Cofreeness in Real Bordism Theory and the Segal Conjecture
aa r X i v : . [ m a t h . A T ] J a n COFREENESS IN REAL BORDISM THEORY AND THE SEGALCONJECTURE
CHRISTIAN CARRICK
Abstract.
We prove that the genuine C n -spectrum N C n C MU R is cofree,for all n . Our proof is a formal argument using chromatic hypercubes and theSlice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a newproof of the Segal conjecture for C , independent of Lin’s theorem. Introduction
In this paper, we establish the following result:
Theorem 1.1.
For all n > , the C n -spectrum N C n C M U R is cofree, i.e. the map N C n C M U R → F ( EC n + , N C n C M U R ) is an equivalence. The equivariant spectra N C n C M U R play a central role in the solution to the Ker-vaire Invariant One problem by Hill, Hopkins, and Ravenel [7]. Their detecting spec-trum Ω is the homotopy fixed point spectrum of a localization Ω O := D − N C C M U R of N C C M U R . An essential piece of their argument is the Homotopy Fixed PointTheorem ([7], 1.10), which states that this homotopy fixed point spectrum coincideswith the genuine fixed point spectrum, i.e. that Ω O is cofree. Our result showsthat this holds even before localization away from D .In the case n = 1 , Hu and Kriz show that M U R is cofree via direct computation[8]. They compute the C -homotopy fixed point and Tate spectral sequences for BP R , and deduce that ( BP R ) tC = H F , so that the result is an immediate con-sequence of the Tate square for BP R . We give a new, more conceptual proof oftheir result that generalizes readily to n > . The idea is that BP R [ v i − ] is cofreefor formal reasons, so one can take an approach via local cohomology and formcartesian cubes ˜ L BP R BP R [ v − ] BP R [ v − ] BP R [( v v ) − ] BP R [ v − ] BP R [( v v ) − ]˜ L BP R BP R [ v − ] BP R [( v v ) − ] BP R [( v v v ) − ] BP R [ v − ] BP R [( v v ) − ] and so on, and ˜ L n BP R is cofree for all n . Applying the slice tower to each vertex BP R [( v i · · · v i j ) − ] , one forms a cartesian cube in filtered C -spectra, and the limitterm gives a modified slice filtration of ˜ L n BP R . It is then a formal consequence ofthe Hill-Hopkins-Ravenel (HHR) slice theorem [7] that, taking the limit in n , onerecovers the slice tower of BP R .The n = 1 case may then be used as the base case for an induction argumentwhich allows us to reduce to showing that ( N C n C M U R ) C n → ( N C n C M U R ) hC n is an equivalence. This map may be analyzed by a separate induction argumentthat originates in various inductive proofs of versions of the Segal Conjecture. In[17], Ravenel showed that the Segal conjecture for C p n follows from the case n = 1 ;he provided both a computational approach via a modified Adams spectral sequenceas well as an approach using explicit geometric constructions. Bokstedt, Bruner,Lunoe-Nielsen, and Rognes [2] generalized the geometric approach and proved thefollowing: Theorem 1.2. ( [2] , Theorem 2 . Let X be a C p n -spectrum. Suppose for each Y ∈ { X, Φ C p ( X ) , . . . , Φ C pn − ( X ) } that π ∗ ( Y ) is bounded below, H ∗ ( Y ) is of finitetype, and Y C p → Y hC p is a p -complete equivalence. Then X G → X hG is a p -complete equivalence. In [16], Nikolaus and Scholze strengthen this result by giving a description ofthe subcategory of genuine C p n -spectra whose geometric fixed points spectra arebounded below in terms of iterated pullbacks and gluing maps. In the case of M U (( C n )) , we identify these gluing maps with either the map in the n = 1 case,or maps of the form M O ∧ k → ( N C e ( M O ∧ k )) tC Each of these in the latter case is an equivalence by the Segal conjecture:
Theorem 1.3.
For any bounded below spectrum X , the Tate diagonal X → ( N C e ( X )) tC is a -complete equivalence. This theorem was shown for X with finitely generated homotopy groups byLunøe-Nielsen and Rognes ([11], 5.13) and for all X bounded below by Nikolausand Scholze ([16], III.1.7). These both rest on the the case X = S , due to Lin: OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 3
Theorem 1.4. [10]
Let γ denote the canonical line bundle over RP ∞ , and for eachinteger n > , let RP ∞− n denote the Thom spectrum of − nγ . Then there is anequivalence of spectra RP ∞−∞ = holim n RP ∞− n ≃ ( S − ) ˆ2 We refer the reader to the introduction of [5] for a discussion of the differentforms of the Segal conjecture for C and their relation to Lin’s theorem. Lin’stheorem follows from a difficult calculation of a continuous Ext group d Ext A ( H ∗ ( RP ∞−∞ ; F ); F ) where A is the Steenrod algebra. Nikolaus and Scholze showed, however, that 1.3follows formally for all X bounded below from the case X = H F . Hahn andWilson [5] used this to show that 1.3 can be established by analysis of the descentspectral sequence for the map N C e H F → H F which reduces to a continuous Ext group calculation over a much smaller polynomialcoalgebra F [ x ] . We use Lin’s theorem to prove the following, from which 1.1follows. Theorem 1.5.
Let Y be a bounded below C -spectrum. If Y ∧ k is a cofree C -spectrum for all ≤ k < n , then N C n C Y is cofree. Our argument for 1.1 may be reversed: knowing that ( M U (( C n )) ) C n → ( M U (( C n )) ) hC n is an equivalence may be used to show that the corresponding gluing maps areequivalences, and, using the reduction of Nikolaus and Scholze to the case of X = H F , we may deduce the Segal conjecture for C . This gives a proof of the Segalconjecture for C that involves no homological algebra - apart from the Tate orbitlemma of Nikolaus and Scholze - and proceeds from a chromatic approach. Inparticular, the main piece of our argument that is not formal is the use of the HHRslice theorem. Remark . Essential to our proof is the identification Φ C ( N C C BP R ) ≃ N C e H F .In [15], Meier, Shi, and Zeng use this identification to deduce differentials in thehomotopy fixed point spectral sequence of N C e H F from differentials in the slicespectral sequence of N C C BP R . Our results should shed light on these spectralsequences.In particular, the map from the Slice SS of N C C BP R to its HFPSS is an isomor-phism below a line of slope 3 (see [18]). The Slice SS vanishes above this line, butthere are many classes above this line in the HFPSS. By Theorem 1.1, the mapbetween them must give an isomorphism on their E ∞ -pages, so there must be somepattern of differentials killing all the classes above this line in the HFPSS. Summary.
In Section 2, we show that the cofreeness of
M U (( G )) follows formallyfrom (and is equivalent to) the Hu-Kriz n = 1 case together with Lin’s theorem.This is the most direct way to Theorem 1.1, using these known results. In Section 3,we withhold knowledge of these theorems and give a different proof - via chromatichypercubes - that BP (( C )) is cofree. In turn, this result implies the n = 1 case andLin’s Theorem, which then gives the result for n > by the same induction usedin Section 2. CHRISTIAN CARRICK
Notation and Conventions.
We use Sp G to denote the category of orthogonal G -spectra or the associated ∞ -category given by taking the homotopy coherent nerveof bifibrant objects in the stable model structure of Mandell and May [14]. We usethe notation M U (( G )) and BP (( G )) to denote N GC M U R and N GC BP R respectively,as in HHR. Acknowledgments.
The n = 1 case of our chromatic hypercubes result - namelythat BP R = holim n ˜ L n BP R - is due to Mike Hill. It was his idea to use this approachto establish the n > cases. We thank him for introducing us to this problem andfor his guidance throughout the project.2. Cofreeness and Gluing Maps
Cofreeness.
We begin by reviewing the notion of cofreeness for a genuine G -spectrum. Proposition 2.1.
For X ∈ Sp G , the following are equivalent (1) X → F ( EG + , X ) is an equivalence of G -spectra. (2) X H → X hH is an equivalence of spectra for all H ⊂ G . (3) X is G + -local.Proof. For ⇐⇒ , it suffices to show that L G + ( X ) = F ( EG + , X ) . The map X → F ( EG + , X ) becomes an equivalence after smashing with G + by the Frobenius relation, and thetarget is G + -local because if Z ∧ G + ≃ ∗ , then [ Z, F ( EG + , X )] G = [ Z ∧ EG + , X ] G = 0 as EG + is in the localizing subcategory generated by G + . ⇐⇒ follows fromthe fact that the fixed point functors ( − ) H are jointly conservative, and i GH ( F ( EG + , X )) = F ( EH + , i GH X ) as can be seen from the more general statement i GH ( L E ( X )) = L i GH E ( i GH X ) (see [3], 3.2). (cid:3) Definition 2.2.
We say a G -spectrum X is cofree if any of the equivalent conditionsin 2.1 hold. Corollary 2.3.
The category of cofree G -spectra is closed under homotopy limits.Proof. This is true of any category of E -locals. (cid:3) Remark . Cofree G -spectra are often called Borel complete , or just
Borel . Thesource of this terminology is the fact that there is a forgetful functor Sp G → Fun(
BG, Sp ) from genuine G -spectra to so-called Borel G -spectra. For formal reasons, this func-tor admits a right adjoint, and it is not hard to show that this right adjoint is anequivalence onto the full subcategory of cofree G -spectra. OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 5
We will make use of the slice filtration on G -spectra, introduced for C -spectraby Dugger [4] and generalized to all finite groups G by HHR [7]. To fix notions,we use the regular slice filtration, as in Ullman [18], although for the G -spectra weconsider, using the original slice filtration in HHR would not change anything. Let X ≥ n denote that a G -spectrum is slice ≥ n , i.e. X is slice ( n − -connected. Weneed the following useful lemma: Lemma 2.5.
Suppose { X i } i ∈ N is a family of G -spectra such that, for all n ∈ Z ,all but finitely many X i have the property that X i ≥ n . Then the canonical map _ i X i → Y i X i is an equivalence.Proof. It suffices to show that, for all n ∈ Z , the map of Mackey functors M i π n ( X i ) ∼ = π n (cid:18) _ i X i (cid:19) → π n (cid:18) Y i X i (cid:19) ∼ = Y i π n ( X i ) is an isomorphism. This follows immediately from the observation that for all butfinitely many i , π n ( X i ) = 0 . Indeed, by ([7], 4.40), if Y ≥ n , then π k ( Y ) = 0 for k < ⌈ n/ | G |⌉ . (cid:3) Proposition 2.6. If M U R is cofree, then M U ∧ n R is cofree for all n ≥ , andsimilarly for BP ∧ n R .Proof. We proceed by induction on n . Since M U ∧ ( n − R is Real-oriented, we have M U ∧ n R = M U ∧ ( n − R [ b , b , . . . ] = _ m ∈ M S | m | ρ ∧ M U ∧ ( n − R where M is a monomial basis of Z [ b , b , . . . ] . By the lemma, the canonical map _ m ∈ M S | m | ρ ∧ M U ∧ ( n − R → Y m ∈ M S | m | ρ ∧ M U ∧ ( n − R is an equivalence, as M U ∧ ( n − R ≥ and S kρ ≥ k , so that S kρ ∧ M U ∧ ( n − R ≥ k by ([7], 4.26). This completes the proof, as the category of cofree C -spectra isclosed under limits and smashing with a dualizable C -spectrum, hence the targetis cofree. (cid:3) Gluing maps and cofreeness.
We set up an inductive argument to proveTheorem 1.5. To fix notation, we use Φ C k to denote the functor Sp C n → Sp and e Φ C k to denote the functor Sp C n → Sp C n − k , so that i C n − k e ◦ e Φ C k = Φ C k .Nikolaus and Scholze use a result of Hesselholt and Madsen ([6], 2.1) along withtheir Tate orbit lemma, to show: CHRISTIAN CARRICK
Proposition 2.7. ( [16] , Corollary II . . If X ∈ Sp G has the property that Φ C k X ∈ Sp is bounded below for all ≤ k < n , there is a homotopy limit diagram X C n Φ C n X ( e Φ C n − X ) hC ( e Φ C n − X ) tC ( e Φ C X ) hC n − · · · (cid:18)e Φ C X (cid:19) hC n − (cid:18) ( e Φ C X ) tC (cid:19) hC n − X hC n (cid:18) X tC (cid:19) hC n − Theorem 2.8.
Let Y be a bounded below C -spectrum. If Y ∧ k is a cofree C -spectrum for all ≤ k < n , then N C n C Y is cofree.Proof. Set X := N C n C Y . We proceed by induction on n , with the base case n = 1 being tautological. For all ≤ k < n , i C n C n − k X = N C n − k C ( Y ∧ k ) is cofree by induction, so it suffices to show the map X C n → X hC n is an equiva-lence. Since Y is bounded below, so is X , and this map is an equivalence if all ofthe short vertical maps in 2.7 are equivalences. Each such map is of the form ( f ) hC n − k : (cid:18)e Φ C k X (cid:19) hC n − k → (cid:18) ( e Φ C k − X ) tC (cid:19) hC n − k for k > , which is induced by the map in Sp C n − k f : e Φ C k X → ( e Φ C k − X ) tC It therefore suffices to show that for all k > , f is an equivalence of Borel C n − k -spectra, which by definition is simply an underlying equivalence. The underlyingmap is the natural map Φ C (cid:18) i C n − k +1 C ˜Φ C k − X (cid:19) → (cid:18) i C n − k +1 C ˜Φ C k − X (cid:19) tC so it suffices to show i C n − k +1 C ˜Φ C k − X is a cofree C -spectrum. When k = 1 , wehave i C n − k +1 C ˜Φ C k − X ≃ Y ∧ n − and for k > , one has i C n − k +1 C ˜Φ C k − X ≃ i C n − k +1 C ( N C n − k +1 e (Φ C Y )) ≃ N C e (cid:18) Φ C ( Y ∧ n − k ) (cid:19) OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 7 using the identification e Φ C k X ≃ N C n − k e (Φ C Y ) (see [15], Theorem 2.2). N C e Φ C ( Y ∧ n − k ) is cofree by Lin’s theorem: since Y ∧ n − k is bounded below and cofree, Φ C ( Y ∧ n − k ) ≃ ( Y ∧ n − k ) tC is bounded below and 2-complete. (cid:3) Remark . This result has various converses. For example, if Y is a boundedbelow C -spectrum, then N C k C Y is cofree for all ≤ k ≤ n if and only if Y ∧ k is a cofree C -spectrum for all ≤ k < n . The other direction follows because if N C k +1 C Y is cofree, then Y ∧ k = i C k +1 C N C k +1 C Y is also cofree.If Y is also a ring spectrum, then the direct converse of 2.8 is true: N C n C Y iscofree if and only if Y ∧ k is a cofree C -spectrum for all ≤ k < n . This followsbecause Y ∧ k is a retract of Y ∧ n − = i C n C N C n C Y in this case. Corollary 2.10.
For all n ≥ , M U (( C n )) is cofree, and similarly for BP (( C n )) .Proof. M U R is bounded below, so this follows immediately from 2.6, the Hu-Kriz n = 1 case, and the theorem. (cid:3) We have shown that the case n = 1 , due to Hu and Kriz, along with Lin’stheorem, implies that M U (( C n )) is cofree for all n ≥ . The argument can bereversed to point to another proof of Lin’s theorem, namely: Proposition 2.11.
For any n > , the cofreeness of M U (( C n )) implies both Lin’stheorem and the n = 1 case.Proof. If for any n > , M U (( C n )) is cofree, then a smash power of BP (( C )) iscofree, and it follows that BP (( C )) is cofree, as a retract; similarly for BP R andtherefore for its smash powers by 2.6. In this case, the limit diagram in 2.7 is asfollows: ( BP (( C )) ) C Φ C ( BP (( C )) )( e Φ C BP (( C )) ) hC ( e Φ C BP (( C )) ) tC ( BP (( C )) ) hC (( BP (( C )) ) tC ) hC The lefthand vertical arrow is an equivalence by assumption, and the middle arrowis an equivalence since BP R ∧ BP R is cofree. We find that the righthand verticalmap is an equivalence, and this is the Tate diagonal H F → ( N C e H F ) tC , whichis an equivalence if and only if Lin’s theorem holds, by ([16], III.1.7). (cid:3) Chromatic Hypercubes
Generalities on Hypercubes.
We give some general results on hypercubesthat look like (summands of) our chromatic hypercubes. In this section, we usethe language of ∞ -categories following [13]; in particular, we work in the model CHRISTIAN CARRICK of quasicategories, and use stable ∞ -categories following [12]. For a discussion ofcubical diagrams in the context of ∞ -categories, see ([12], Section 6) or [1].We fix C a stable ∞ -category that has all finite limits. Let [ n ] denote the totallyordered set { , . . . , n } . For T a totally ordered set, let P ( T ) denote its power set,regarded as a poset under inclusion. Let P ( T ) denote the sub-poset P ( T ) \ {∅} . Definition 3.1. An n -cube X in C is a functor X : P ([ n ]) → C , and a partial n -cube is a functor P ([ n ]) → C . We say an n -cube X is cartesian if the map X ( ∅ ) → holim T ∈P ([ n ]) X ( T ) is an equivalence. Construction 3.2.
Suppose for each T ∈ P ([ n ]) , one has an object C T ∈ C . Weconstruct inductively an n -cube X in C as follows: (1) When n = 1 , X is the canonical inclusion C ∅ → C ∅ ⊕ C { } . (2) We may assume inductively that we have constructed ( n − -cubes Y and Y with Y ( T ) = M S ≤ T C S and Y ( T ) = M S ≤ T C S ∪{ n } for T ∈ P ([ n − , where the maps in Y and Y are the canonical inclu-sions. X is then given by the canonical inclusion of ( n − -cubes Y →Y ⊕Y , via the identification Fun( P ([ n ]) , C ) = Fun(∆ , Fun( P ([ n − , C )) . Definition 3.3.
Suppose for each T ∈ P ([ n ]) , one has an object C T ∈ C and C ∅ = ∗ . Let X be the associated n -cube as in 3.2, and define a partial n -cube Y : P ([ n ]) ֒ → P ([ n ]) X −→ C We say a partial n -cube in C is built from disjoint split inclusions if it is equivalentto Y for some choice of objects { C T } T ∈P ([ n ]) . If X is a cartesian n -cube such thatthe corresponding partial n -cube is built from disjoint split inclusions, we say X isa cartesian n -cube built from disjoint split inclusions.To make this definition clearer, note that any partial -cube built from disjointsplit inclusions is equivalent to one of the form C C C ⊕ C ⊕ C and any partial -cube built from disjoint split inclusions is equivalent to one of theform C C ⊕ C ⊕ C C C ⊕ C ⊕ C C ⊕ C ⊕ C ⊕ C ⊕ C ⊕ C ⊕ C C C ⊕ C ⊕ C OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 9 where the inclusions are the canonical ones. We want to identify the limit of adiagram of this form, and we use a result of Antolín-Camarena and Barthel oncomputing limits of cubical diagrams inductively:
Proposition 3.4. ( [1] , . Let X : P ([ n ]) → C be a partial n -cube in C . One hasa pullback square holim S ∈P ([ n ]) X ( S ) holim S ∈P ([ n − X ( S ) X ( { n } ) holim S ∈P ([ n − X ( S ∪ { n } ) Proposition 3.5.
Let X be a partial n -cube in C built from disjoint split inclusionsso that it is equivalent to Y for some choice of objects { C T } T ∈P ([ n ]) as in 3.3. Then X satisfies (1) holim S ∈P ([ n ]) X ( S ) ≃ Ω n − C { ,...,n } (2) The map holim S ∈P ([ n ]) X ( S ) → holim S ∈P ([ n − X ( S ) is nullhomotopic.Proof. We proceed by induction on n . For n = 1 , a cartesian -cube is an equiva-lence holim S ∈P ([1]) X ( S ) ≃ −→ X ( { } ) and the map in (2) is the map to the terminal object. It is straightforward to showthat the partial ( n − -cube P ([ n − → P ([ n ]) X −→ C is built from disjoint split inclusions, and P ([ n − −∪{ n } −−−−→ P ([ n ]) X −→ C is of the form C { n } ⊕ Z where Z is a partial ( n − -cube built from disjoint splitinclusions using the objects { C T ⊕ C T ∪{ n } } T ∈P ([ n − , as in 3.3. By induction, 3.4gives a pullback square holim S ∈P ([ n ]) X ( S ) Ω n − C { ,...,n − } C { n } C { n } ⊕ Ω n − C { ,...,n − } ⊕ Ω n − C { ,...,n } which is a cartesian -cube built from disjoint split inclusions. It therefore sufficesto prove the proposition in the case n = 2 , which is the claim that for objects C , C , C ∈ C , there is a pullback square of the form Ω C C C C ⊕ C ⊕ C One may form a morphism of partial 2-cubes ∗∗ C = ⇒ C C C ⊕ C ⊕ C via 3.2 which, taking limits, constructs such a square. Taking fibers along thevertical maps, one has the identity map of Ω C ⊕ Ω C ; the square is thereforecartesian by ([1], 2.2). (cid:3) Chromatic Hypercubes and Slice Towers.
We introduce the chromatic n -cubes we need to prove Theorem 1.1 and show they split as a summand that isconstant in n and a cartesian n -cube built from disjoint split inclusions. Definition 3.6.
Consider the following hypercubes:(1) Let H n be the cartesian n -cube so that for { i , . . . , i j } ∈ P ([ n ]) H n ( { i , . . . , i j } ) = BP (( C )) [( N ( t i ) · · · N ( t i j )) − ] One may form this cube inductively in a manner similar to 3.2 by working inthe category of
M U (( C ))(2) -modules and applying the functors ( − )[ N ( t i ) − ] .See ([1], 3.1) for a similar construction.(2) Let S n,d be the cartesian n -cube defined on P ([ n ]) by S n,d : P ([ n ]) H n −−→ Sp C P d d −−→ Sp C where P d d is the d -slice functor.To understand the n -cubes S n,d , we need to determine the slices of BP (( C )) i ,...,i j := BP (( C )) [( N ( t i ) · · · N ( t i j )) − ] and this follows as expected from the HHR slice theorem for BP (( C )) . We followthe discussion in ([7], Section 6) and use their notation: in π u ∗ ( BP (( C )) ) = π ∗ ( BP ∧ BP ) , there are classes { t i } i ≥ with the property that π u ∗ ( BP (( C )) ) = Z (2) [ t i , γ ( t i ) : i ≥ as a C -algebra, where γ is the generator of C and γ ( t i ) = − t i . Inverting theclasses above, we see that π u ∗ ( BP (( C )) i ,...,i j ) = Z (2) [ t i , γ ( t i ) : i ≥ t i · · · t i j γ ( t i ) · · · γ ( t i j )) − ] and the restriction map π C ∗ ρ ( BP (( C )) i ,...,i j ) → π u ∗ ( BP (( C )) i ,...,i j ) is a split surjection (in fact an isomorphism, but we don’t need this). Lifting theclasses t i along this map, we have an associative algebra map A := S [ t i : i ≥ t i · · · t i j ) − ] → i C C BP (( C )) i ,...,i j Using the method of twisted monoid rings (and the C -commutative ring structureon M U (( C ))(2) ), this gives a map S [ C · t i : i ≥ C · ( t i · · · t i j ) − ] → BP (( C )) i ,...,i j OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 11
Proposition 3.7.
The above map S [ C · t i : i ≥ C · ( t i · · · t i j ) − ] → BP (( C )) i ,...,i j gives a refinement of homotopy. Let M d be the monomial ideal in A consisting ofthe slice sphere summands of underlying dimension ≥ d , and set K d = BP (( C )) i ,...,i j ∧ A M d Then the cofiber sequences P d +1 ( BP (( C )) i ,...,i j ) → BP (( C )) i ,...,i j → P d ( BP (( C )) i ,...,i j ) are equivalent to K d +2 → BP (( C )) i ,...,i j → BP (( C )) i ,...,i j /K d +2 ,P d +1 BP (( C )) i ,...,i j ≃ P d BP (( C )) i ,...,i j , and K d /K d +2 ≃ H Z (2) ∧ M d /M d +2 Proof.
The proof is identical to that of ([7], Section 6), where the last identificationfollows from the key computation: the reduction theorem. (cid:3)
Remark . The previous proposition should be interpreted as follows: the slicetower for BP (( C )) i ,...,i j forgets to the ordinary Postnikov tower of i C e BP (( C )) i ,...,i j , whichhas P d − d − ≃ ∗ and P d d ≃ H Z (2) ∧ W d where W d is a wedge of S d ’s over the set of monomials of degree d in π u ∗ ( BP (( C )) i ,...,i j ) = Z (2) [ t i , γ ( t i ) : i ≥ t i · · · t i j γ ( t i ) · · · γ ( t i j )) − ] The slice tower is an equivariant refinement of this wherein the odd slices vanish, H Z (2) is replaced with H Z (2) , the spheres in W d corresponding to a summand ofthe above C -module with stabilizer C are grouped with their conjugates in a C ∧ C S dρ , the spheres corresponding to a C -fixed summand are replaced with S d ρ , andthere are no free summands. For BP (( C )) i ,...,i j , we let c W i ,...,i j d denote the quotient M d /M d +2 as above, and c W d the corresponding quotient for BP (( C )) .Note that for any i , . . . , i j , c W i ,...,i j d has c W d as a split summand, correspondingto the split inclusion π u d ( BP (( C )) ) ֒ → π u d ( BP (( C )) i ,...,i j ) This splitting is natural in { i , . . . , i j } , so we see that there is a splitting S n,d ≃ ( H Z (2) ∧ c W d ) ⊕ X n,d where X n,d is a cartesian n -cube satisfying X n,d ( { i , . . . , i j } ) = H Z (2) ∧ ( c W i ,...,i j d / c W d ) We have the following connection to the generalities in 3.1:
Proposition 3.9. X n,d is a cartesian n -cube built from disjoint split inclusions. Proof. X n,d is cartesian by definition. The result - and the terminology - followsfrom the fact that for any { i , . . . , i j } , the maps π u ∗ ( BP (( C )) i k ) ֒ → π u ∗ ( BP (( C )) i ,...,i j ) are split inclusions, and after factoring out π u ∗ ( BP (( C )) ) , the maps ι k : π u ∗ ( BP (( C )) i k ) π u ∗ ( BP (( C )) ) ֒ → π u ∗ ( BP (( C )) i ,...,i j ) π u ∗ ( BP (( C )) ) are split inclusions with the property that im( ι k ) ∩ im( ι k ′ ) = { } for k = k ′ . Inparticular, for T ≤ T ′ in P ( n ) , the map X n,d ( T ) → X n,d ( T ′ ) is the split inclusion of the free H Z (2) -module on wedges of slice spheres corre-sponding to the split inclusion π u d ( BP (( C )) T ) π u d ( BP (( C )) ) ֒ → π u d ( BP (( C )) T ′ ) π u d ( BP (( C )) ) so the claim follows from the fact that π u ∗ ( BP (( C )) i ,...,i j ) π u ∗ ( BP (( C )) ) = (cid:18) M T < { i ,...,i j } T ∈P ( n ) π u ∗ ( BP (( C )) T ) π u ∗ ( BP (( C )) ) (cid:19) ⊕ ( t i · · · t i j γ ( t i ) · · · γ ( t i j )) − π u ∗ ( BP (( C )) ) π u ∗ ( BP (( C )) ) where the latter summand denotes the subgroup of π u ∗ ( BP (( C i ,...,ij ) π u ∗ ( BP (( C ) generated by mono-mials containing ( t i · · · t i j γ ( t i ) · · · γ ( t i j )) − . (cid:3) The following is an immediate consequence of 3.5 and 3.9:
Corollary 3.10.
The map S n,d ( ∅ ) → S n − ,d ( ∅ ) can be identified with ( H Z (2) ∧ c W d ) ⊕ X n,d ( ∅ ) −−−−−−→ ( H Z (2) ∧ c W d ) ⊕ X n − ,d ( ∅ ) Proof of Main Theorem.
The canonical map BP (( C )) → BP (( C )) i ,...,i j , byuniversal property, determines compatible maps BP (( C )) → H n ( ∅ ) so that there isa map BP (( C )) → holim n H n ( ∅ ) We will show this map is an equivalence, and this will complete the proof that BP (( C )) is cofree by the following: Proposition 3.11. holim n H n ( ∅ ) is cofree.Proof. The category of cofree C -spectra is closed under limits, hence it suffices toshow that each H n ( ∅ ) is cofree. There is by definition an equivalence H n ( ∅ ) ≃ −→ holim T ∈P ([ n ]) H n ( T ) = holim { i ,...,i j }∈P ([ n ]) BP (( C )) i ,...,i j so it suffices to show each BP (( C )) i ,...,i j is cofree. This is as in ([7], Section 10): wehave that Φ C ( BP (( C )) i ,...,i j ) ≃ Φ C ( BP (( C )) i ,...,i j ) ≃ ∗ , as Φ C ( N C C ( t i )) = Φ C ( t i ) = 0 OFREENESS IN REAL BORDISM THEORY AND THE SEGAL CONJECTURE 13 and similarly Φ C ( N C C ( t i )) = Φ C ( i C C N C C ( t i )) = Φ C ( t i · γ ( t i )) = 0 (cid:3) To show that the map BP (( C )) → holim n H n ( ∅ ) is an equivalence, we use the slice filtration to work in the ∞ -category Fun( Z op , Sp G ) of filtered G -spectra (see [12], 1.2.2). We refer to [19] for a treatment of the slicefiltration in an ∞ -categorical context. Let T : Sp G → Fun( Z op , Sp G ) be the functor which associates to a G -spectrum its slice tower, which may beobtained as in ([12], 1.2.1.17). Functoriality gives a map of filtered C -spectra f : T ( BP (( C )) ) → holim n (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19) Theorem 3.12. BP (( C )) is cofree, independent of Lin’s theorem.Proof. Let holim : Fun( Z op , Sp C ) → Sp C be the functor sending a tower to itshomotopy limit. Limits are computed pointwise in functor categories, so we findthat holim (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19) ≃ H n ( ∅ ) It therefore suffices to show that f is an equivalence. Note that the filtrationinduced on H n ( ∅ ) is not its slice filtration, hence we use the notation e P k : Fun ( Z op , Sp C ) ev k −−→ Sp C and e P kk = fib ( e P k → e P k − ) Note that e P kk (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19) ≃ holim { i ,...,i j }∈P ([ n ]) e P kk (cid:18) T ( BP (( C )) i ,...,i j ) (cid:19) ≃ ( ∗ k = 2 d − S n,d ( ∅ ) k = 2 d The map e P d d ( f ) is then identified with the map H Z (2) ∧ c W d → holim n (( H Z (2) ∧ c W d ) ⊕X n,d ) ≃ holim n ( H Z (2) ∧ c W d ) ⊕ holim n X n,d ( ∅ ) By 3.10, the lefthand summand is constant in n , and the righthand summand ispro-zero, hence the map is an equivalence.To establish that f is an equivalence, it therefore suffices to show that colim k e P k (cid:18) holim n (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19)(cid:19) ≃ ∗ i.e. that the filtration on holim n H n ( ∅ ) strongly converges. Note that by ([7], 4.40),if X ∈ Sp C , then π l ( P k X ) = 0 for l ≥ ⌈ ( k + 1) / ⌉ . Taking limits, it follows that π l (cid:18) e P k (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19)(cid:19) = 0 for l ≥ ⌈ ( k + 1) / ⌉ , and so π l (cid:18) e P k (cid:18) holim n (cid:18) holim { i ,...,i j }∈P ([ n ]) T ( BP (( C )) i ,...,i j ) (cid:19)(cid:19)(cid:19) = 0 for l ≥ ⌈ ( k + 1) / ⌉ by the Milnor sequence. It follows that, for any l , taking thecolimit as k → −∞ of π l gives zero. (cid:3) Remark . This result recovers the Hu-Kriz result that BP R is cofree: since BP (( C )) is cofree, i C C BP (( C )) = BP R ∧ BP R is cofree, hence so is the retract BP R .Alternatively, as discussed in the introduction, one may argue similarly to 3.12 toshow that BP R is cofree, and the result in this case is due to Mike Hill. References [1] O. Antolín-Camarena and Tobias Barthel.
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