Additive power operations in equivariant cohomology
Peter J. Bonventre, Bertrand J. Guillou, Nathaniel J. Stapleton
aa r X i v : . [ m a t h . A T ] J a n ADDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY
PETER J. BONVENTRE, BERTRAND J. GUILLOU, AND NATHANIEL J. STAPLETON
Abstract.
Let G be a finite group and E be an H ∞ -ring G -spectrum. For any G -space X and positive integer m , we give an explicit description of the smallest Mackey ideal J in E ( X × B Σ m ) for which the reduced m th power operation E ( X ) Ð→ E ( X × B Σ m )/ J is a map of Green functors. We obtain this result as a special case of a general theoremthat we establish in the context of G × Σ m -Green functors. This theorem also specializesto characterize the appropriate ideal J when E is an ultra-commutative global ringspectrum. We give example computations for the sphere spectrum, complex K -theory,and Morava E -theory. Contents
1. Introduction 21.1. Conventions 41.2. Organization 51.3. Acknowledgments 52. The Borel equivariant case 52.1. Overview 52.2. Stabilizers of elements in ( G / H ) × m J G J GH for normal subgroups H ⊴ G G -Green functor 163.3. Power operations on Green functors 203.4. Power operations on G -spectra 223.5. Power operations on global spectra 234. Examples 244.1. Ordinary cohomology 244.2. The sphere spectrum 244.3. Global KU Date : January 31, 2020.2000
Mathematics Subject Classification.
Key words and phrases. power operations, Mackey functors, equivariant cohomology.Guillou was supported by NSF grant DMS-1710379. Stapleton was supported by NSF grant DMS-1906236. Introduction An H ∞ -ring structure on a spectrum E gives rise to power operations on the E -cohomologyof any space. These operations have played an important role in both our theoretical andour computational understanding of essentially all naturally-occurring cohomology theories.The most useful power operations, Steenrod operations and Adams operations, are both ad-ditive power operations. Additive power operations are all built from the universal additivepower operation P m ∶ E ( X ) → E ( X × B Σ m )/ I Tr , (1.1)where Σ m is the symmetric group and I Tr is a specific transfer ideal that can be definednaturally for any spectrum E . Given the effectiveness of these additive power operations,it is desirable to understand their analogues in other contexts. In this paper we focus onthe case of equivariant cohomology theories. In particular, in Section 2 we study Borelequivariant cohomology theories, and in Section 3 we tackle genuine and global equivariantcohomology theories.For any finite group G , the G -Mackey functor of coefficients of a genuine G -spectrum E is given by the formula G / H ↦ E ( G / H ) = [ G / H + , E ] G . More generally, given a G -space X , the E -cohomology of X is a G -Mackey functor by theformula G / H ↦ E ( X )( G / H ) = E ( G / H × X ) = [( G / H × X ) + , E ] G If we further assume that E is equipped with the structure of an H ∞ -ring in the categoryof genuine G -spectra, then the associated power operation is a map P m ∶ E ( G / H ) → E ( B Σ m )( G / H ) , where B Σ m is a G -space with trivial action. In this case, the Mackey functors E and E ( B Σ m ) are both G -Green functors, so both E ( G / H ) and E ( B Σ m )( G / H ) are com-mutative rings. The map P m is multiplicative, but not additive, and it does not respectthe induction maps in the G -Mackey functors. The additivity of P m reduces to a classi-cal problem, which was solved for spectra in complete generality. [BMMS86, PropositionVIII.1.4(iv)] identifies an ideal I Tr ⊆ E ( B Σ m )( G / H ) , generated by the image of the transfermaps E ( B Σ i × Σ j )( G / H ) → E ( B Σ m )( G / H ) for i, j > i + j = m , with the propertythat the composite P m / I Tr ∶ E ( G / H ) → E ( B Σ m )( G / H ) → E ( B Σ m )( G / H )/ I Tr is a map of commutative rings that respects the restriction maps in the G -Mackey functorstructure. However, these maps do not necessarily respect the induction maps in the G -Mackey functor structure. The goal of this paper is to identify and study the minimalMackey ideal J ⊆ E ( B Σ m ) so that the composite P m / J ∶ E → E ( B Σ m ) → E ( B Σ m )/ J is a map of Green functors.The ideal J ( G / H ) ⊆ E ( B Σ p )( G / H ) is built out of transfer maps. Given a surjectivemap of finite G × Σ m -sets X → Y , applying homotopy orbits for the Σ m -action gives a coverof G -spaces X h Σ m → Y h Σ m . This gives rise to a transfer map in E -cohomologyTr ∶ E ( X h Σ m ) → E ( Y h Σ m ) . If Y = G / H , with trivial Σ m -action, then the target of this transfer map is E ( G / H × B Σ m ) = E ( B Σ m )( G / H ) . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 3
In the case that m = p is a prime, the ideal J ( G / H ) ⊆ E ( B Σ p )( G / H ) is defined to bethe ideal generated by the transfer maps induced by the maps of G × Σ m -sets(i) ( G × Σ p )/( H × Σ i × Σ j ) → G / H for i + j = p and i, j > ( G × Σ p )/ Γ ( a ) → G / H , for all subgroups S ≤ H and homomorphisms a ∶ S → Σ p withimage containing a p -cycle, where Γ ( a ) ⊆ G × Σ p is the graph subgroup of a .By construction, I Tr is contained in J ( G / H ) , and J is natural in the cohomology theory E . The following result is the special case of the main theorems of this paper when E is an H ∞ -ring G -spectra and when m = p . It makes use of Proposition 3.22 and is a special caseof Theorem 3.24 and Corollary 3.35. Proposition.
Assume that E is an H ∞ -ring in genuine G spectra. The ideals J ( G / H ) ,defined above, assemble to a Mackey ideal J ⊆ E ( B Σ p ) , minimal with the property that thecomposite P p / J ∶ E → E ( B Σ p )/ J is a map of G -Green functors. In fact, this result holds much more generally. Let R be a G × Σ m -Green functor. From R we may form the induced G -Green functor R ↑ GG × Σ m given by the formula R ↑ GG × Σ m ( G / H ) = R (( G × Σ m )/( H × Σ m )) . As R is a G × Σ m -Green functor, it may be viewed as a functor from the category of finite G × Σ m -sets to commutative rings that admits transfers along surjections. In this situation,we define J ( G / H ) ⊆ R ↑ GG × Σ m ( G / H ) to be the ideal generated by the images of certaintransfer maps generalizing the maps in (i) and (ii). These maps are described explicitly inSection 3.2 and make use of a group extension of Γ ( a ) by a product of symmetric groupsdescribed in Section 2.2. The following result is Theorem 3.24. Theorem.
Let R be a G × Σ m -Green functor. The ideals J ( G / H ) ⊆ R ↑ GG × Σ m ( G / H ) assembleinto a G -Mackey ideal J ⊆ R ↑ GG × Σ m . When E is a homotopy commutative ring G -spectrum, we get a G × Σ m -Green functor R via the formula R (( G × Σ m )/ Λ ) = E ((( G × Σ m )/ Λ ) h Σ m ) . In this case, the restriction of R to G satisfies R ↓ G × Σ m G = E , the induced G -Green functorsatisfies R ↑ GG × Σ m = E ( B Σ m ) and, when m = p , the Mackey ideals called J in the propositionand theorem above agree.Global equivariant homotopy theory furnishes us with further examples of G × Σ m -Greenfunctors. If E is a homotopy commutative ring global spectrum in the sense of [Sch18],there is an associated G × Σ m -functor R given by R (( G × Σ m )/ Λ ) = π Λ0 E. In this case, the induced G -Green functor is given by R ↑ GG × Σ m ( G / H ) = π H × Σ m E. The above theorem furnishes us with a Mackey ideal J ⊆ R ↑ GG × Σ m . The restricted G -Greenfunctor satisfies R ↓ G × Σ m G ( G / H ) = π H E .In case E is either an H ∞ -ring in genuine G -spectra or an ultra-commutative globalspectrum, then the m th power operation is a map P m ∶ R ↓ G × Σ m G ( G / H ) → R ↑ GG × Σ m ( G / H ) . P. J. BONVENTRE, B. J. GUILLOU, AND N. J. STAPLETON
In Section 3.3, we introduce the notion of an m th total power operation on a G × Σ m -Greenfunctor that captures the m th power operation in each of these examples. Although boththe source and target of P m are Green functors, the operation P m is not a map of Greenfunctors before passing to a quotient. The proposition above is then a special case of thefollowing theorem (see Corollary 3.35 and Corollary 3.41). Theorem.
Let E be an H ∞ -ring in genuine G -spectra or an ultra-commutative global spec-trum and let R be the associated G × Σ m -Green functor defined above. The composite P m / J ∶ R ↓ G × Σ m G Ð→ R ↑ GG × Σ m Ð→ R ↑ GG × Σ m / J is a map of G -Green functors. The general case of a G × Σ m -Green functor with m th power operation is treated inTheorem 3.30.Since Borel equivariant cohomology theories are examples of genuine equivariant coho-mology theories, if E is an ordinary H ∞ -ring spectrum, then the proposition above maybe applied to the Borel equivariant cohomology theory associated to E . However, in thissetting, it is also natural to ask for the smallest ideal with the property that the transferfrom a specific subgroup commutes with the power operation after taking the quotient bythe ideal. If H ⊆ G , then BH → BG is equivalent to a finite cover of spaces. We would likethe smallest ideal J GH ⊆ E ( BG × B Σ m ) such that the following diagram commutes E ( BH ) / / Tr (cid:15) (cid:15) E ( BH × B Σ m )/ I TrTr (cid:15) (cid:15) E ( BG ) / / E ( BG × B Σ m )/ J GH . Specializing the case of genuine G × Σ m -spectra to Borel equivariant G × Σ m -spectra, the G × Σ m -Green functor associated to E is given by the formula R (( G × Σ m )/ Λ ) = E ((( G × Σ m )/ Λ ) hG × Σ m ) ≅ E ( B Λ ) and R ↑ GG × Σ m ( G / H ) ≅ E ( BH × B Σ m ) . For the case where H is a normal subgroup of G , inSection 2.4 we explicitly describe a subset of the transfer maps that go into the constructionof J ( G / G ) ⊆ R ↑ GG × Σ m ( G / G ) and show that J GH is the sub-ideal generated by the image ofthis subset. As a consequence of this description of J GH , we learn that, when H is normal, if m and ∣ G / H ∣ are relatively prime, then J GH = I Tr ⊆ E ( BG × B Σ m ) .1.1. Conventions. ● By G , we will always mean a finite group. ● By a graph subgroup Γ ≤ G × Σ m , we will mean a subgroup such that Γ ∩ Σ m = { e } .Such a subgroup is the graph of a homomorphism K Ð→ Σ m , where K = π G ( Γ ) and π G ∶ G × Σ m Ð→ G is the projection. ● We will use the notation n = { , . . . , n } . ● A G -spectrum will always mean in the “genuine” sense. In other words, our G -spectra are indexed over a complete G -universe. ● We will use transfer maps in cohomology throughout, and in order to help orientthe reader, we always display such transfer maps in orange. ● In Section 3.2, we abbreviate an induced Mackey functor R ↑ GG × Σ m to R ↑ . ● In Section 4, we abbreviate an induced Mackey functor ( A G × Σ m ) ↑ GG × Σ m to A ↑ GG × Σ m . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 5
Organization.
We begin Section 2 by considering the Borel equivariant case. Keyresults about the ideal J are given in Section 2.3 and Section 2.4; these results are specializedto the case m = p is prime in Section 2.5. Our main results about power operations appearin Section 3. We introduce the notion of an m th total power operation for a G × Σ m -Greenfunctor in Section 3.3. One of the central results of the article, Theorem 3.24, is that J isa Mackey ideal. Section 4 gives a number of examples. We consider the sphere spectrum, KU -theory, Eilenberg-Mac Lane spectra, and height 2 Morava E -theory.1.3. Acknowledgments.
It is a pleasure to thank Sune Precht Reeh and Tomer Schlankfor helpful comments. 2.
The Borel equivariant case
The purpose of this section is to understand the relationship between the additive poweroperations for an H ∞ -ring spectrum E and transfers along finite covers of the form BH → BG for H < G a subgroup. In particular, the goal is to describe, as explicitly as possible,the smallest ideal J GH ⊂ E ( BG × B Σ m ) containing I Tr such that the diagram E ( BH ) E ( BH × B Σ m )/ I Tr E ( BG ) E ( BG × B Σ m )/ J GHP m / I Tr P m / J GH (2.1)commutes and the horizontal maps are additive, so that in particular this a commutingsquare of ring maps.We will also describe the absolute ideal J G ⊆ E ( BG × B Σ m ) , which is the sum of theideals J GH as H varies. In terms of the notation from Section 1, J G is what was denotedthere as J ( G / G ) . This provides the smallest ideal such that the reduced power operation P m / J G commutes with transfers BH → BG for all subgroups H ≤ G .In Section 2.1, we outline this story. In the remaining subsections, we describe explicitlythe ideals J GH and J G in various cases. First, in Section 2.2, we study the stabilizers forthe G × Σ m -action on ( G / H ) m for H ≤ G and m ≥
0. In Section 2.3, we provide an explicitdescription of J G . In Section 2.4, we provide an explicit description of J GH in the case where H ⊴ G is a normal subgroup. In Section 2.5, we provide a more concrete description in thecase that m is prime. Finally, in Section 2.6, we consider the simpler case where m and ∣ G ∣ are relatively prime.2.1. Overview.
We have two tasks: first, to ensure that the power operation is additive,and second, to ensure it commutes with transfer maps. As we saw in Section 1, in order forthe power operation to be additive and thus a map of commutative rings, one must quotientby the ideal generated by transfers along the proper partition subgroups G × Σ i × Σ j ⊂ G × Σ m ;that is, we must have I Tr ⊆ J GH . However, this ideal is not necessarily sufficient to makediagram (2.1) commute, as we demonstrate in Section 4. The problem boils down to therelationship between the transfer along the inclusion H ≀ Σ m ⊆ G ≀ Σ m and the diagonal map G × Σ m → G ≀ Σ m . We are particularly interested in studying the power operation thatlands in the product because of the relationship to power operations for genuine equivariantcohomology theories. P. J. BONVENTRE, B. J. GUILLOU, AND N. J. STAPLETON
In [BMMS86], the power operation P m ∶ E ( BH ) → E ( BH × B Σ m ) is defined as a com-posite P m ∶ E ( BH ) P m ÐÐ→ E ( BH ≀ Σ m ) ∆ ∗ Ð→ E ( BH × B Σ m ) where P m is the total power operation . The operation P m is functorial on all stable maps,and thus every subgroup H ⊆ G gives rise to a commutative diagram E ( BH ) E ( BH ≀ Σ m ) E ( BG ) E ( BG ≀ Σ m ) . P m Tr Tr P m (2.2)After composing the bottom arrow with the map in E -cohomology induced by the diagonal BG × B Σ m → BG ≀ Σ m , we may extend the diagram above by considering a homotopypullback. We may do this by making use of the fact (see [Ada78, Chapter 4], for instance)that, given a homotopy pullback of spaces Y (cid:15) (cid:15) H o o (cid:15) (cid:15) X A o o in which Y → X is a finite cover, there is a commutative diagram E ( Y ) E ( H ) E ( X ) E ( A ) , Tr Tr (2.3)where the horizontal maps are restriction maps and the vertical maps are transfer maps. Forany subgroups
H, K ⊆ G , the homotopy pullback of the span BH → BG ← BK is equivalentto ( G / H ) hK .Making use of the isomorphism of Σ m × G -sets ( G ≀ Σ m ) / ( H ≀ Σ m ) ≅ ( G / H ) × m , we get the following proposition: Proposition 2.4.
We have the following homotopy pullback of spaces BH ≀ Σ m ( G / H ) × mh ( G × Σ m ) BG ≀ Σ m BG × B Σ m . Applying E -cohomology and composing the resulting diagram (2.3) with the total poweroperation diagram (2.2) gives the commutative diagram E ( BH ) E (( G / H ) × mh ( G × Σ m ) ) E ( BG ) E ( BG × B Σ m ) . Tr Tr P m (2.5) DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 7
The subset ∆ ( G / H ) ∶ = {( gH, . . . , gH ) ∣ g ∈ G } ⊂ ( G / H ) × m is closed under the action of G × Σ m , and there is an equivalence∆ ( G / H ) h ( G × Σ m ) ≃ BH × B Σ m . Thus we have a decomposition of spaces ( G / H ) × mh ( G × Σ m ) ≃ ( BH × B Σ m ) ∐ Z G,Hh ( G × Σ m ) , where Z G,H = ( G / H ) × m ∖ ∆ ( G / H ) . (2.6)Applying E -cohomology, there is an isomorphism E (( G / H ) × mh ( G × Σ m ) ) ≅ E ( BH × B Σ m ) × E ( Z G,Hh ( G × Σ m ) ) . (2.7)We can obtain E ( BH × B Σ m )/ I Tr from this product by taking the quotient by the idealgenerated by transfers along H × Σ i × Σ j ⊆ H × Σ m for i, j > i + j = m and also thetransfer along the component Z G,Hh ( G × Σ m ) ⊆ ( G / H ) × mh ( G × Σ m ) (ie. the entire right factor). Wethus make the following definition. Definition 2.8.
Define J GH ⊆ E ( BG × B Σ m ) to be the ideal generated by the image of thetransfers along(i) G × Σ i × Σ j ⊆ G × Σ m for i, j > i + j = m , and(ii) the composite Z G,Hh ( G × Σ m ) ⊆ ( G / H ) × mh ( G × Σ m ) → BG × B Σ m . (2.9)The following result is then immediate from the above discussion. Proposition 2.10.
Let J GH ⊆ E ( BG × B Σ m ) be the ideal defined above. After taking thequotient by J GH , the transfer and additive power operation are compatible in the sense thatthe following diagram commutes: E ( BH ) E ( BH × B Σ m )/ I Tr E ( BG ) E ( BG × B Σ m )/ J GH . Tr P m / I Tr Tr P m / J GH Proof.
Consider the commutative diagram (2.5). According to (2.7), the top right vertexdecomposes as a product, one factor of which is the desired E ( BH × B Σ m ) . Thus in order forthe right vertical transfer in (2.5) to factor through the projection onto E ( BH × B Σ m )/ I Tr ,we must collapse the image in E ( BG × B Σ m ) of the complementary factor E ( Z G,Hh ( G × Σ m ) ) and I Tr ; these desiderata motivated the definition of J GH . (cid:3) We give a complete description of J GH in the case where H ⊴ G is a normal subgroup: thefollowing is a direct consequence of Proposition 2.34. See Notation 2.19 for a description ofthe group Σ q ≀ n Γ ( a S / H ) . Theorem 2.11.
Fix a normal subgroup H ◁ G . Then J GH ⊆ E ( BG × B Σ m ) is the idealgenerated by I Tr and the images of the transfers along Σ q ≀ n Γ ( a S / H ) Ð→ G × Σ m (2.12) for all m = nq and H < S ≤ G with [ S ∶ H ] = n ≠ , where a S / H ∶ S → Aut
Set ( S / H ) ≅ Σ n isthe action map by left multiplication. P. J. BONVENTRE, B. J. GUILLOU, AND N. J. STAPLETON
Note that although the definition of a S / H depends on a choice of ordering of S / H , thechoice will not affect the image of the transfer.We also consider the related absolute ideal, to ensure compatibility with transfers fromall subgroups of G . Definition 2.13.
Define J G ⊆ E ( BG × B Σ m ) to be the ideal generated by J GH for all H ≤ G . More explicitly, J G is the ideal generated by the image of the transfers along(i) G × Σ i × Σ j ⊆ G × Σ m for i, j > i + j = m , and(ii) the composites Z G,Hh ( G × Σ m ) ⊆ ( G / H ) × mh ( G × Σ m ) → BG × B Σ m (2.14)for all H < G .Proposition 2.10 implies the following. Corollary 2.15.
Let J G ⊆ E ( BG × B Σ m ) be the ideal defined above. Taking the quotientby J G , the additive power operation is compatible with all transfers in the sense that thefollowing diagram commutes for all H < G : E ( BH ) E ( BH × B Σ m )/ I Tr E ( BG ) E ( BG × B Σ m )/ J G . Tr P m / I Tr Tr P m / J G (2.16)The following description of J G is a consequence of Proposition 2.30. Theorem 2.17.
The ideal J G ⊆ E ( BG × B Σ m ) is generated by I Tr and the images of thetransfers along Σ q ≀ n Γ ( a S / K ) Ð→ G × Σ m (2.18) for all m = nq and K < S ≤ G with [ S ∶ K ] = n ≠ , where a S / K ∶ S → Aut
Set ( S / K ) ≅ Σ n isthe action map by left multiplication. Stabilizers of elements in ( G / H ) × m . Our goal is to understand the ideals J GH and J G appearing in Proposition 2.10 and Corollary 2.15 and defined in Definitions 2.8 and 2.13using two collections of transfers. In general, there can be overlap between these transfersin the following sense: sometimes the map from a component of Z G,Hh ( G × Σ m ) factors through BG × B Σ i × B Σ j for some choice of i and j . It then suffices to describe the components of Z G,Hh ( G × Σ m ) that do not factor through BG × B Σ i × B Σ j for any i, j > i + j = m .To start, we note that Z G,Hh ( G × Σ m ) of (2.9) is equivalent to the disjoint union of classifyingspaces of the form B Λ for Λ ≤ G × Σ m the stabilizer of some element in Z G,H (2.6). Moreover,the associated component of Z G,Hh ( G × Σ m ) does not factor through some BG × B Σ i × B Σ j ifand only if the image π Σ m ( Λ ) ≤ Σ m is a transitive subgroup, where π Σ m ∶ G × Σ m → Σ m isthe projection. Thus, it suffices to analyze the stabilizers of the G × Σ m -action on ( G / H ) × m that have transitive image in Σ m .Elements of the diagonal ∆ ( G / H ) = {( gH, . . . , gH ) ∣ g ∈ G } have the simplest stabilizers:Stab G × Σ m ( gH, . . . , gH ) = gHg − × Σ m . However, the stabilizers of the elements of Z G,H canbe quite complicated.In this section, we establish some group-theoretic results regarding these stabilizers andset up notation for describing these groups in the sections ahead.
DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 9
Notation 2.19.
Given Λ ≤ G × Σ n and a group H , let H ≀ n Λ ≤ G × ( H ≀ Σ n ) denote thepreimage H ≀ n Λ = π − Λ , (2.20)where G × ( H ≀ Σ n ) π Ð→ G × Σ n is the canonical map.Note that the group H ≀ n Λ is isomorphic to H n ⋊ Λ, where Λ acts on H n via its projectionto Σ n . This follows from the canonical isomorphism G × ( H ≀ Σ n ) ≅ H n ⋊ ( G × Σ n ) . Notation 2.21.
Let X be a G -set, and Y ⊆ X a finite subset. We write S Y ≤ G to denotethe set-wise stabilizer of Y , S Y = { s ∈ G ∣ s ⋅ y ∈ Y for all y ∈ Y } . A choice of total ordering Y = { y , y , . . . , y n } induces an associated action map a Y ∶ S Y Ð→ Aut
Set ( Y ) ≅ Σ n . (2.22)Different choices of ordering on Y give conjugate action homomorphisms.In general, S Y , a Y , and ker ( a Y ) can be difficult to compute. We give one primaryexample. Example 2.23.
Let H ≤ G , X = G / H , and Y = { g H, . . . , g n H } ⊆ G / H . Then we have S Y = ⋃ σ ∈ im ( a Y ) n ⋂ i = g σ ( i ) Hg − i and ker ( a Y ) = n ⋂ i = g i Hg − i . If Y = K / H for some H ≤ K ≤ G , then S K / H = K . Indeed, if g ∈ G satisfies g ⋅ eH = kH ,then g = g ⋅ e lies in kH ⊆ K . Lemma 2.24.
Let Y ⊆ X be a finite subset of a G -set, equipped with a total ordering Y = { y , . . . , y n } .(i) Let ⇀ y = ( y , y , . . . , y n ) ∈ X × n . Then Stab G × Σ n ( ⇀ y ) = Γ ( a Y ) , where Γ ( a Y ) is the graph subgroup associated to a Y ∶ S Y Ð→ Σ n .(ii) Let ⇀ y ∗ q = ( y , . . . , y ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ q , y , . . . , y ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ q , . . . , y n , . . . , y n ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ q ) be the q -fold shuffle of ⇀ y . Then the stabilizer of ⇀ y ∗ q is Stab G × Σ qn ( ⇀ y ∗ q ) = Σ q ≀ n Γ ( a Y ) ≤ G × ( Σ q ≀ Σ n ) ≤ G × Σ qn . Proof.
For (i), ( g, σ ) is in Stab ( ⇀ y ) if and only if gy i = y σi for all i . This defines an action of g on Y , and thus g is in S Y and σ = a Y ( g ) .For (ii), it is clear that ( Σ q ) × n stabilizes the q -fold shuffle ⇀ y ∗ q . The group Σ n acts by per-muting the blocks of size q . Given ( g, ( ⇀ τ , σ )) ∈ G × ( Σ q ≀ Σ n ) , we have (( g, ( ⇀ τ , σ )) ⋅ ⇀ y ∗ q ) i + kq = gy σ − k for 1 ≤ i ≤ q . Thus if ( g, ( ⇀ τ , σ )) is in the stabilizer, we must have ( g, σ ) ∈ Γ ( a Y ) ,while the τ i ∈ Σ q have no restrictions. (cid:3) The absolute transfer ideal J G . In Proposition 2.30, we give a complete descriptionof the subgroups of G × Σ m that we must transfer along to form J G . This description willbe used in the proof of Theorem 3.24.First, we establish a restricted case. Let ∆ fat ( G / H ) ⊆ ( G / H ) × n be the fat diagonal,which consists of tuples of cosets in which two or more of the cosets are identical. Proposition 2.25.
Fix n ≥ . Then the assignment of the stabilizer Stab G × Σ n ( − ) admitsa section ζ { ⇀ gH ∈ ( G / H ) × n ∖ ∆ fat ( G / H ) H ≤ G, π Σ n ( Stab ( ⇀ gH )) ≤ Σ n transitive }{ Γ ( φ ) ≤ G × Σ n φ ∶ S → Σ n , S ≤ G, im ( φ ) ≤ Σ n transitive } Stab ζ (2.26) and is therefore surjective.Proof. According to Lemma 2.24(i), the stabilizer is a graph subgroup. Since the cosets g i H are all distinct and G / H is a transitive G -set, the image of φ in Σ n must be a transitivesubgroup.Now, any φ as above encodes a transitive action of S on { , . . . , n } . In particular, letting K = Stab S ( ) , the action provides a bijection S / K ≅ { , . . . , n } , which specifies an orderingof S / K . Thus we may consider S / K as an element of ( G / K ) × n , and the assignment Γ ( φ ) ↦ S / K ∈ ( G / K ) × n is a section of (2.26) by Example 2.23. (cid:3) Remark 2.27.
In light of the description of the section ζ to (2.26) given in the proof above,after passing to Σ n -conjugacy classes, we may replace the target of Stab in (2.26) with { [ Γ ( a S / K ) ≤ G × Σ n ] K ≤ S ≤ G, [ S ∶ K ] = n } , (2.28)where a S / K ∶ S Ð→ Σ n is the action homomorphism specified in (2.22). Note that differentchoices of orderings S / K ≅ Ð→ n induce the same Σ n -conjugacy class [ Γ ( a S / K )] .We note that the source in (2.26) runs over all subgroups of G . One might hope for asimilar result with a fixed H ≤ G . However, K = Stab S ( ) = a − ⇀ gH ( Σ × Σ n − ) = g Hg − ∩ S ⇀ gH need not equal H , as we show in Example 2.29. Therefore, we cannot expect a section ifwe first fix H ≤ G . We will show in Proposition 2.32 that such a section does exist if H isnormal in G . Example 2.29.
Let G = D , generated by a rotation r and a reflection s . Let H = ⟨ rs ⟩ ,and consider ( eH, sH ) ∈ ( G / H ) × . We check by hand that S = ⟨ s ⟩ and K = e , so the section ζ sends Γ ( a ( eH,sH ) ) to the S -set S / e = ( e, s ) .Note in particular that K ≠ H , m = r ∈ H . Thus in general we cannot apriori fix H ≤ G in (2.26).We will now give a complete description of the stabilizers which appear in J G . Recallthat Z G,H is ( G / H ) × m ∖ ∆ ( G / H ) . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 11
Proposition 2.30.
Fix m ≥ . Then the assignment of the stabilizer Stab G × Σ m ( − ) gives asurjection { ⇀ gH ∈ Z G,H H < G, π Σ m ( Stab ( ⇀ gH )) ≤ Σ m transitive }{ [ Σ q ≀ n Γ ( a S / K ) ≤ G × Σ m ] m = nq, K < S ≤ G, [ S ∶ K ] = n ≠ } , Stab (2.31) where Σ q ≀ n Γ ( a S / K ) is defined as in (2.20) and [ Σ q ≀ n Γ ( a S / K )] denotes the Σ m -conjugacyclass of the subgroup in G × Σ m .Proof. Suppose given ⇀ gH in Z G,H ⊂ ( G / H ) × m . For ( g, σ ) ∈ Stab G × Σ m ( ⇀ gH ) , we have g i H = g j H ⇔ g ⋅ g i H = g ⋅ g j H ⇔ g σ ( i ) H = g σ ( j ) H. Since π Σ m ( Stab ( ⇀ gH )) is a transitive subgroup of Σ m , we conclude that after reordering ifnecessary, ⇀ gH is the q -fold shuffle of ( g H, . . . , g n H ) , where the cosets g H , . . . , g n H aredistinct. Using both parts of Lemma 2.24, we see thatStab G × Σ m ( ⇀ gH ) = Σ q ≀ n Stab G × Σ n (( g H, . . . , g n H )) , and so the arrow (2.31) is well-defined.We wish to show that (2.31) is surjective. A choice of a S / K ∶ S → Σ n specifies an orderingof S / K . Given Σ q ≀ n Γ ( a S / K ) , the assignment ζ ∶ Σ q ≀ n Γ ( a S / K ) z→ ( S / K ) ∗ q , is a section of (2.31) by an argument similar to that used in the proof of Proposition 2.25. (cid:3) The relative ideal J GH for normal subgroups H ⊴ G . The relative transfer ideal J GH was defined in Definition 2.8 and appears in Proposition 2.10. Propositions 2.25 and 2.30can be used to obtain a description of the subgroups that we must transfer along to form J GH in the case that H is normal in G . Proposition 2.32.
Fix n ≥ and H ◁ G normal. The map (2.26) restricts to give asurjection { ⇀ gH ∈ ( G / H ) × n ∖ ∆ fat ( G / H ) π Σ n ( Stab ( ⇀ gH )) ≤ Σ n transitive }{ [ Γ ( a S / H ) ≤ G × Σ n ] H ≤ S ≤ G, [ S ∶ H ] = n } , Stab (2.33) where [ − ] denotes the Σ n -conjugacy class of the subgroup in G × Σ n .Proof. Since H is normal, the G -stabilizer of each g i H ∈ G / H is H . Thus, if S denotes theset-wise stabilizer of ⇀ gH ⊂ G / H and K ≤ S denotes the stabilizer in S of g H ∈ G / H , then K = H . Furthermore, since K = H , the section ζ of (2.26) restricts to a section for fixed,normal H . It follows that (2.33) is surjective. (cid:3) The next result is an analogue of Proposition 2.30.
Proposition 2.34.
Fix a normal subgroup H ◁ G . Then the assignment of the stabilizer (2.31) restricts to a surjection { ⇀ gH ∈ Z G,H π Σ m ( Stab ( ⇀ gH )) ≤ Σ m transitive }{ [ Σ q ≀ n Γ ( a S / H ) ≤ G × Σ m ] m = nq, H < S ≤ G, [ S ∶ H ] = n ≠ } Stab (2.35) where Σ q ≀ n Γ ( φ ) is defined as in (2.20) and [ − ] denotes the Σ m -conjugacy class of thesubgroup in G × Σ m . Specializing to a prime.
In this section, we give more explicit identifications of thesubgroups of G × Σ p which appear in J G and J GH , where p is a prime. We identify the relevanttuples in Z G,H (Proposition 2.36) and give closed-form descriptions of their stabilizers.Working at a prime has the advantage that transitive subgroups of Σ p are exactly thosewhich contain a p -cycle σ p . This is an immediate consequence of Cauchy’s theorem.We first classify, for general m , those tuples ⇀ gH ∈ ( G / H ) × m for which π Σ m ( Stab ( ⇀ gH )) contains a long cycle. Proposition 2.36.
Assume that ⇀ gH = ( eH, g H, . . . , g m − H ) , and let σ m = ( . . . m ) bethe long cycle. Then ( g, σ m ) lies in Stab ( ⇀ gH ) if and only if g m ∈ H and ⇀ gH = ( eH, gH, g H, . . . , g m − H ) . Proof.
First we will prove the forward direction. By direct observation, we see g ∈ ( Hg − m − ) ∩ ( g m − Hg − m − ) ∩ . . . ∩ ( g H ) . Thus, there exists h i ∈ H such that g = g i h i g − i − when 0 < i ≤ m (where we set g = g m = e ).With this convention we see that g i H = g i h i g − i − ⋅ g i − h i − g − i − ⋯ g H = g i H. Thus we have that ( g, σ m ) stabilizes ⇀ gH = ( eH, gH, g H, . . . , g m − H ) , so g m ∈ H .For the reverse direction, it suffices to note that under the condition that g m is in H , ( g, σ m ) ⋅ ( eH, gH, g H, . . . , g m − H ) = ( gg m − H, geH, ggH, . . . , gg m − H ) = ( eH, gH, g H, . . . , g m − H ) . (cid:3) Corollary 2.37.
Using the notation of Proposition 2.25, the subgroup π Σ p ( Stab ( ⇀ gH )) ≤ Σ p is transitive if and only if ⇀ gH ∈ ( G / H ) × p lies in the same G × Σ p -orbit as the p -tuple ( eH, gH, g H, . . . g p − H ) for some g ∈ G such that g p ∈ H . Proposition 2.25 then specializes to the following.
Corollary 2.38.
Fix a prime p . Then the assignment of the stabilizer Stab G × Σ p ( − ) givesa surjection { ( g i H ) ∈ Z G,H ⊂ ( G / H ) × p H < G, g p ∈ H, g ∉ H }{ [ Γ ( a S / K ) ≤ G × Σ p ] K < S ≤ G, [ S ∶ K ] = p } , Stab (2.39) where [ − ] denotes the Σ p -conjugacy class of the subgroup in G × Σ p . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 13
Remark 2.40.
The codomain of (2.39) can be described more simply as { [ Γ ( S a Ð→ Σ p ) ≤ G × Σ p . ] S ≤ G, im ( a ) contains a p -cycle } Example 2.29 shows that we still cannot restrict to a fixed H < G on either side forgeneral subgroups H . However, a cleaner description of Γ ( a S / K ) does occur for H normalin G . Notation 2.41.
For ⇀ gH = ( g i H ) ∈ ( G / H ) × p with g p ∈ H and H ◁ G normal, we write a g ∶ S g → Σ m for the action map (so a g = a ( g i H ) , S g = S ( g i H ) ).The following result is a specialization of Proposition 2.32. Corollary 2.42.
Fix a prime p and a normal subgroup H ⊴ G . The assignment ( g i H ) ↦ Stab G × Σ p (( g i H )) = Γ ( a g ) induces a surjection { ( g i H ) ∈ Z G,H ⊂ ( G / H ) × p g p ∈ H, g ∉ H }{ [ Γ ( a S / H ) ≤ G × Σ p ] H < S ≤ G, [ S ∶ H ] = p } . Stab (2.43)
Moreover, the action map a g is precisely a g ∶ ⟨ H, g ⟩ Ð→ C p ≤ Σ p , g i H z→ σ ip with σ p = ( . . . p ) the long cycle.Proof. It remains to describe the action map a g . First, note that, given ( g i H ) , the set-wisestabilizer of ( g i H ) ⊂ G / H is the subgroup ⟨ H, g ⟩ ≤ G . Thus S = S g = ⟨ H, g ⟩ . The formulathen follows from the fact that S / H = ⟨ H, g ⟩/ H is isomorphic to C p . (cid:3) Generally speaking, the action maps a S / K are difficult to understand. However, inCorollary 2.42, we explicitly describe the action map when H is normal in G and m = p isprime. This will be useful later when we compute power operations.2.6. The relatively prime case.
In this section, we record conditions on the integers m , ∣ G ∣ , and ∣ G / H ∣ that force every component of Z G,Hh ( G × Σ m ) to factor through BG × B Σ i × B Σ j for some i, j > i + j = m . Corollary 2.44.
Suppose that m and ∣ G ∣ are relatively prime. Then J G = I Tr .Proof. By Theorem 2.17, it suffices to show that the codomain of (2.31) is empty. Supposenot; then we would have subgroups K < S ≤ G with [ S ∶ K ] ≠ m . But [ S ∶ K ] divides ∣ S ∣ and hence ∣ G ∣ , a contradiction. (cid:3) When H ◁ G is normal, we have the following specification of Corollary 2.44. Corollary 2.45.
Let H ◁ G be normal, and suppose m and ∣ G / H ∣ are relatively prime.Then J GH = I Tr .Proof. Similarly, by Theorem 2.11 it suffices to show that the codomain of (2.35) is empty.Suppose not; then there exists H < S ≤ G such that [ S ∶ H ] is larger than 1 and divides m .But [ S ∶ H ] also divides [ G ∶ H ] = ∣ G / H ∣ , a contradiction. (cid:3) Remark 2.46.
We record that this result fails if H is not a normal subgroup. Consider G = Σ with H = { e, ( )} so that ∣ G / H ∣ =
3, and let m =
2. We note that (( ) , ( )) ∈ G × Σ is in the stabilizer of ( eH, ( ) H ) ∈ ( G / H ) × . Therefore, letting Γ ≤ G × Σ be the ordertwo subgroup generated by the element (( ) , ( )) , the ideal J GH contains the image of thetransfer along Γ Ð→ G × Σ , which is not contained in I Tr .3. Additive power operations and Green functors
Making use of the group-theoretic results in Section 2, we provide in this section a generalframework for additive power operations in the equivariant setting. In Section 3.1 we recallthe notion of a G -Green functor and describe two sources of examples from equivarianthomotopy theory. Motivated by the discussion in Section 2.1, in Section 3.2 we prove thatthe induced G -Green functor associated to a G × Σ m -Green functor contains a canonicalMackey ideal J . In Section 3.3, we introduce the notion of a G × Σ m -Green functor with m th total power operation and show in Theorem 3.30 that taking the quotient by the Mackeyideal J leads to a reduced power operation that is a map of Green functors. In the finaltwo subsections, we show that H ∞ -rings in G -spectra and ultra-commutative ring spectraprovide two classes of examples of G × Σ m -Green functors with m th total power operation.3.1. Reminder on Mackey functors and Green functors.
Let G be a finite group.Recall that a G -Mackey functor M consists of abelian groups M ( G / H ) for each subgroup H ≤ G , together with restriction and induction mapsRes ∶ M ( G / K ) M ( G / H ) and Tr ∶ M ( G / H ) M ( G / K ) for each map G / H → G / K , satisfying a number of axioms. The most notable axiom is thedouble-coset formula, which we describe in Remark 3.1 below.A G -Mackey functor can be extended (uniquely up to canonical isomorphism) to all finite G -sets via M ( A ∐ . . . A n ) = ⊕ i M ( A i ) ≅ ⊕ i M ( G / H i ) for G -orbits A , . . . , A n . Remark 3.1.
The double-coset formula says that for any pullback of G -sets A PC B, in which the vertical maps are surjective, the diagram of abelian groups M ( A ) M ( P ) M ( C ) M ( B ) ResTr TrRes (3.2)commutes.
Definition 3.3. A G -Green functor is a G -Mackey functor R such that each R ( G / H ) is acommutative ring, each restriction map is a ring homomorphism, and each induction mapTr ∶ R ( G / H ) Ð→ R ( G / K ) is an R ( G / K ) -module map. The condition that induction is amodule map is also referred to as “Frobenius reciprocity”. A Mackey ideal in a Greenfunctor is a sub-Mackey functor which is levelwise an ideal.
DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 15
Equivalently, a G -Green functor is a commutative monoid in the category of G -Mackeyfunctors under the box product; see e.g. [Lew, Prop. 1.4], [Shu10, Lemma 2.17]. Example 3.4. If E is a homotopy-commutative ring (genuine) G -spectrum, then the G -Green functor of coefficients is given by E ( G / H ) = E ( G / H ) . The restriction and transfer maps are defined via naturality on the maps of G -sets G / K → G / H and the G -transfers Σ ∞ G G / H + Tr Ð→ Σ ∞ G G / K + . Moreover, the double coset formula (3.2)is a special case of Nishida’s push-pull property for equivariant cohomology [Nis78, Prop.4.4] (see also [LMSM86, IV.1]).We will pay particular attention to G × Σ m -Green functors and to the following associated G -Green functors. Definition 3.5. If R is any G × Σ m -Green functor, we define R ↑ GG × Σ m to be the G -Greenfunctor given by the induction of R along the projection G × Σ m → G (see [TW95, Lemma5.4(ii)]). The value of R ↑ GG × Σ m on G / H is given by R ↑ GG × Σ m ( G / H ) = R (( G × Σ m )/( H × Σ m )) . In other words, given a G -set A , we define R ↑ GG × Σ m ( A ) to be R ( A ) , where in R ( A ) weequip A with a trivial Σ m -action. For example, there is an isomorphism of G × Σ m -sets ( G × Σ m )/( H × Σ m ) ≅ G / H , where G × Σ m acts on G / H through the projection G × Σ m → G . Definition 3.6. If R is any G × Σ m -Green functor, we define R ↓ G × Σ m G to be the restrictionof R along the inclusion G ≅ G × { e } ≤ G × Σ m . Explicitly, R ↓ G × Σ m G ( G / H ) = R (( G × Σ m )/( H × e )) . Example 3.7. If E is a homotopy-commutative ring G -spectrum, then the assignment ( G × Σ m )/ Λ ↦ E ((( G × Σ m )/ Λ ) h Σ m ) , (3.8)for Λ a subgroup of G × Σ m , is a G × Σ m -Green functor. In this case, the induced G -Greenfunctor as in Definition 3.5 is given by E ( B Σ m ) , or explicitly the assignment G / H ↦ E ( G / H × B Σ m ) , (3.9)where B Σ m has a trivial G -action. The restricted G -Green functor as in Definition 3.6 isnaturally isomorphic to E , as G / H ↦ E (( G / H × Σ m ) h Σ m ) = E ( G / H × E Σ m ) ≅ E ( G / H ) . (3.10) Example 3.11. If E is a homotopy commutative global ring spectrum, there is an associatedglobal Green functor E ([Sch18, Definition 5.1.3, Theorem 5.1.11], [Gan13, Definition 3.1]).By [Gan13, Lemma 2.10], this gives a G -Green functor E G and a G × Σ m -Green functor E G × Σ m (see Section 3.5). In this case, the induced G -Green functor as in Definition 3.5 isgiven by the assignment G / H ↦ E G × Σ m ( G / H ) , where G / H is given a trivial Σ m -action.We highlight a particular case of the double-coset formula. Corollary 3.12.
For any G × Σ m -Green functor R , the following square of abelian groupscommutes R (( G / H ) × m ) R (( G / H ) × G / L m ) R (( G / L ) × m ) R ( G / L ) i ∗ Tr Tr∆ ∗ for any map of G -sets G / H Ð→ G / L .Proof. This follows from Remark 3.1, as we have a pullback square of G × Σ m -sets ( G / H ) × m ( G / H ) × G / L m ( G / L ) × m G / L, i ∆ in which the vertical maps are surjective. (cid:3) Certain ideals in R ↑ GG × Σ m . Given a G × Σ m -Green functor R , Definition 3.5 producesa G -Green functor R ↑ GG × Σ m . Notation 3.13.
Since the induced Mackey functor R ↑ GG × Σ m will appear many times in thissubsection, we will abbreviate it to R ↑ .In this subsection, we describe two Mackey ideals in the G -Green functor R ↑ that dependon the fact that R ↑ is induced from R . The definitions of these Mackey ideals are motivatedby considerations coming from power operations as in Section 2; however, they make sensein any G -Green functor of the form R ↑ .We begin with the transfer Mackey ideal. Definition 3.14.
Fix a G × Σ m -Green functor R . Define I Tr ( G / H ) ⊆ R ↑ ( G / H ) to be theimage of the transfers ⊕ i + j = mi,j > R (( G × Σ m )/( H × Σ i × Σ j )) Tr Ð→ R (( G × Σ m )/( H × Σ m )) . We note that the target is R ↑ ( G / H ) . Lemma 3.15.
The ideals I Tr ( G / H ) of Definition 3.14 fit together to define a Mackey idealof R ↑ .Proof. Frobenius reciprocity implies that I Tr ( G / H ) is an ideal of R ↑ ( G / H ) .It remains to show that I Tr is a sub-Mackey functor. To see that I Tr is closed underrestriction maps, note that G / H × Σ m /( Σ i × Σ j ) G / K × Σ m /( Σ i × Σ j ) G / H × Σ m / Σ m G / K × Σ m / Σ m is a pullback square of G × Σ m -sets and apply Remark 3.1. Finally, I Tr is closed underinductions since the composition of inductions is again an induction. (cid:3) DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 17
Now we will define a Mackey ideal J ⊆ R ↑ , inspired by the ideal J G of Section 2, withthe property that I Tr ⊆ J .We have a diagonal inclusion of G × Σ m -sets G / H ∆ Ð→ ( G / H ) × G / L m with complementary G × Σ m -set (cf. (2.6)) Z L,HG = Z L,H = ( G / H ) × G / L m ∖ ∆ ( G / H ) . In other words, there is a decomposition of G × Σ m -sets ( G / H ) × G / L m ≅ G / H ∐ Z L,HG , (3.16)where G / H has a trivial Σ m -action. Note that Z G,HG is what was previously called Z G,H in(2.6). We will often suppress the subscript G in the notation when there is no likelihood forconfusion. The following description of Z L,HG will be useful below.
Proposition 3.17.
The G × Σ m -set Z L,HG is induced from the subgroup L × Σ m : Z L,HG ≅ G × L [( L / H ) × m ∖ ∆ ( L / H )] = G × L Z L,HL . Proof.
Since the G -set induction functor G × L ( − ) ∶ L Set Ð→ G Set preserves pullbacks, itfollows that ( G / H ) × G / L m is isomorphic to G × L (( L / H ) × m ) . Since G × L L / H ≅ G / H , applyinginduction to the L = G case of the decomposition (3.16) produces a decomposition G × L Z L,HL ≅ G / H ∐ G × L Z L,HH . (cid:3) The decomposition (3.16) induces an isomorphism of commutative rings R (( G / H ) × G / L m ) ≅ R ↑ ( G / H ) × R ( Z L,H ) . (3.18)We may obtain R ↑ ( G / H )/ I Tr ( G / H ) from R (( G / H ) × G / L m ) by taking the quotient by I Tr ( G / H ) in the first factor of (3.18) and taking the quotient with respect to the entiresecond factor. This inspires the definition of J : Definition 3.19.
Fix a G × Σ m -Green functor R . Define J ( G / L ) ⊆ R ↑ ( G / L ) to be theideal generated by the images of the transfers along: ● the quotients G / L × Σ m /( Σ i × Σ j ) → G / L × Σ m / Σ m for i + j = m and ● the composition Z L,H ↪ ( G / H ) × G / L m → G / L × Σ m / Σ m for H ≤ L .By construction, we have the following compatibility. Proposition 3.20.
For H subconjugate to L , we have the following commutative diagram R (( G / H ) × G / L m ) R ↑ ( G / H ) / I Tr ( G / H ) R ↑ ( G / L ) R ↑ ( G / L )/ J ( G / L ) , Tr Tr in which the top map is given by first projecting to the left factor in (3.18) and then takingthe quotient by the ideal I Tr ( G / H ) . The analogue of Theorem 2.17 in this context reads as follows. Like Theorem 2.17, it isan immediate consequence of Proposition 2.30.
Proposition 3.21.
Let R be a G × Σ m -Green functor and fix m ≥ . Then J ( G / L ) ⊆ R ↑ ( G / L ) is generated by I Tr ( G / L ) and the images of the transfers R ( G × Σ m / Σ q ≀ n Γ ( a S / K ) ) Tr Ð→ R ( G / L ) = R ↑ ( G / L ) for all m = nq , K < S ≤ L with [ S ∶ K ] = n ≠ , and a S / K ∶ S → Aut
Set ( S / K ) ≅ Σ n the actionmap by left multiplication. In the case that m is prime, Corollary 2.38 gives the following simplified form. Proposition 3.22.
Let R be a G × Σ m -Green functor and let m = p be prime. Then J ( G / L ) ⊆ R ↑ ( G / L ) is generated by I Tr ( G / L ) and the images of the transfers R ( G × Σ p / Γ ( a ) ) Tr Ð→ R ( G / L ) = R ↑ ( G / L ) for all subgroups S ≤ L and homomorphisms a ∶ S Ð→ Σ p whose images contain a p -cycle. The proof of the following corollary is the same as for Corollary 2.44.
Corollary 3.23. If m is relatively prime to the order of G , then J = I Tr . Now that we have described the ideals J ( G / L ) for fixed L , we turn to the question ofhow they interact as L varies. Theorem 3.24.
The ideals J ( G / L ) of Definition 3.19 fit together to define a Mackey idealof R ↑ .Proof. It suffices to show that J is a sub-Mackey functor, i.e. that the image of the transfersin Definition 3.19 is closed under restriction and induction. By Lemma 3.15, it suffices toshow that if H ≤ K ≤ L (up to conjugacy), then(1) the image of R ( Z L,H ) R ( G / L ) = R ↑ ( G / L ) R ↑ ( G / K ) Tr Res lands in J ( G / K ) ⊂ R ↑ ( G / K ) and(2) the image of R ( Z K,H ) R ( G / K ) = R ↑ ( G / K ) R ↑ ( G / L ) Tr Tr lands in J ( G / L ) ⊂ R ↑ ( G / L ) .We begin with (2), as it is much simpler to verify. We have a commutative diagram of G × Σ m -sets Z K,H G / H × G / K m G / KZ L,H G / H × G / L m G / L, which yields the commutative diagram R ( Z K,H ) R ( G / K ) R ↑ ( G / K ) R ( Z L,H ) R ( G / L ) R ↑ ( G / L ) . Tr Tr TrTr
It follows that J is closed under Mackey induction. DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 19
We now turn to (1), which is more difficult to handle. It suffices to show that we have afactorization ⊕ H < L R ( Z L,H ) R ↑ ( G / L ) ⊕ i + j = m R ( G / K × Σ m /( Σ i × Σ j )) ⊕ ⊕ H ′ < K R ( Z K,H ′ ) R ↑ ( G / K ) . Tr ResTr (3.25)We have a pullback diagram of G × Σ m -sets Z L,H G / L ( Z L,H × G / L G / K ) G / K q p (3.26)in which all maps are surjective. Then Remark 3.1 gives a commutative diagram R ( Z L,H ) R ↑ ( G / L ) R ( Z L,H × G / L G / K ) R ↑ ( G / K ) . q ∗ Tr Res = p ∗ Tr It remains to show that the bottom transfer map factors through the sum ⊕ i + j = m R ( G / K × Σ m /( Σ i × Σ j )) ⊕ ⊕ H ′ < K R ( Z K,H ′ ) . We may decompose Z L,H × G / L G / K into G × Σ m -orbits, and it suffices to produce thefactorization at the level of G × Σ m -sets on each orbit.Thus let U ⊂ Z L,H × G / L G / K be such an orbit, and choose ( ⇀ x, y ) ∈ U . Then if Λ ≤ G × Σ m is the stabilizer of ( ⇀ x, y ) , the presence of the factor G / K forces π G ( Λ ) to be subconjugateto K . We now consider two cases.In the first case, suppose that π Σ m ( Λ ) is not a transitive subgroup. Then π Σ m ( Λ ) issubconjugate to Σ i × Σ j for some positive i and j summing to m . It follows that there existsa map of G × Σ m -sets of the form U ( G × Σ m )/ Λ ( G × Σ m )/( K × Σ i × Σ j )( ⇀ x, y ) e Λ ( g, σ )( K × Σ i × Σ j ) ≅ where y = gK . Then composing this map with the projection onto G / K produces a mapof G × Σ m -sets sending ( ⇀ x, y ) to gK = y . It follows that the image of the R -transfer along U Ð→ G / K is contained in the image of the R -transfer along the G -cover G / K × Σ m /( Σ i × Σ j ) Ð→ G / K .In the second case, we suppose that π Σ m ( Λ ) is a transitive subgroup of Σ m . We furtherassume for simplicity that H and K are subgroups of L , rather than merely subconjugate.The general case is similar but notationally more complex.We now reduce to the case L = G : recall from Proposition 3.17 that the G × Σ m -set Z L,HG is induced up from the subgroup L × Σ m . Since the G -set induction G × L ( − ) ∶ L Set Ð→ G Set preserves pullbacks, the pullback square of L -sets Z L,HL × L / K L / KZ L,HL L / L gives rise to an isomorphism of G × Σ m -sets Z L,HG × G / L G / K ≅ G × L ( Z L,HL × L / K ) . Moreover, the projection Z L,HG × G / L G / K Ð→ G / K is the induction from L to G of theprojection Z L,HL × L / K Ð→ L / K . We may therefore restrict to the case L = G .We now return to the G × Σ m -orbit U with chosen point ( ⇀ x, y ) and Λ = Stab G × Σ m ( ⇀ x, y ) .Up to Σ m -conjugacy, the tuple ⇀ x is a q -fold shuffle. We have assumed that π Σ m ( Λ ) ≤ Σ m istransitive. Let y = gK . ThenΛ = Stab ( ⇀ x, y ) = Stab ( ⇀ x ) ∩ Stab ( y ) = Stab ( ⇀ x ) ∩ ( gKg − × Σ m ) . Lemma 2.24 explicitly describes Stab ( ⇀ x ) as Σ q ≀ n Γ ( a ) for some homomorphism a ∶ S Ð→ Σ n .The intersection Λ = Stab ( ⇀ x ) ∩ ( gKg − × Σ m ) is now Σ q ≀ n Γ ( a ∣ S ∩ gKg − ) , where a ∣ S ∩ gKg − is the restriction of a to S ∩ gKg − . This is a subgroup of gKg − × Σ m that projects ontoa transitive subgroup of Σ m by assumption. We may use the surjectivity statement ofProposition 2.30 to describe this as a stabilizer of some element of ( gKg − / H ′ ) × m ∖ ∆ ( gKg − / H ′ ) for some H ′ ≤ gKg − . It follows that U ≅ ( G × Σ m )/ Λ appears as an orbit of Z gKg − ,H ′ ≅ Z K,g − H ′ g . Therefore the image of the R -transfer from U is contained in the image of the R -transfer from Z K,g − H ′ g . (cid:3) Remark 3.27.
By construction, a map of G × Σ m -Green functors R Ð→ S gives rise to amap of G -Green functors R ↑ / J Ð→ S ↑ / J .3.3. Power operations on Green functors.
For any m ≥
0, let F m ∶ G − Set Ð→ G × Σ m − Setbe the m th power functor F m ( A ) = A × m , where Σ m permutes the factors and G actsdiagonally. Note that F m preserves pullbacks. It follows that, for any G × Σ m -Greenfunctor R , the composition R ○ F m is a G -Green functor.Following Definition 3.14, let I Tr ⊆ R ○ F m be the Mackey ideal defined by letting I Tr ( G / H ) be the image of the transfers ⊕ i + j = mi,j > R (( G / H ) × m × Σ m /( Σ i × Σ j )) Tr Ð→ R (( G / H ) × m × Σ m / Σ m )) . Note that the target is R ○ F m ( G / H ) . Definition 3.28. An m th total power operation on a G × Σ m -Green functor R is a naturaltransformation P m ∶ R ↓ G × Σ m G Ð→ R ○ F m of G -Mackey functors of sets which preserves the multiplicative structure and such that P m / I Tr is a map of G -Green functors. DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 21
We will consider two sources of G × Σ m -Green functors with m th total power operation inthe following two subsections. These are H ∞ -rings in genuine G -spectra (Section 3.4) andultra-commutative ring spectra in the sense of [Sch18, Chapter 5] (Section 3.5).For any G -set A , the diagonal inclusion ∆ ∶ A ↪ A × m = F m ( A ) is G × Σ m -equivariant.Pulling back along ∆ defines a map of coefficient systems of commutative rings R ○ F m ∆ ∗ Ð→ R ↑ GG × Σ m . The image of I Tr under ∆ ∗ is I Tr (Definition 3.14). We then define the power operation P m as the composition P m ∶ R ↓ G × Σ m G P m ÐÐ→ R ○ F m ∆ ∗ Ð→ R ↑ GG × Σ m . In general, the power operation P m is not additive, and it does not commute with the Mackeyinduction maps and is therefore not a map of Mackey functors. Making use of [BMMS86,VIII.1.4] and Lemma 3.15, additivity can be arranged by taking the quotient with respectto I Tr . Proposition 3.29.
Let R be a G × Σ m -Green functor with an m th total power operation.Then the composition R ↓ G × Σ m G P m ÐÐ→ R ↑ GG × Σ m ↠ R ↑ GG × Σ m / I Tr is a map of coefficient systems of commutative rings. Further passing to the quotient with respect to J produces a map of G -Green functors. Theorem 3.30.
Let R be a G × Σ m -Green functor with an m th total power operation, andlet J be as in Definition 3.19. Then the reduced m th power operation P m / J ∶ R ↓ G × Σ m G P m ÐÐ→ R ○ F m ∆ ∗ Ð→ R ↑ GG × Σ m ↠ R ↑ GG × Σ m / J is a map of G -Green functors. The argument below follows the strategy outlined in Section 2.1.
Proof.
By Theorem 3.24, J is a Mackey functor ideal in R ↑ GG × Σ m . Thus R ↑ GG × Σ m / J is a G -Green functor. Since I Tr is contained in J , Proposition 3.29 implies that it remains toshow that P m / J commutes with induction maps. Thus suppose that H ≤ G is subconjugateto L ≤ G .Since the total power operation P m is a map of Mackey functors (of sets), Corollary 3.12implies that we have the following commuting diagram: R ↓ G × Σ m G ( G / H ) R (( G / H ) × m ) R (( G / H ) × G / L m ) R ↓ G × Σ m G ( G / L ) R (( G / L ) × m ) R ( G / L ) . Tr P m i ∗ Tr Tr P m ∆ ∗ (3.31)Note that in the bottom right corner, G / L has a trivial Σ m -action, so that R ( G / L ) is R ↑ GG × Σ m ( G / L ) . Proposition 3.20 states that we have a commuting diagram R (( G / H ) × G / L m ) R ↑ GG × Σ m ( G / H )/ I Tr ( G / H ) R ↑ GG × Σ m ( G / L ) R ↑ GG × Σ m ( G / L )/ J ( G / L ) . Tr Tr2 P. J. BONVENTRE, B. J. GUILLOU, AND N. J. STAPLETON
The result then follows by the factorization R ↑ GG × Σ m ( G / H )/ I Tr ( G / H ) R ↑ GG × Σ m ( G / H )/ J ( G / H ) R ↑ GG × Σ m ( G / L )/ J ( G / L ) R ↑ GG × Σ m ( G / L )/ J ( G / L ) , Tr Tr which occurs since J is a Mackey ideal containing I Tr . (cid:3) Power operations on G -spectra. We first consider H ∞ -ring spectra, in the sense of[BMMS86], in the equivariant category. Definition 3.32. An H ∞ -ring G -spectrum is a G -spectrum E equipped with G -equivariantmaps E ∧ mh Σ m Ð→ E which make the diagrams of [BMMS86, I.3] commute in the equivariantstable homotopy category.As noted in Example 3.4, every homotopy-commutative ring G -spectrum E induces aGreen functor-valued equivariant cohomology theory on G -spaces, defined by E ( X )( G / H ) = E ( G / H × X ) = [( G / H × X ) + , E ] G , where [ − , − ] G denotes the abelian group of maps in the equivariant stable homotopy cate-gory.If E is moreover an H ∞ -ring G -spectrum, more is true: Proposition 3.33. If E is an H ∞ -ring G -spectrum, then the G × Σ m -Green functor givenby R ( A ) = E ( X × A h Σ m ) , as in Example 3.7, has an m th total power operation.Proof. In this case, R ↓ G × Σ m G ( G / H ) = E ( X × G / H ) and R ○ F m ( G / H ) = E ( X × ( G / H ) mh Σ m ) . We define P m levelwise to be the composite [( X × G / H ) + , E ] G → [( X × G / H ) × mh Σ m , + , E ∧ mh Σ m ] G µ Ð→ [( X × G / H ) × mh Σ m , + , E ] G ∆ ∗ X ÐÐ→ [( X × ( G / H ) × mh Σ m ) + , E ] G . The map P m satisfies the requirements of Definition 3.28: it is natural in all stable maps byconstruction, and so in particular is a map of G -Mackey functors of sets; it is multiplicative;and it is additive after passing to the quotient by I Tr by [BMMS86, VIII.1.1]. (cid:3) Remark 3.34.
We note that the function spectrum E X is an H ∞ -ring G -spectrum whenever E is an H ∞ -ring G -spectrum and X is a G -space. Thus the power operation P m for E defined at X in Proposition 3.33 agrees with the power operation P m for E X at G / G . Thus,without loss of generality we may assume X = G / G throughout.The target of the power operation P m is E ( B Σ m ) (3.9). Thus, Definition 3.19 yields anideal J ( G / L ) ⊆ E ( G / L × B Σ m ) generated by the images of the transfers along: ● the covers G / L × B Σ i × B Σ j → G / L × B Σ m for i + j = m and ● the composition Z L,Hh Σ m ↪ ( G / H ) × G / L mh Σ m → G / L × B Σ m for H ≤ L . This is false if we replace X with a G -spectrum Y , as the diagonal map X → X × n plays a key role inthis structure. DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 23
The following is then an immediate corollary of Theorem 3.30.
Corollary 3.35.
For any H ∞ -ring G -spectrum E , the reduced m th power operation P m / J ∶ E P m ÐÐ→ E ( B Σ m ) ↠ E ( B Σ m )/ J is a map of G -Green functors. Remark 3.36.
By Remark 3.34, for any G -space X , we get a map of G -Green functors P m / J ∶ E ( X ) Ð→ E ( X × B Σ m )/ J, where J ⊂ E ( X × B Σ m ) is defined as above by crossing with X .3.5. Power operations on global spectra.
Let E be an ultra-commutative ring spectrumin the sense of [Sch18, Chapter 5]. By [Sch18, Theorem 5.1.11], E is a global power functor.Following [Gan13], we will view E as a contravariant functor from the category of finitegroupoids to commutative rings satisfying the properties of [Gan13, Sections 2.2, 3.1, and4.1]. Most notably, applying E to fibrations of finite groupoids gives rise to transfer mapssatisfying the push-pull identity of Remark 3.1. Notation 3.37.
For a group G and G -set A , we write A // G for the action groupoid G // G × G A .The objects of A // G are the elements of A , and a morphism a Ð→ b in A // G consists of anelement g ∈ G such that g ⋅ a = b .We note that geometric realization translates action groupoids to homotopy orbits: ∣ C // G ∣ ≃ ∣ C ∣ hG . (3.38)In particular, we have equivalences ∣ ∗ // G ∣ ≃ BG and ∣ G / H // G ∣ ≃ BH . Proposition 3.39. If E is an ultra-commutative ring spectrum, then the G × Σ m -Greenfunctor given by R ( A ) = E G × Σ m ( A ) = E ( A //( G × Σ m )) , as in Example 3.11, has an m thtotal power operation.Proof. Given a finite groupoid G , there is an operation E ( G ) → E ( G ≀ Σ m ) , (3.40)where G ≀ Σ m is the finite groupoid of [Gan13, Definition 3.3]. In the case G = G / H // G , thegroupoid G ≀ Σ m is the action groupoid ( G / H ) × m //( G ≀ Σ m ) . From (3.40) we can constructthe power operation P m ∶ E ( G / H // G ) Ð→ E (( G / H ) × m //( G ≀ Σ m )) → E (( G / H ) × m //( G × Σ m )) by restricting the action along the diagonal map G × Σ m → G ≀ Σ m . Note that the source is E G ( G / H ) = ( E G × Σ m ) ↓ G × Σ m G ( G / H ) and the target is ( E G × Σ m ○ F m )( G / H ) .The requirements of Definition 3.28 now follow from [Sch18, Section 5.1]. (cid:3) In this case, Definition 3.19 specializes to an ideal J ( G / L ) ⊆ E G × Σ m ( G / L ) generated bythe images of the transfers along: ● the functors ( G / L )//( G × Σ i × Σ j ) → ( G / L )//( G × Σ m ) for i + j = m and ● the composition Z L,H //( G × Σ m ) ↪ ( G / H ) × G / L m //( G × Σ m ) → ( G / L )//( G × Σ m ) for H ≤ L .The following is then an immediate corollary of Theorem 3.30. Corollary 3.41.
Let E be an ultra-commutative ring spectrum. Then the reduced m thpower operation E G P m ÐÐ→ ( E G × Σ m ) ↑ GG × Σ m ↠ ( E G × Σ m ) ↑ GG × Σ m / J is a map of G -Green functors. Examples
In this section, we calculate the ideal J in a number of examples. In certain cases, we iden-tify J for all finite groups G . Throughout this section, Mackey induction homomorphismswill be displayed in orange, whereas power operations will be displayed in blue. We willdeal with many global Green functors in this section, and we will abbreviate an inductionsuch as ( A G × Σ m ) ↑ GG × Σ m to A ↑ GG × Σ m .4.1. Ordinary cohomology.
We begin with ordinary cohomology. This case turns out tobe degenerate, in the sense that J is equal to I Tr .Let R be a G -Green functor and HR the G -equivariant Eilenberg-Mac Lane spectrum.We will show that the composition R = H R P m ÐÐ→ H R ( B Σ m ) Ð→ H R ( B Σ m )/ I Tr is already a map of Mackey functors in the case of ordinary cohomology, and thus J = I Tr .At a G -orbit G / K , the m th power operation P m is R ( G / K ) (−) m ÐÐÐ→ R ( G / K ) . Lemma 4.1. If m = p r , where p is prime, then for any K ⊂ G , the ideal I Tr ( G / K ) ⊂ R ( G / K ) is the principal ideal ( p ) . On the other hand, if m has multiple prime factors, then I Tr ( G / K ) = ( ) .Proof. Let m = p r . We will abbreviate I Tr ( G / K ) simply to I Tr . For i + j = m , the cover B Σ i × B Σ j Ð→ B Σ m induces a transfer mapH R ( B Σ i × B Σ j )( G / K ) ≅ R ( G / K ) Tr Ð→ H R ( B Σ m )( G / K ) ≅ R ( G / K ) , which is multiplication by the index of the cover, the binomial coefficient ( mi ) = m ! i ! j ! . Since0 < i < p r , all of these coefficients are multiples of p , and we conclude that I Tr ⊂ ( p ) . Taking i = p r ∈ I Tr . In the case i = p r − , the coefficient ( p r p r − ) is congruent to p modulo p r . It follows that p ∈ I Tr .Suppose, on the other hand, that m is divisible by distinct primes p i . By Lucas’ Theorem,if r i is the largest integer such that p r i i divides m , then ( mp rii ) is prime to p i . It follows thatthe collection of coefficients {( mp rii )} are relatively prime to ( m ) = n , and therefore generatethe ideal ( ) . (cid:3) Thus in the interesting case m = p r , the composition R (−) m ÐÐÐ→ R Ð→ R / I Tr is equivalentto the quotient map R Ð→ R / p , which is a map of Mackey functors.4.2. The sphere spectrum.
We start with the global sphere spectrum S , which is an ultra-commutative ring spectrum [Sch18, Example 4.2.7]. Recall that π G = S G is isomorphic tothe Burnside ring Mackey functor, A G [Seg71, Corollary to Proposition 1]. Thus π G ( G / H ) ≅ A G ( G / H ) ≅ A ( H ) DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 25 is the commutative ring of isomorphism classes of finite H -sets. The restriction and inductionmaps, in the case that H ≤ K , are given by considering a finite K -set as a finite H -set, onthe one hand, and by inducing an H -set up to a K -set, on the other.The m th power operation associated to the ultra-commutative ring spectrum S as inSection 3.5 takes the form P m ∶ A G Ð→ A ↑ GG × Σ m , where we recall that A ↑ GG × Σ m is the G -Green functor given by G / H ↦ A ( H × Σ m ) . On theother hand, the m th power operation associated to the H ∞ -ring G -spectrum S G takes theform P m ∶ A G Ð→ π G ( B Σ m ) as in Section 3.4. The map of G × Σ m -spaces E Σ m Ð→ ∗ induces a map of G -green functors A ↑ GG × Σ m Ð→ π G ( B Σ m ) which is a completion map in the sense that, at an orbit G / H , π G ( B Σ m )( G / H ) ≅ π H ( B Σ m ) ≅ A ( Σ m × H ) ∧ I Σ m ≅ A ↑ GG × Σ m ( G / H ) ∧ I Σ m . This observation appears without proof in [May96, Chapter XX]; we provide a proof forcompleteness. Recall that for groups L and W , the Burnside module A ( L, W ) ⊂ A ( L × W ) is free abelian on ( L × W ) -sets for which the action of W is free. Proposition 4.2.
For any groups H and L , there is an isomorphism A ( L × H ) ≅ ⊕ [ K ⊂ H ] A ( L, W H ( K )) of A ( L ) -modules, where the sum runs over conjugacy classes of subgroups. This induces anisomorphism of commutative rings π H ( BL ) ≅ A ( L × H ) ∧ I L , where I L is the augmentation ideal of A ( L ) .Proof. We define an A ( L ) -module mapΦ ∶ A ( L × H ) Ð→ ⊕ [ K ⊂ H ] A ( L, W H ( K )) by Φ (( L × H )/ Γ ) = ( L × W H ( K Γ ))/( Γ / K Γ ) , where K Γ = Γ ∩ H . Writing p H ∶ L × H Ð→ H for the projection, we have K Γ ⊴ p H ( Γ ) , so thatΓ / K Γ is a subgroup of L × W H ( K Γ ) . For the reverse direction, we send a ( L × W H ( K )) -set Y to Y × W H ( K ) H / K . Here, L acts on Y , and the quotient (coequalizer) is formed in thecategory of H -sets, where H is acting trivially on Y . This assignment is inverse to Φ, andit follows that Φ is an isomorphism of A ( L ) -modules. We therefore deduce an isomorphism A ( L × H ) ∧ I L ≅ ⊕ [ K ⊂ H ] A ( L, W H ( K )) ∧ I L upon completion.Now consider the ring map A ( L × H ) Ð→ π H ( BL ) defined by sending the ( L × H ) -set X to the composition Σ ∞ H ( BL ) + Tr Ð→ Σ ∞ H ( X hL ) + Ð→ S H . This ring map factors as in the commutative diagram A ( L × H ) π H ( BL ) ⊕ [ K ⊂ H ] A ( L, W H ( K )) ⊕ [ K ⊂ H ] [ Σ ∞ BL + , Σ ∞ BW H ( K ) + ] . Φ ≅ tom Dieck splitting ≅ The lower horizontal arrow is completion at I L according to the version of the Segal conjec-ture given in [LMM82]. (cid:3) The m th power operation P m ∶ A G Ð→ A ↑ GG × Σ m , when evaluated at an orbit G / H , takesthe form P m ∶ A ( H ) Ð→ A ( H × Σ m ) . On an H -set X , it is given by X ↦ X × m , where the output is considered as an ( H × Σ m ) -set.In the case m =
2, we have a complete description of J as follows. Proposition 4.3.
In the case m = , the Mackey ideal J ⊂ A ↑ GG × Σ is the kernel of the Σ -fixed point homomorphism A ↑ GG × Σ Ð→ A G , and the composition A G P Ð→ A ↑ GG × Σ Ð→ A ↑ GG × Σ / J ≅ A G is the identity map.Proof. It suffices to consider the case H = G . The kernel of the Σ -fixed point homomor-phism has generators ( G × Σ )/ Γ where Γ is a graph subgroup. If Γ is a non-transitive graphsubgroup, then it is of the form K × Σ × Σ and therefore in J ( G / G ) . On the other hand,if it is transitive, then it is in J ( G / G ) according to Proposition 3.22.Now we consider the composition of the power operation P and fixed points. The Σ -fixed points of P ( G / H ) = ( G / H ) × are simply the diagonal G / H ≅ ∆ ( G / H ) ⊂ ( G / H ) × .This shows that the composition is the identity as claimed. (cid:3) When m = p > Proposition 4.4. If p > is prime and L ≤ G , then the orbit ( L × Σ p )/ Γ lies in the ideal J ( G / L ) ⊂ A ( L × Σ p ) if and only if either Γ is a graph subgroup of L × Σ p such that π Σ p ( Γ ) contains some C p , or Γ is subconjugate to L × Σ i × Σ j , for i and j positive and summing to p .Proof. This follows from Proposition 3.22. (cid:3)
Example 4.5.
Consider the case G = C and m =
2. We will describe the power operation P ∶ A C Ð→ A ↑ C C × Σ , which is only a map of coefficient systems (of sets) over C . We writeΓ = C × Σ and D < Γ for the diagonal subgroup. Writing 1 for the one-point orbit (of anygroup), we have A ( C ) = Z { C , } ,A ( Σ ) = Z { Σ , } , and A ( C × Σ ) = Z { Γ , Γ / C , Γ / Σ , Γ / D, } . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 27
Figure 1.
The ( C × Σ ) -set ( nC + k ) × ( nC + k ) nC knC k k ∐ k − k Γ / C kn Γ ( n − n ) Γ ∐ n Γ / Σ ∐ n Γ / D Proposition 4.6.
The power operations P e ∶ A ( e ) Ð→ A ( Σ ) and P C ∶ A ( C ) Ð→ A ( C × Σ ) are given by P e ( k ) = k − k + k and P C ( nC + k ) = ( n − n + kn ) Γ + k − k / C + n Γ / Σ + n Γ / D + k, respectively.Proof. For P e , this is simply a matter of observing that the diagonal of k × k is fixed by theΣ -action, and the rest is free. The case of P C is displayed in Figure 1. The key here isthat C × C ≅ Γ / Σ ∐ Γ / D . (cid:3) The images of the transfer maps A ( e ) Ð→ A ( Σ ) and A ( C ) Ð→ A ( C × Σ ) are Z { Σ } ⊂ A ( Σ ) and Z { Γ , Γ / C } ⊂ A ( C × Σ ) . Thus, after modding out by the images of thesetransfer maps, we have Z { C , } Z ( ) ( ) Z { Γ / Σ , Γ / D, } Z ( ) ⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ The power operation does not commute with the Mackey induction homomorphisms. Butafter collapsing out the image of the additional transfer A ( D ) Ð→ A ( C × Σ ) , the poweroperation becomes the identity map of the Burnside Green functor, as it must be accordingto Proposition 4.3. Example 4.7.
Consider G = Σ and m =
2. We will write ρ for a 3-cycle and τ for atransposition in Σ . We will let σ ∈ Σ be the transposition. We begin by restrictingattention to C < Σ . The decomposition of ( C × Σ ) -sets C × ≅ ( C × Σ )/ Σ ∐ ( C × Σ ) gives the following result. Proposition 4.8.
Writing Γ = C × Σ , the second power operation P C ∶ A ( C ) Ð→ A ( C × Σ ) is given by P C ( nC + k ) = ( nk + n − n ) Γ + k − k / C + n Γ / Σ + k. On the other hand, now writing Γ = Σ × Σ , we haveΣ × ≅ Γ / Σ ∐ Γ ∐ ( Γ / D ) , where D < Γ is the order two subgroup generated by the element ( τ, σ ) . We have A ( Σ ) = Z { Σ , Σ / C , Σ / C , } and A ( Σ × Σ ) = Z { Γ , Γ / C , Γ / C , Γ / Σ , Γ / Σ , Γ /( C × Σ ) , Γ /( C × Σ ) , Γ / D, Γ / DC , } . Proposition 4.9.
Writing Γ = Σ × Σ , the second power operation P Σ ∶ A ( Σ ) Ð→ A ( Σ × Σ ) is given by P Σ ( n Σ ∐ i Σ / C ∐ j Σ / C ∐ k ) = ( n − n + ni + nj + nk + ij + j − j ) Γ ∐ ( i − i + ik ) Γ / C ∐ ( j − j + jk ) Γ / C ∐ k − k / Σ ∐ n Γ / Σ ∐ i Γ /( C × Σ ) ∐ j Γ /( C × Σ ) ∐ ( n + j ) Γ / D ∐ i Γ / DC ∐ k. In addition to the transfers already discussed in Example 4.5, the images of the transfers A ( C ) Ð→ A ( C × Σ ) and A ( Σ ) Ð→ A ( Σ × Σ ) are Z { C × Σ , ( C × Σ )/ C } ⊂ A ( C × Σ ) and Z { Γ , Γ / C , Γ / C , Γ / Σ } ⊂ A ( Σ × Σ ) . Thus, after modding out by the images of these transfer maps, we have
DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 29 Z { Σ , Σ / C , Σ / C , } Z { C , } Z { C , } Z ( ) ( )( ) ( ) ( ) ⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠( ) ⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠ Z { Γ / Σ , Γ /( C × Σ ) , Γ /( C × Σ ) , Γ / D, Γ / DC , } Z {( C × Σ )/ Σ , } Z {( C × Σ )/ Σ , ( C × Σ )/ D, } Z ( ) ⎛⎜⎝ ⎞⎟⎠( ) ( ) ⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠( ) ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛⎜⎝ ⎞⎟⎠( ) ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ In order to make the power operations commute with Mackey induction, we must furthercollapse Z {( C × Σ )/ D } ⊂ A ( C × Σ ) and Z { Γ / D, Γ / DC } ⊂ A ( Σ × Σ ) . The result-ing power operation of Green functors is the identity on A Σ , as it must be according toProposition 4.3. Example 4.10.
Consider G = C and m =
3. Then A ( Σ ) = Z { Σ , Σ / C , Σ / C , } . For Γ = C × Σ , we write C R for the order 3 subgroup of Σ , and we write C for a choiceof order two subgroup of Σ and ∆ for the order 3 subgroup generated by ( ρ, σ ) , where ρ generates C and σ is a 3-cycle. Then A ( C × Σ ) ≅ Z { Γ , Γ / C , Γ / C , Γ / Σ , Γ / C R , Γ /( C × C ) , Γ / C × , Γ / ∆ , Γ / ∆ C , } , where ∆ C is the internal product in C × Σ . Proposition 4.11.
Writing Γ = C × Σ , the power operations P e ∶ A ( e ) Ð→ A ( Σ ) and P C ∶ A ( C ) Ð→ A ( C × Σ ) are given by P e ( k ) = ( k ) Σ + k ( k − ) Σ / C + k Σ / Σ and P C ( nC + k ) = [ n ( k ) + k n − n + ( n ) + ( n )] Γ + ( k ) Γ / C + n Γ / Σ + k ( k − ) Γ /( C × C ) + [ nk + n + ( n )] Γ / C + n Γ / ∆ + k Γ / Γ respectively. By Proposition 4.4, J ( C / C ) ⊂ A ( C × Σ ) is generated by the orbits Γ, Γ / C , Γ /( C × C ) , Γ / C , and Γ / ∆, which are precisely the terms appearing in Proposition 4.11 with a nonlinear coefficient. The resulting power operation of Green functors is then an inclusion A C P / J ÐÐÐ→ A ↑ C C × Σ / J ≅ A C ⊕ A C ⊕ Z { Γ / ∆ } Here the first copy of A C contains the orbits Γ / Σ and 1 = Γ / Γ, whereas the second copycontains the orbits Γ / C R and Γ /( C ) .4.3. Global KU . Consider the ultra-commutative ring spectrum KU ([Sch18, 6.4.9]). Theassociated G -Green functor KU G is the representation ring Green functor, with KU G ( G / H ) ≅ RU ( H ) . The restriction and induction maps correspond to restriction and induction of rep-resentations, respectively. The m th power operation associated to the ultra-commutativering spectrum KU as in Section 3.5 takes the form P m ∶ RU G Ð→ RU ↑ GG × Σ m and is given at G / H by V ↦ V ⊗ m , where the latter is consider as a ( H × Σ m ) -representation.On the other hand, the m th power operation associated to the H ∞ -ring G -spectrum KU G takes the form P m ∶ RU G = KU G Ð→ KU G ( B Σ m ) . as in Section 3.4. The map of G × Σ m -spaces E Σ m Ð→ ∗ induces a map of G -green functors RU ↑ GG × Σ m Ð→ KU G ( B Σ m ) . At an orbit G / H , this is the map RU ↑ GG × Σ m ( G / H ) ≅ RU ( H × Σ m ) ( − ) h Σ m ÐÐÐÐ→ KU H ( B Σ m ) ≅ KU G ( B Σ m )( G / H ) which takes an H × Σ m -representation and passes to homotopy orbits with respect to theΣ m -action. As we show in the following proposition, this map is a completion, in the sensethat KU G ( B Σ m )( G / H ) ≅ KU H ( B Σ m ) ≅ RU ( H × Σ m ) ∧ I Σ m ≅ RU ↑ GG × Σ m ( G / H ) ∧ I Σ m . Proposition 4.12.
For any groups H and L , the map ( − ) hL ∶ RU ( H × L ) Ð→ KU H ( BL ) is completion at the augmentation ideal I L ⊂ RU ( L ) .Proof. We have a commuting square RU ( H × L ) KU H ( BL ) RU ( H ) ⊗ RU ( L ) RU ( H ) ⊗ KU ( BL ) , ≅ ≅ where the right vertical map is an isomorphism because H acts trivially on BL . Now theAtiyah-Segal Completion Theorem [AS69] states that RU ( L ) Ð→ KU ( BL ) is completionat I L . The isomorphism RU ( H ) ⊗ RU ( L ) ∧ I L ≅ RU ( H ) ⊗ RU ( L ) ⊗ RU ( L ) RU ( L ) ∧ I L ≅ ( RU ( H ) ⊗ RU ( L )) ⊗ RU ( L ) RU ( L ) ∧ I L finishes the proof. (cid:3) DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 31
As in Section 4.2, we focus on the power operation with target RU ↑ GG × Σ m . Denote byev σ ∶ RU ( Σ m ) Ð→ Z the homomorphism that evaluates the character of a representation atan m -cycle. This homomorphism is precisely the quotient map KU m ( ∗ ) Ð→ KU m ( ∗ )/ I Tr by the transfer ideal. Proposition 4.13 ([Ati66]) . The composition KU G ( X ) P m ÐÐ→ KU G × Σ m ( X ) ≅ KU G ( X ) ⊗ RU ( Σ m ) id ⊗ ev σ ÐÐÐÐ→ KU G ( X ) ⊗ Z is the Adams operation ψ m . In other words, RU G P m / I Tr ÐÐÐÐ→ RU ↑ GG × Σ m / I Tr ≅ RU G is levelwise the Adams operation ψ m . As in Proposition 3.29, this is a map of coefficientsystems of commutative rings but not a map of Green functors. Example 4.14.
Consider the case G = C and m =
2. Then RU ( C ) = Z { , s } , where s isthe sign representation, satisfying s =
1. Then the Mackey induction sends 1 ∈ RU ( e ) = Z to the regular representation 1 + s . The ring homomorphism ψ squares both of the 1-dimensional representations 1 and s , so the diagram RU ( C ) RU ( C ) RU ( e ) = Z RU ( e ) = Z ψ ψ = id does not commute, since 1 + s + s = ≠ + s . (4.15)Following Proposition 3.22, we observe that J is generated by I Tr as well as an additionaltransfer: we must further collapse the image of RU ( D ) RU ( C × Σ ) ≅ RU ( C ) ⊗ RU ( Σ ) RU ( C ) ⊗ Z , + ss − s ⊗ ev σ where D ≤ C × Σ is once again the diagonal subgroup and s ∈ RU ( Σ ) is the sign represen-tation. Thus, collapsing the ideal J imposes the relation s ∼
1, in particular making (4.15)commute. The resulting quotient Green functor is the constant Mackey functor Z , and thepower operation of Green functors P / J ∶ RU C Ð→ RU ↑ C C × Σ / J ≅ Z is the augmentation, given by restricting to the trivial subgroup. Example 4.16.
Consider now G = Σ and m =
2. We have RU ( Σ ) = Z [ s, W ]/( s − , sW − W, W − W − s − ) , where W is the reduced standard representation, and RU ( C ) = Z [ λ ]/( λ − ) . By Proposition 3.22, the ideal J is generated by I Tr and three additional transfers: RU ( D ) RU ( Σ × Σ ) ≅ RU ( Σ ) ⊗ RU ( Σ ) RU ( Σ ) ⊗ Z , + s ¯ s + W + W ¯ s − sRU ( Γ ( sgn )) RU ( Σ × Σ ) ≅ RU ( Σ ) ⊗ RU ( Σ ) RU ( Σ ) ⊗ Z , and1 1 + s ¯ s − sRU ( D ) RU ( C × Σ ) ≅ RU ( C ) ⊗ RU ( Σ ) RU ( C ) ⊗ Z , + s ¯ s − s ⊗ ev σ ⊗ ev σ ⊗ ev σ where C ≤ Σ is a choice of order two subgroup, D ≤ C × Σ ≤ Σ × Σ is the diagonal order2 subgroup, sgn ∶ Σ → Σ is the sign homomorphism, and Γ ( sgn ) ≤ Σ × Σ the associatedgraph subgroup. As in Example 4.14, we conclude that we must impose the relation s ∼ RU ( Σ ) . The resulting power operation of Green functors P / J ∶ RU Σ Ð→ RU ↑ Σ Σ × Σ / J is given by Z { , s, W } Z { , λ, λ } Z { , s } Z ( ) ( )( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ⎛⎜⎝ ⎞⎟⎠⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ Z { , W } Z { , λ, λ } ZZ . ( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ( )⎛⎜⎝ ⎞⎟⎠ ( ) ( )⎛⎜⎝ ⎞⎟⎠( ) Here, the value of ψ ( W ) may be deduced by using the (character) embedding of the repre-sentation ring into the ring of class functions and using the formula for the Adams operation ψ given in Proposition 4.19. Since all other representations that appear are 1-dimensional,the operation ψ simply squares them. Example 4.17.
Consider now G = Σ and m =
3. We begin with the same source Greenfunctor as in Example 4.16. As in Example 4.16, J is generated by I Tr and three additional DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 33 transfers: RU ( D C ) RU ( Σ × Σ ) ≅ RU ( Σ ) ⊗ RU ( Σ ) RU ( Σ ) ⊗ Z , + s + ¯ s + s ¯ s + W ¯ W + s − WRU ( D Σ ) RU ( Σ × Σ ) ≅ RU ( Σ ) ⊗ RU ( Σ ) RU ( Σ ) ⊗ Z , and1 1 + s ¯ s + W ¯ W + s − WRU ( D C ) RU ( C × Σ ) ≅ RU ( C ) ⊗ RU ( Σ ) RU ( C ) ⊗ Z , + ¯ s → λ ¯ W + λ ¯ W − λ − λ ⊗ ev σ ⊗ ev σ ⊗ ev σ where D C ≤ C × Σ is the order 3 subgroup generated by ( ρ, σ ) for ρ a generator of C and σ a 3-cycle, and D Σ ≤ Σ × Σ is the diagonal subgroup. Thus we collapse the ideals ( W − s − ) ⊂ RU ( Σ ) and ( λ + λ − ) ⊂ RU ( C ) . The quotients are RU ( Σ )/( W − s − ) ≅ Z { , s } and RU ( C )/( λ + λ − ) ≅ Z { } ⊕ Z / { ¯ λ } , where ¯ λ = λ − P / J ∶ RU Σ Ð→ RU ↑ Σ Σ × Σ / J is given by Z { , s, W } Z { , λ, λ } Z { , s } Z ( ) ( )( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ⎛⎜⎝ ⎞⎟⎠⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ Z { , s } Z ⊕ Z / { ¯ λ } Z { , s } Z . ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( )( ) Again, the value of ψ on representations may be deduced via the character embedding.4.4. Class functions.
Rings of class functions appear in homotopy theory as approxima-tions to cohomology theories. In particular, equivariant KU -theory is approximated by thering of class functions Cl ( G, C ) , which is the ring of C -valued functions on the set of con-jugacy classes of G . Further, Hopkins, Kuhn, and Ravenel [HKR00] have shown that theMorava E -theories, which are generalizations of p -adic KU -theory, all admit similar approx-imations by a ring of “generalized class functions”. We introduce this ring for the “height2” Morava E -theories in Section 4.4.2.The rings of class functions fit together to give Green functors with restriction and in-duction maps compatible with the restriction and induction maps for the equivariant coho-mology theory that they approximate. C -valued class functions. We begin by considering Cl ( G ) = Cl ( G, C ) , the ring of classfunctions that arises in the representation theory of finite groups. For any group G , we havea Mackey functor Cl G defined by Cl G ( G / H ) = Cl ( H ) . For H ≤ K , the restriction map isgiven by simply restricting class functions along the map H / conj → K / conj. For H ≤ K , theinduction homomorphism Tr ∶ Cl ( H ) Ð→ Cl ( K ) is (cf. [Ser77, Theorem 12]) Tr ( f )( k ) = ∑ gH ∈ ( K / H ) k f ( g − kg ) , (4.18)where ( K / H ) k denotes the k -fixed points under the left action of K on K / H .Conjugacy classes in Σ m correspond to partitions of m . Let { m , . . . , m j } be a partitionof m , so that m + . . . + m j = m . The power operation P m ∶ Cl ( G ) Ð→ Cl ( G × Σ m ) is givenby P m ( f )( g, { m , . . . , m j }) = j ∏ i = f ( g m i ) . The following result is well-known.
Proposition 4.19.
The quotient Cl ( G × Σ m )/ I Tr is isomorphic to Cl ( G ) , and the compo-sition Cl ( G ) P m ÐÐ→ Cl ( G × Σ m ) Ð→ Cl ( G × Σ m )/ I Tr ≅ Cl ( G ) is the Adams operation ψ m , given by ψ m ( f )( g ) = f ( g m ) .Proof. For any proper partition { m , . . . , m j } of m and conjugacy class g in G , class func-tions on G × Σ supported on ( g, { m , . . . , m j }) are in the image of the transfer along G × ∏ i Σ m i ↪ G × Σ m . It follows that the quotient Cl ( G × Σ m )/ I Tr can be identifiedwith functions supported on ( g, σ ) , where σ = ( ⋯ m ) . This identifies the quotient with Cl ( G ) .By the previous discussion, after passing to the quotient by the transfer ideals for sub-groups G × Σ i × Σ j , only the value of the class function on the long cycle is retained, and P m ( f )( g, ( ⋯ m )) = f ( g m ) = ψ m ( f )( g ) . (cid:3) Let G p -div ⊂ G denote the set of elements whose orders are divisible by p , and let G p -prime ⊂ G denote the set of elements whose orders are not divisible by p . Similarly, we denoteby Cl p -div and Cl p -prime the rings of C -valued functions on G p -div / conj and G p -prime / conj,respectively. Then the decomposition G = G p -div ∐ G p -prime induces an isomorphism ofcommutative rings Cl ( G ) ≅ Cl p -div ( G ) × Cl p -prime ( G ) . Proposition 4.20.
The Green functor structure on Cl G descends to Green functor struc-tures on Cl p - div G and Cl p - prime G .Proof. This follows from the fact that any subgroup inclusion H ↪ G induces inclusions H p -div ↪ G p -div and H p -prime ↪ G p -prime . (cid:3) Proposition 4.21. If m = p is prime, the image of J under the isomorphism of Proposition 4.19is the Mackey ideal Cl p - div G . The power operation of Green functors P p / J ∶ Cl G Ð→ Cl ↑ GG × Σ p / J ≅ Cl p - prime G is the composition Cl G restrict ÐÐÐÐ→ Cl p - prime G ψ p Ð→ Cl p - prime G . DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 35
Proof.
By Proposition 4.19, it suffices to show that for any g ∈ G p -div , any class function f on G × Σ p supported at [ g, σ ] is in the image of the transfer from a graph subgroup as inCorollary 2.38. Let S be the cyclic subgroup of G generated by g , and let a ∶ S Ð→ Σ p send g to σ . Then (4.18) shows that Tr ( f ) is also supported at [ g, σ ] . Furthermore, the value ofTr ( f ) at [ g, σ ] is a positive integer multiple of f ([ g, σ ]) . (cid:3) If G is a p -group, we get the following result. Corollary 4.22.
Suppose that G is a p -group. Then the G -Green functor Cl ↑ GG × Σ p / J isisomorphic to the constant Mackey functor at C . The power operation of Green functors Cl G P p / J ÐÐÐ→ Cl ↑ GG × Σ p / J ≅ C is given by evaluating a character at the identity element. The following examples all follow from Proposition 4.21. We include them for comparisonwith the examples of Section 4.3.
Example 4.23.
For the group G = Z / m =
2, we have Cl ( Z / ) Cl ( Z / × Σ )/ I Tr ≅ Cl ( Z / ) Cl ( ) = C Cl ( Σ )/ I Tr ≅ C . ψ ψ Tracing the diagram using induction maps gives a a . ↦ a ↦ , σ ↦ a , σ ↦ a ≠ , σ ↦ a , σ ↦ If we further quotient by the image of the transfer from the diagonal subgroup C Ð→ Z / × Σ , then the induction diagram commutes, so that we have a map of Green functors.The quotient Green functor is constant at C , and the resulting map of Green functors P ∶ Cl Z / Ð→ C is given by restriction of class functions to the identity element. Example 4.24.
Consider G = Σ and m =
2. We use ρ to denote a 3-cycle and τ to denote(any choice of) transposition. We write C and C for the subgroups generated by ρ and τ Then by Proposition 4.21, in order for the power operation to be a map of Mackeyfunctors, we must quotient Cl ( Σ ) by the values on the conjugacy class of τ , and we mustalso quotient Cl ( C ) by the same conjugacy class. The resulting power operation of Greenfunctors P / J ∶ Cl Σ Ð→ Cl ↑ Σ Σ × Σ / J is given by C { e, ρ, τ } C { e, ρ, ρ } C { e, τ } C ( ) ( )( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ⎛⎜⎝ ⎞⎟⎠⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ C { e, ρ } C { e, ρ, ρ } CC ( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ( )⎛⎜⎝ ⎞⎟⎠ ( ) ( )⎛⎜⎝ ⎞⎟⎠( ) The target Green functor has been made constant on the 2-torsion subgroup.
Example 4.25.
Consider again G = Σ , but with m =
3. We again use ρ to denote a 3-cycleand τ to denote (any choice of) transposition. We continue to abuse notation by writing C and C for the subgroups generated by ρ and τ Then by Proposition 4.21, in order for the power operation to be a map of Mackeyfunctors, we must quotient Cl ( Σ ) by the values on the conjugacy classes of ρ and ρ , andwe must also quotient Cl ( C ) by the (collapsed) conjugacy class of ρ . The resulting poweroperation of Green functors P / J ∶ Cl Σ Ð→ Cl ↑ Σ Σ × Σ / J is given by C { e, ρ, τ } C { e, ρ, ρ } C { e, τ } C ( ) ( )( ) ⎛⎜⎝ ⎞⎟⎠ ( ) ⎛⎜⎝ ⎞⎟⎠⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ C { e, τ } C C { e, τ } C ( ) ( )( ) ( ) ( ) ( ) ( )( )( ) The target Mackey functor has been made constant on the 3-torsion subgroup.4.4.2.
Height . We now turn our attention to height 2. Let E be height 2 Morava E -theoryat the prime p , so that E ≅ W ( k )[[ u ]] , where k is a perfect field of characteristic p .Hopkins, Kuhn, and Ravenel [HKR00] introduced a rational E -algebra C and producedan isomorphism of C -algebras C ⊗ E E ( BG ) ≅ Cl ,p ( G, C ) , DDITIVE POWER OPERATIONS IN EQUIVARIANT COHOMOLOGY 37 where Cl ,p ( G ) = Cl ,p ( G, C ) denotes the character ring of C -valued functions on conju-gacy classes of commuting pairs of p -power order elements of G . This extends to a Greenfunctor Cl ,pG , where the restriction and induction maps are similar to those described inSection 4.4.1. See also [HKR00, Theorem D].A description of the power operation P m ∶ Cl ,p ( G ) Ð→ Cl ,p ( G × Σ m ) can be found in the introduction to [Sta], which is a specialization of the main result of[BS17]. One important point is that the power operation on class functions depends on a choice of ordered set of generators for sublattices of Z × p . Example 4.26.
Let G = Z / p =
2, and m =
2. We are to consider Cl ,p ( Z / ) Cl ,p ( Z / × Σ ) Cl ,p ( e ) Cl ,p ( Σ ) . P P A class function f ∈ Cl ,p ( Σ ) can be displayed as a table of values ( e, e ) ( e, σ ) ( σ, e ) ( σ, σ ) a b c d, where a, b, c, d ∈ C . For simplicity, we will assume that a, b, c, d ∈ Z ⊂ C so that certainring-automorphisms of C that appear in the general formula in [Sta] do not appear here.Following [Sta], the power operation Cl ,p ( e ) P Ð→ Cl ,p ( Σ ) is given by a ↦ ( e, e ) ( e, σ ) ( σ, e ) ( σ, σ ) a a a a. Collapsing the transfer from the subgroup Σ × Σ of Σ will eliminate the (nonlinear) valueat ( e, e ) .We will similarly describe a class function f ∈ Cl ,p ( Z / × Σ ) via a table of values ( e, e ) ( e, σ ) ( σ, e ) ( σ, σ )( , ) a b c d ( , ) e f g h ( , ) i j k l ( , ) m n o p. Then a choice of power operation P ∶ Cl ,p ( Z / ) Ð→ Cl ,p ( Z / × Σ ) , compatible with thesecond power operation on height 2 Morava E -theory at the prime 2, is ( , ) ( , ) ( , ) ( , ) a b c d ↦ ( e, e ) ( e, σ ) ( σ, e ) ( σ, σ )( , ) a a a a ( , ) b a b b ( , ) c c a b ( , ) d c b a , where all of the values that depend on a choice are displayed in color. Collapsing the transferfrom the subgroup Z / × Σ × Σ of Z / × Σ will eliminate the first column of values, whilecollapsing the transfer from the diagonal subgroup of Z / × Σ will eliminate the a ’s on the(slope negative one) diagonal.Then the diagrams of restriction maps and power operations a ( , ) ( , ) ( , ) ( , ) a b c d ( e, σ ) ( σ, e ) ( σ, σ ) a a a , ( e, σ ) ( σ, e ) ( σ, σ )( , ) a a a ( , ) b b ( , ) c b ( , ) c b and the diagram of induction maps and power operations a ( , ) ( , ) ( , ) ( , ) a ( e, σ ) ( σ, e ) ( σ, σ ) a a a , ( e, σ ) ( σ, e ) ( σ, σ )( , ) a a a ( , ) ( , ) ( , ) References [Ada78] John Frank Adams,
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Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
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