and D. C. Ravenel. On the nonexistence of elements of Kervaire invari-ant one.
Ann. of Math. (2) , 184(1):1–262, 2016.[HK01] Po Hu and Igor Kriz. Real-oriented homotopy theory and an analogue of the Adams-Novikovspectral sequence.
Topology , 40(2):317–399, 2001.[HM17] Michael Hill and Lennart Meier. The C2–spectrum Tmf1(3) and its invertible modules.
Algebr.Geom. Topol. , 17(4):1953–2011, 2017.[HS17] Jeremy Hahn and Xiaolin Danny Shi. Real orientations of Morava E -theories. arXiv:1707.03413 ,2017.[JW73] David Copeland Johnson and W. Stephen Wilson. Projective dimension and Brown-Peterson ho-mology. Topology , 12:327–353, 1973.[KLW16] Nitu Kitchloo, Vitaly Lorman, and W. Stephen Wilson. Landweber flat real pairs and ER ( n ) -cohomology. arXiv:1603.06865, 2016.[KLW17] Nitu Kitchloo, Vitaly Lorman, and W. Stephen Wilson. The ER (2) -cohomology of B Z / (2 q ) and C P n . Canadian Journal of Mathematics , 2017.[KS04] Igor Kriz and Hisham Sati. M-theory, type IIA superstrings, and elliptic cohomology.
Adv. Theor.Math. Phys. , 8(2):345–394, 2004.[KW07a] Nitu Kitchloo and W. Stephen Wilson. On fibrations related to real spectra. In
Proceedings of theNishida Fest (Kinosaki 2003) , volume 10 of
Geom. Topol. Monogr. , pages 237–244. Geom. Topol. Publ.,Coventry, 2007.[KW07b] Nitu Kitchloo and W. Stephen Wilson. On the Hopf ring for ER ( n ) . Topology Appl. , 154(8):1608–1640, 2007.[KW08a] Nitu Kitchloo and W. Stephen Wilson. The second real Johnson-Wilson theory and nonimmer-sions of RP n . Homology, Homotopy Appl. , 10(3):223–268, 2008.[KW08b] Nitu Kitchloo and W. Stephen Wilson. The second real Johnson-Wilson theory and nonimmer-sions of RP n . II. Homology, Homotopy Appl. , 10(3):269–290, 2008.[KW13] Nitu Kitchloo and W. Stephen Wilson. Unstable splittings for real spectra.
Algebr. Geom. Topol. ,13(2):1053–1070, 2013.[KW14] Nitu Kitchloo and W. Stephen Wilson. The ER ( n ) -cohomology of BO ( q ) , and Real Johnson-Wilson orientations for vector bundles. arXiv:1409.1281 , 2014.[Lan68] Peter S. Landweber. Conjugations on complex manifolds and equivariant homotopy of M U . Bull.Amer. Math. Soc. , 74:271–274, 1968.[Lor16] Vitaly Lorman. The Real Johnson–Wilson cohomology of CP ∞ . Topology Appl. , 209:367–388, 2016.[May96] J. P. May.
Equivariant homotopy and cohomology theory , volume 91 of
CBMS Regional Conference Seriesin Mathematics . Published for the Conference Board of the Mathematical Sciences, Washington,DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole,G. Comeza ˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza,G. Triantafillou, and S. Waner.[Sch] Stefan Schwede. Lectures on equivariant stable homotopy theory. Available at . EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
ALTIMORE , USA
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF R OCHESTER , R
OCHESTER , USA
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , J
OHNS H OPKINS U NIVERSITY , B
ALTIMORE , USA
E-mail address : [email protected]@math.jhu.edu