Farrell–Tate cohomology

In this section, we briefly mention a generalization of the Tate construction to general groups G, including compact and infinite discrete groups. In fact, most of what we do works for a general Kan complexS in place ofBG.

Recall that Farrell [34] had generalized Tate cohomology to infinite discrete groups of finite virtual cohomological dimension. He essentially constructed a norm map

(D_BG⊗X)_hG−!X^hG

for a certain G-equivariant object DBG, but he worked only on the level of abelian groups. Klein generalized this to spectra [59], and gave a universal characterization of the resulting cohomology theory [60].

Here, we prove the following general result. It is closely related to Klein’s axioms in [60]. We have also been informed by Tobias Barthel that he has obtained similar results in current work in progress.

Theorem I.4.1. Let S be a Kan complex, and consider the ∞-category Sp^S= Fun(S,Sp). Let p:S!∗ be the projection to the point. Then p^∗: Sp!Sp^S has a left adjoint p!: Sp^S!Sp given by homology, and a right adjoint p∗: Sp^S!Spgiven by coho-mology.

(i) The ∞-category Sp^S is a compactly generated presentable stable ∞-category.

For every s∈S, the functor s_!: Sp!Sp^S takes compact objects to compact objects, and for varying s∈S,the objects s_!S generate Sp^S.

(ii) There is an initial functor p^T_∗: Sp^S!Spwith a natural transformation p_∗!p^T_∗, with the property that p^T_∗ vanishes on compact objects.

(iii) The functor p^T_∗: Sp^S!Sp is the unique functor with a natural transformation p_∗!p^T_∗ such that p^T_∗ vanishes on all compact objects,and the fiber of p_∗!p^T_∗ commutes with all colimits.

(iv) The fiber of p_∗!p^T_∗ is given by X7!p_!(D_S⊗X)for a unique object D_S∈Sp^S. The object D_S is given as follows. Considering S as an ∞-category, one has a functor Map:S×S!S,sending a pair (s, t)of points in S to the space of paths between sand t.

Then, DS is given by the composition DS:S−!Fun(S,S) ^Σ

∞

−−−+!Fun(S,Sp) = Sp^S−−−^p∗!Sp.

(v) The map p!(DS⊗−)!p∗ is final in the category of colimit-preserving functors from Sp^S toSpover p_∗,i.e. it is an assembly map in the sense of Weiss–Williams, [91].

(vi) Assume that, for all s∈S and X∈Sp, one has p^T_∗(s_!X)'0. Then, there is a unique lax symmetric monoidal structure on p^T_∗ which makes the natural transformation p_∗!p^T_∗ lax symmetric monoidal.

We note that, by [59, Corollary 10.2], the condition in (vi) is satisfied if S=BG, where G is a compact Lie group. In fact, in the case of groups, the functor s_!, for s:∗!BGbeing the inclusion of the base point, is given by induction, and the condition is asking that the generalized Tate cohomology vanishes on induced representations.

This theorem suggests the following definition.

Definition I.4.2. TheSpivak–Klein dualizing spectrum ofS is the object D_S∈Sp^S mentioned in TheoremI.4.1(iv).

In classical language, this is a parameterized spectrum overS in the sense of May–

Sigurdsson [78], and it has been discussed in this form by Klein [62, §5]. By Theo-remI.4.1(iv), the fiber over any points∈S is given by

limt∈SΣ^∞₊ Map(s, t).

Klein observed that this gives a homotopy-theoretic definition of the Spivak fibration for finite CW complexes [88]. In particular, ifS is a closed manifold of dimension d, then, by Poincar´e duality, DS is fiberwise a sphere spectrum shifted into degree−d. In the caseS=BG for a topological group G, the dualizing spectrum has also been described by Klein [59]. In that case, one has the formula

DBG= (Σ^∞₊G)^hG,

regarded as a spectrum with G-action via the remaining G-action from the other side.

This is a spectral version of the Γ-module

H^∗(Γ,Z[Γ])

that appears in Bieri–Eckmann duality of discrete groups [15]. The norm map takes the form

((Σ^∞₊G)^hG⊗X)_hG−!X^hG.

If the functor p_∗: Sp^S!Sp commutes with all colimits, for example if S is a finite CW complex, then, by Theorem I.4.1(v), the map p!(DS⊗X)!p∗X is an equivalence for all X∈Sp^S, as p∗ itself is colimit-preserving. This is a topological version of Bieri–

Eckmann duality [15], which states that for certain discrete groups Γ, there are functorial isomorphisms

H_d−i(Γ, D_Γ⊗X)∼=Hⁱ(Γ, X)

for the Γ-module D_Γ=H^d(Γ,Z[Γ]), and some d depending on Γ. In general, the norm map encodes a “generalized Poincar´e duality onS”, and the cofiberp^T_∗ of the norm map can be regarded as a “failure of generalized Poincar´e duality onS”. In the proof, we will ignore all set-theoretic issues. These can be resolved by passing to-compact objects as in the proof of TheoremI.3.1.

Proof of Theorem I.4.1. By [69, Proposition 5.5.3.6] and [71, Proposition 1.1.3.1], Sp^S is presentable and stable. As s^∗ commutes with all colimits (and in particular filtered colimits), it follows that s! preserves compact objects. Now a map f:X!Y of objects in Sp^S is an equivalence if and only if for alls∈S and i∈Z, the induced map πis^∗X!πis^∗Y is an isomorphism. Butπis^∗X=Hom(S[i], s^∗X)=Hom(s!S[i], X), so that the objectss!Sare generators.

Now part (ii) follows from TheoremI.3.3(iii), applied toC=Sp^S andD=(Sp^S)ωthe compact objects inC. By TheoremI.3.3(iii), the functorp^T_∗ can be computed as follows.

It sends anyX∈Sp^S to

colim_Y∈(SpS)_ω/Xp_∗cofib(Y !X).

For part (iii), we claim first that fib(p∗!p^T_∗) does indeed commute with all colimits. By definition, all functors are exact, so we have to show that the fiber commutes with all filtered colimits. For this, assume thatX=colim_iX_i is a filtered colimit. In that case, as all objects of (Sp^S)_ωare compact,

(Sp^S)_ω/X= colimi(Sp^S)_ω/Xi.

Commuting colimits using [69, Proposition 4.2.3.8 and Corollary 4.2.3.10], one sees that fib(p∗(X)!p^T_∗(X)) = colim_Y∈(SpS)_ω/Xp∗Y = colimicolim_Y∈(SpS)_ω/Xip∗Y

= colimifib(p∗(Xi)!p^T_∗(Xi)),

as desired. For the uniqueness, note that colimit-preserving functors from Sp^S to Sp are equivalent to exact functors from (Sp^S)ω to Sp by [69, Proposition 5.5.1.9]; but we are given an exact functor from (Sp^S)ω to Sp, namely the restriction of p∗. The same argument proves (v).

For part (iv), note that colimit-preserving functors Sp^S!Sp are equivalent to colimit-preserving functors S^S!Sp by [71, Corollary 1.4.4.5]. These, in turn, are given by functorsS!Sp by [69, Theorem 5.1.5.6], i.e. objects of Sp^S. Unraveling, any colimit-preserving functorF: Sp^S!Sp is given byF(X)=p_!(D⊗X) for a unique objectD∈Sp^S. To identifyD_S in our case, we look at the family of compact objectsF:S!(Sp^S)_ω send-ings∈S tos_!S, i.e. we consider ∆_!S∈Fun(S×S,Sp) as a functorS!Fun(S,Sp) for the diagonal ∆:S!S×S. The functorp^T_∗ vanishes identically, so

p!(DS⊗F)'p_∗F

as functors S!Sp. Unraveling definitions, the left side is just D_S, and the right side gives the desired formula.

Finally, in part (vi), note that if p^T_∗ vanishes on all objects of the forms!X, then we can use TheoremI.3.6(ii) to produce the desired lax symmetric monoidal structure as in the proof of TheoremI.3.1, by applying it to C=Sp^S and the full subcategory D generated bys!X for alls∈S,X∈Sp, which is a⊗-ideal.

Corollary I.4.3. Consider the ∞-category Sp^BT of spectra with T-action. There is a natural transformation Σ(−hT)!−^hTwhich exhibitsΣ(−hT)as the universal colimit preserving functor mapping to the target. The cofiber −^tTadmits a unique lax symmetric monoidal structure making −^hT!−^tT lax symmetric monoidal.

Moreover, for any n>1, consider the functors −^hCn,−^tCn: Sp^BT!Spwhich factor over Sp^BCn. Then,there is a unique (lax symmetric monoidal) natural transformation

−^tT!−^tCn making the diagram

−^hT //

−^tT

−^hCn //−^tCn

of (lax symmetric monoidal)functors commute.

Proof. The first part follows from TheoremI.4.1and [59, Theorem 10.1 and Corol-lary 10.2]. For the second part, we first check the result without lax symmetric monoidal structures. Then, by the universal property of−^tT, it suffices to show that−^tCnvanishes on the spectra s!S[i], where s:∗!BT is the inclusion of a point, and i∈Z. For this, it suffices to show thats!S[i] is compact as an object of Sp^BCn, i.e.f^∗: Sp^BT!Sp^BCn preserves compact objects, wheref:BCn!BTis the canonical map. For this, it suffices to show thatf_∗commutes with all colimits. But the fibers off_∗are given by cohomology ofT/Cn, which is a finite CW complex (a circle).

To get the result with lax symmetric monoidal structures, we need to check that−^tCn vanishes on all induced spectra s!X for X∈Sp. Although one can give more abstract reasons, let us just check this by hand. First, note that s_! commutes with all limits, as its fibers are given by homology of T, which is a finite CW complex; in fact, s_! decreases coconnectivity by at most 1, and so also the composition (s_!)_hCn commutes with Postnikov limits as in Lemma I.2.6(ii). Thus, by taking the limit over allτ_6nX, we may assume that X is bounded above, say X is coconnective. Now all functors commute with filtered colimits, so one may assume that X is bounded, and then that X is an Eilenberg–MacLane spectrum, in fact X=HZ. It remains to compute −^tCn onHZ[S¹]=s_!HZ. As the homotopy of (HZ[S¹])^tCn is a module over the homotopy of (HZ)^tCn which is a 2-periodic ring, it suffices to see that π_i(HZ[S¹])_hCn vanishes for

i>1. But this is given by πi(HZ[S¹/Cn])∼=πiHZ[S¹], which is non-zero only in degrees 0 and 1.

We will need the following small computation. It also follows directly from the known computation of π∗(HZ)^tT as the ring Z((X)), but we argue directly with our slightly inexplicit definition of (HZ)^tT.

Lemma I.4.4. Consider HZ∈Sp^BT endowed with the trivial T-action. The natural map (HZ)^tT!(HZ)^tCn induces isomorphisms

πi(HZ)^tT/n'πi(HZ)^tCn.

Proof. The result is classical on π_i for i60, as it becomes a result about group cohomology. Note that, by adjunction, there is a natural map HZ[S¹]!HZ, and −^tT and−^tCn vanish on HZ[S¹] (the latter by the proof of the previous corollary). On the other hand, the cone ofHZ[S¹]!HZis given byHZ[2] with necessarily trivialT-action.

Inductively, we may replaceHZbyHZ[2i] for alli>0, and then the result is clear in any given degree by takingilarge enough.