Manifolds, K-theory and the calculus of functors
MMANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS
GREGORY ARONE AND MICHAEL CHING
Abstract.
The Taylor tower of a functor from based spaces to spectra can be classified accordingto the action of a certain comonad on the collection of derivatives of the functor. We describevarious equivalent conditions under which this action can be lifted to the structure of a moduleover the Koszul dual of the little L -discs operad. In particular, we show that this is the case whenthe functor is a left Kan extension from a certain category of ‘pointed framed L -manifolds’ andpointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen’salgebraic K-theory of spaces functor is classified by an action of the Koszul dual of the little 3-discsoperad. In previous work [2, 3] we described the structure possessed by the Goodwillie derivatives of afunctor from the category of based spaces, T op ∗ , to the category of spectra, S p . This structureis that of a symmetric sequence with coaction of a certain comonad C . The comonad C can bedescribed as the homotopy colimit of a sequence C → C → C → . . . of comonads where a coaction by C L gives a symmetric sequence precisely the structure of a rightmodule over the operad KE L (the Koszul dual of the stable reduced little L -discs operad).It follows that any right KE L -module M gives rise to a C -coalgebra, via the canonical comonadmap C L → C , and hence to some functor F whose derivatives are precisely the original M . If thisis the case, let us say that the derivatives of F are a KE L -module . This construction is natural,so that a morphism of KE L -modules induces a natural transformation between the correspondingfunctors.The above construction suggests several tasks: most notable being to decide which functors F , andwhich natural transformations, arise in this way for a given L . One answer to this is given by thefollowing theorem, which is the main result of Section 2. Theorem 0.1.
Let F : T op ∗ → S p be a finitary polynomial functor and L ≥ an integer. Thenthe following are equivalent: (1) the derivatives of F are a KE L -module; (2) there is an E L -comodule N such that, for X ∈ T op ∗ : F ( X ) (cid:39) N ∧ E L Σ ∞ X ∧∗ ; where the symmetric sequence Σ ∞ X ∧∗ becomes an E L -module via the diagonal maps on X ; (3) F is the left Kan extension, along the inclusion Γ L → T op ∗ , of a functor G : Γ L → S p ,where Γ L is the subcategory of T op ∗ described in Definition 2.2 below. In particular, one can view the formula in part (2) as an explicit description of how a polynomialfunctor can be pieced together from homogeneous pieces. The special case where L = 0 tells usthat the C -coalgebra structure on the derivatives of F is trivial if and only if F splits as a product Date : September 1, 2018. a r X i v : . [ m a t h . A T ] O c t GREGORY ARONE AND MICHAEL CHING of the layers of its Taylor tower. Part (3) of the theorem then tells us that this is the case if andonly if F is the left Kan extension of a functor defined on Γ , which, as we shall see, is the categoryof finite pointed sets and pointed functions that are injective away from the basepoint.In Section 3 we use Theorem 0.1 to give conditions under which the derivatives of a representablefunctor Σ ∞ Hom T op ∗ ( X, − ) are a KE L -module. We show that this is the case when X is a ‘pointedframed manifold’ of dimension L . This basically means that X is a finite based cell complex suchthat removing the basepoint leaves behind a framed manifold. For example, the sphere S L has thisproperty and so the derivatives of the functor Σ ∞ Ω L ( − ) are a KE L -module.Our construction of a KE L -module structure on the derivatives of Σ ∞ Hom T op ∗ ( X, − ) is functorialin ‘pointed framed embeddings’ in the variable X . This means that if f : X → Y is a pointed mapwhose restriction to f − ( Y − { y } ) preserves the framing (up to scalar multiples), then the inducedmap f ∗ : ∂ ∗ (Σ ∞ Hom T op ∗ ( Y, − )) → ∂ ∗ (Σ ∞ Hom T op ∗ ( X, − ))can be realized as a map of KE L -modules. We then have the following consequence which extendspart (3) of Theorem 0.1. Theorem 0.2.
Let fMfld L ∗ denote the category of pointed framed manifolds and pointed framedembeddings. Let F : T op ∗ → S p be a finitary homotopy functor. If F is the homotopy left Kanextension of a pointed functor G : fMfld L ∗ → S p along the forgetful functor fMfld L ∗ → T op ∗ , then thederivatives of F are a KE L -module. If F is polynomial, the converse of the above statement alsoholds. Section 4 contains an application of the previous results, where we show that the derivatives ofWaldhausen’s algebraic K-theory of spaces functor A ( X ) are a KE -module. Acknowledgements.
The second author would like to thank Andrew Blumberg for useful conver-sations concerning Waldhausen’s algebraic K-theory of spaces, and David Ayala for sharing someof his work with John Francis on zero-pointed manifolds. Work for this article was carried outwhile the second author was a Research Member at the Mathematical Sciences Research Institutein Spring 2014. The second author was supported by the National Science Foundation via grantDMS-1308933. 1.
Background
Let T op f ∗ be the category of finite based CW-complexes and S p the category of S -modules of EKMM[9]. We let [ T op f ∗ , S p ] denote the category of pointed, simplicially-enriched functors F : T op f ∗ → S p .Recall that a functor F is n -excisive if it takes strongly homotopy cocartesian ( n + 1)-cubes tohomotopy cartesian cubes, and we say that F is polynomial if it is n -excisive for some integer n .Each functor F ∈ [ T op f ∗ , S p ] has a Taylor tower F → · · · → P n F → P n − F → . . . P F → P F = ∗ where P n F ∈ [ T op f ∗ , S p ] is n -excisive and the natural transformation F → P n F is initial, up tohomotopy, among maps from F to an n -excisive functor.The layers of the Taylor tower are the functors D n F := hofib( P n F → P n − F ) ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 3 and these can be expressed as D n F ( X ) (cid:39) ( ∂ n F ∧ (Σ ∞ X ) ∧ n ) h Σ n where ∂ n F is a spectrum with (naive) Σ n -action called the n th derivative of F .In [3] we described three related classifications of polynomial functors from based spaces to spectra:in terms of(1) coalgebras over a comonad C on symmetric sequences;(2) comodules over the commutative operad Com ;(3) Kan extensions from the category of pointed finite sets and pointed functions.We recall the details of each of these approaches.Let E L be the standard little L -discs operad in based spaces, where E L ( n ) is the space of ‘standard’embeddings f : (cid:96) n D L → D L , with a disjoint basepoint, where operad composition is given bycomposition of embeddings. Let E L denote the stable reduced version of E L given by E L ( n ) := (cid:40) Σ ∞ E L ( n ) if n > ∗ if n = 0 . Let KE L be (a cofibrant model for) the Koszul dual operad of E L as in [7].Associated to the operad KE L is a comonad C L , on the category of symmetric sequences (in S p ),given by C L ( A )( k ) := (cid:89) n (cid:89) n (cid:16) k Map( KE L ( n ) ∧ . . . ∧ KE L ( n k ) , A ( n )) Σ n . The comonad structure is determined by the operad structure on KE L and a C L -coalgebra isprecisely a right KE L -module.The standard inclusions D ⊆ D ⊆ D ⊆ . . . determine a sequence of operads E → E → E → · · · which, in turn, induces a dual sequence of operads KE ← KE ← KE ← · · · and a sequence of comonads C → C → C → · · · . Let C be the objectwise homotopy colimit of this sequence. Then C inherits a comonad structure.In [3] we showed that the derivatives ∂ ∗ F of a functor F : T op f ∗ → S p have a coaction by C andthat ∂ ∗ sets up an equivalence between the homotopy theory of polynomial functors F and that ofbounded C -coalgebras.The second classification of polynomial functors T op ∗ → S p (due to Dwyer and Rezk) is via functors Ω → S p where Ω is the category of nonempty finite sets and surjections. We think of such functorsas comodules over the commutative operad Com in the category of spectra, given by
Com ( n ) = S for all n ≥ P of spectra, a P -comodule is a symmetric sequence N together with a structuremap N ( n ) ∧ P ( n ) ∧ . . . ∧ P ( n k ) → N ( k ) GREGORY ARONE AND MICHAEL CHING for each surjection n (cid:16) k . Here n denotes the finite set { , . . . , n } and, for a surjection α : n (cid:16) k ,we write n i := | α − ( i ) | . A Com -comodule is the same as a functor Ω → S p .Associated to any functor F : T op f ∗ → S p is a Com -comodule N [ F ] given by N [ F ]( k ) := Nat X ∈ T op f ∗ (Σ ∞ X ∧∗ , F ( X ))with Com -comodule structure arising from the
Com -module structure on Σ ∞ X ∧∗ that is inducedby the diagonal on the based space X .Dwyer and Rezk showed that a polynomial F can be recovered from the Com -comodule N [ F ] viaan equivalence N [ F ] ˜ ∧ Com Σ ∞ X ∧∗ ˜ −→ F ( X ) . Here ˜ ∧ Com denotes the homotopy coend of
Com -comodule and a
Com -module. This constructionsets up an equivalence between the homotopy theory of polynomial functors T op f ∗ → S p and thatof bounded Com -comodules. (Dwyer and Rezk’s work is not published; an account of this can befound in [3]).For the third classification of polynomial functors from based spaces to spectra, let Γ denotethe category of finite pointed sets and basepoint-preserving functions. Any polynomial functor F : T op f ∗ → S p is equivalent to the homotopy left Kan extension of its restriction to the subcategoryΓ ⊆ T op f ∗ . More precisely, this sets up an equivalence, for each n , between the homotopy theoryof n -excisive functors T op f ∗ → S p , and that of pointed functors Γ ≤ n → S p where Γ ≤ n is the fullsubcategory of Γ whose objects are the finite pointed sets of cardinality (excluding the basepoint)at most n .We can summarize these classifications and their relationships in the following diagram whichcommutes up to natural equivalence, and in which each functor induces an equivalence of homotopytheories:(1.1) Coalg b ( C )Comod b ( Com ) [ T op f ∗ , S p ] poly [Γ , S p ] b (cid:55) (cid:55) hocolim L − ˜ ∧ E L B L (cid:47) (cid:47) − ˜ ∧ Com Σ ∞ X ∧∗ (cid:39) (cid:39) L (cid:103) (cid:103) ∂ ∗ (cid:55) (cid:55) hLKan Here [Γ , S p ] denotes the category of pointed functors Γ → S p , and [Γ , S p ] b is the subcategory ofthose functors (and natural transformations) that are left Kan extensions along Γ ≤ n ⊆ Γ for some n .and the subscripts ‘ ≤ n ’ denote the subcategories of n -truncated objects. The subscripts b on theleft and top objects denote the categories of bounded Com -comodules and bounded C -coalgebrasrespectively. The arrow marked hLKan is homotopy left Kan extension along the inclusion Γ ⊆ T op f ∗ .We have introduced the three arrows ∂ ∗ , − ˜ ∧ Com Σ ∞ X ∧∗ and hLKan in (1.1), so now let us describethe other two functors in this picture.First note that a Com -comodule N inherits the structure of a E L -comodule for any L via pullbackalong the operad map E L → Com . In [3] we introduced a certain bisymmetric sequence B L with an E L -module action on one variable and a KE L -module action on the other. Forming the homotopycoend of an E L -comodule N with this E L -module results in a KE L -module which we think of as a ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 5 certain ‘Koszul dual’ of N . When N is a Com -comodule, there is an induced sequence of homotopycoends N ˜ ∧ E B E → N ˜ ∧ E B E → . . . whose homotopy colimit is a C -coalgebra. This construction gives the top-left hand functor in(1.1).We think of the bottom-left functor in (1.1) as part of a ‘Pirashvili’-type equivalence between thecategory of functors Ω → S p and the category of pointed functors Γ → S p where Γ is the categoryof all finite pointed sets and pointed functions. Explicitly, this functor L is given by L ( N )( J + ) = N ˜ ∧ Com Σ ∞ ( J + ) ∧∗ , which is just the restriction of the middle horizontal functor in (1.1) to the subcategory Γ ≤ n of T op f ∗ .2. Functors whose derivatives are a module over the Koszul dual of the littlediscs operad
We now turn to a description of those functors F : T op f ∗ → S p for which the C -coalgebra structureon ∂ ∗ F arises from a KE L -module structure for some given L . We have three characterizationsthat correspond to the three classifications of polynomial functors in (1.1). Our main result is asfollows. Theorem 2.1.
Let F : T op f ∗ → S p be a polynomial functor and L ≥ an integer. Then thefollowing are equivalent: (1) the derivatives ∂ ∗ F have a KE L -module structure, from which F can then be reconstructedby F ( X ) (cid:39) (cid:103) Map C ( ∂ ∗ R X , U ( ∂ ∗ F )) where U denotes the forgetful functor from KE L -modules to C -coalgebras, and (cid:103) Map C ( − , − ) is the derived mapping spectrum for two C -coalgebras; (2) there is a bounded E L -comodule N and a natural equivalence F ( X ) (cid:39) N ˜ ∧ E L Σ ∞ X ∧∗ ;(3) there is a simplicially-enriched functor G : Γ L → S p and an equivalence hLKan ( G ) (cid:39) F where Γ L is the subcategory of T op f ∗ introduced in Definition 2.2 below, and hLKan denotesthe enriched homotopy left Kan extension along the inclusion Γ L → T op f ∗ . The category Γ L is defined as follows. Definition 2.2.
For an integer L ≥
0, let D L + denote the standard closed unit disc in R L with adisjoint basepoint. Then we denote by Γ L the (non-full) subcategory of T op f ∗ whose objects are thebased spaces of the form (cid:95) I D L + for a finite set I , and whose morphisms are the continuous basepoint-preserving functions f : (cid:95) I D L + → (cid:95) J D L + with the following properties: GREGORY ARONE AND MICHAEL CHING • for each i ∈ I , the restriction f i of f to the corresponding copy of D L is: either a standardaffine embedding (i.e. of the form x (cid:55)→ ax + b for some a > b ∈ D L ) into one of thecopies of D L in the target; or the constant map to the basepoint; • the images of the non-constant maps f i are disjoint.For convenience, we often write I + for the object of Γ L given by the based space (cid:87) I D L + . Thecategory Γ L is topologically-enriched, with mapping spaces inherited from those of T op ∗ . Thesemapping spaces can be written(2.3) Γ L ( I + , J + ) = (cid:95) α : I + → J + (cid:94) j ∈ J E L ( I j )where the coproduct is taken over the set of pointed maps α : I + → J + , where I j := α − ( j ), andwhere E L denotes the non-reduced little L -discs operad in based spaces (with E L (0) = S ). Thisdescription says also that Γ L is the PROP associated to the operad E L .We write [Γ L , S p ] for the category of pointed topologically-enriched functors from Γ L to S p .We prove Theorem 2.1 by constructing a diagram of categories and functors of the following form:(2.4) Mod( KE L )Comod( E L ) [ T op f ∗ , S p ][Γ L , S p ] (cid:36) (cid:36) Map C ( ∂ ∗ ( R X ) , U ( − )) (cid:58) (cid:58) Q (cid:47) (cid:47) − ˜ ∧ E L Σ ∞ X ∧∗ (cid:36) (cid:36) L (cid:58) (cid:58) hLKan This diagram should be compared to (1.1) to which it is closely related.The three functors whose target is [ T op f ∗ , S p ] correspond to the three ways to construct a functor F : T op f ∗ → S p described in the statement of Theorem 2.1. The required result follows by establishingthat the other two functors in this diagram induce equivalences of homotopy categories on boundedobjects, and that the whole diagram commutes up to natural equivalence. Koszul duality between bounded E L -comodules and bounded KE L -modules. First notethat we can conveniently describe modules and comodules over an operad P in terms of functorsdefined on the PROP associated to P . If P is an operad of spectra, we have a S p -enriched category P with objects the nonempty finite sets, and with mapping spectra given by(2.5) P ( I, J ) := (cid:95) I (cid:16) J (cid:94) j ∈ J P ( I j ) . Here we take the coproduct over all surjections α : I (cid:16) J , and, for j ∈ J , write I j := α − ( j ) with α understood. The composition and unit maps for the category P come directly from the compositionand unit for the operad P .A (right) P -module can now be identified with a S p -enriched functor P op → S p , and a P -comodulewith a S p -enriched functor P → S p . We freely use these identifications throughout this paper. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 7
One further note: when the operad P arises by taking the suspension spectrum of an operad P of unbased spaces, i.e. by P ( n ) = Σ ∞ P ( n ) + , we can, similarly, form a T op -enriched category P (defined in the same way as P . A P -module can then be identified with a T op -enriched functor P → S p .We now describe the top-left functor in (2.4) and show that it induces an equivalence of homotopytheories. More precisely, we construct a Quillen adjunction (where both sides have the projectivemodel structure) Q : Comod( E L ) (cid:29) Mod( KE L ) : R and show that it restricts to an equivalence on the subcategories of bounded objects. This adjunc-tion is a form of Koszul duality and is constructed via the methods of [3] which we now recall.The key construction, made in [3, Def. 3.38 and after] is of a certain bisymmetric sequence B L ,associated to the operad E L , that has an E L -module structure on its first variable, and a KE L -module structure on its second variable. Each of these structures, when taken individually, isequivalent to a trivial structure, but when considered together they are not. Explicitly we have(2.6) B L ( I, J ) := (cid:89) I (cid:16) J (cid:94) j ∈ J B ( , E L , E L )( I j )where the product is taken over all surjections between the finite sets I and J , and for such asurjection α and j ∈ J we use I j to denote α − ( j ).We can think of B L as an E L -module in the category of KE L -modules, in other words as a S p -enrichedfunctor E opL → Mod( KE L ), or equivalently, a T op -enriched functor B L : E opL → Mod( KE L ) . The category Mod( KE L ) has a projective model structure that is compatible with its enrichmentin T op , so has a topologically-enriched cofibrant replacement functor c : Mod( KE L ) → Mod( KE L ).Write ˜ B L for the composite E opL (cid:47) (cid:47) B L Mod( KE L ) (cid:47) (cid:47) c Mod( KE L ) . In other words, ˜ B L is a replacement for B L (as an E L -module in the category of KE L -modules) andhas the property that, for each finite set I , the KE L -module ˜ B L ( I, − ) is cofibrant in the projectivemodel structure.We now define our left adjoint Q : Comod( E L ) → Mod( KE L )by Q ( N ) := N ∧ E L ˜ B L . This is an enriched coend formed over the category E L and it inherits a KE L -module structure fromthat on ˜ B L . The functor Q has right adjoint R : Mod( KE L ) → Comod( E L )given by R ( M ) := Map KE L (˜ B L , M )with E L -comodule structure arising from the E L -module structure on ˜ B L . The right adjoint pre-serves fibrations and trivial fibrations since each ˜ B L ( I, − ) is a cofibrant KE L -module, so we have aQuillen adjunction.Note that the left adjoint here is a model for the ‘derived indecomposables’ of an E L -comodule,and the right adjoint can be viewed as a model for the ‘derived primitives’ of a KE L -module. GREGORY ARONE AND MICHAEL CHING
Proposition 2.7.
The Quillen adjunction Q : Comod( E L ) (cid:29) Mod( KE L ) : R restricts to an equivalence between the homotopy categories of bounded E L -comodules and bounded KE L -modules.Proof. We first show that Q and R preserve boundedness (up to equivalence). Suppose N is an n -truncated E L -comodule (i.e. N ( k ) = ∗ for k > n ). Then Q ( N ) is equivalent, as a symmetricsequence, to a derived coend N ˜ ∧ E L whose k th term can be written as the geometric realization of a simplicial object with r -simplices (cid:95) n ≥ ...n r ≥ k N ( n ) ∧ Σ n E L ( n , n ) ∧ Σ n . . . ∧ Σ nr − E L ( n r − , n r ) ∧ Σ nr ( n r , k ) . Here is the ‘trivial’ bisymmetric sequence with ( I, J ) = Σ ∞ Bij ( I, J ) + , where Bij ( I, J ) is the setof bijections from I to J . This k th term is trivial if k > n and so Q ( N ) is also n -truncated.Similarly, if M is an n -truncated KE L -module, then R ( M ) is equivalent to the derived end (cid:103) Map KE L ( , M )which is, by a similar calculation, also n -truncated. It follows that Q and R determine an adjunctionbetween the homotopy categories of bounded E L -comodules and bounded KE L -modules.We now prove that this adjunction is an equivalence by showing that the derived unit and counitof the adjunction are equivalences when applied to bounded objects. First take a bounded E L -comodule N . Our goal is to show that the derived unit η : N → RQ N is an equivalence. To see thislet N ≤ n denote the n -truncation of N , defined by N ≤ n ( k ) := (cid:40) N ( k ) if k ≤ n ; ∗ if k > n. There are natural fibre sequences of E L -comodules N ≤ ( n − → N ≤ n → N = n , where N = n has only its n th non-trivial, and equal to N ( n ). Since both Q and R preserve fi-bre sequences (which are the same as cofibre sequences in the stable categories Comod( E L ) andMod( KE L )), we can use these sequences to reduce to the case that N is concentrated in a singleterm, and, in particular, has a trivial E L -comodule structure.Proposition 3.67 of [3] tells us that when N is a bounded trivial E L -comodule, there is an equivalenceof KE L -modules(2.8) N ∧ E L ˜ B L (cid:39) C L ( N )where the right-hand side can be thought of as the ‘cofree’ KE L -module generated by N . ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 9
Now consider the following diagram (of symmetric sequences):(2.9) N Map KE L (˜ B L , N ∧ E L ˜ B L )Map Σ (˜ B L , N ) Map KE L (˜ B L , C L ( N )) (cid:47) (cid:47) η (cid:15) (cid:15) ∼ (cid:15) (cid:15) ∼ (cid:47) (cid:47) ∼ = where the top map η is the derived unit of the adjunction, the right-hand map is the equivalence of(2.8), the bottom isomorphism is the adjunction between the forgetful and cofree functors betweensymmetric sequences and KE L -modules, and the left-hand map is adjoint to the composite N ∧ E L ˜ B L ˜ −→ C L ( N ) (cid:47) (cid:47) ε N where ε is the counit map for the comonad C L . Since ˜ B L is equivalent to (as an E L -module inits first variable), this adjoint map is an equivalence. It is easy to check that the above diagramcommutes and it follows that η is an equivalence.We use a similar method to prove that the derived counit (cid:15) : QR M → M is an equivalence for abounded KE L -module M , this time reducing to the case of cofree KE L -comodules. Suppose that M is n -truncated and consider the map of symmetric sequences M → M = n . This is adjoint to a map of KE L -modules M → C L ( M ( n ))which is an isomorphism in terms k with k ≥ n . (Those terms are trivial on both sides when k > n ,and are both equal to M ( n ) when k = n .) There is therefore a fibre sequence of KE L -modules ofthe form M (cid:48) → M → C L ( M ( n ))where M (cid:48) is ( n − n to reduce to proving that thederived counit is an equivalence for a bounded cofree KE L -module, i.e. one of the form C L ( A ) fora bounded symmetric sequence A . Note that this covers the base case of the induction because any1-truncated KE L -module is cofree.Now consider the following diagram (of symmetric sequences):(2.10) Map Σ (˜ B L , A ) ∧ E L ˜ B L A ∧ E L ˜ B L Map KE L (˜ B L , C L ( A )) ∧ E L ˜ B L C L ( A ) (cid:47) (cid:47) ∼ (cid:15) (cid:15) ∼ = (cid:15) (cid:15) ∼ (cid:47) (cid:47) (cid:15) analogous to (2.9), where the bottom horizontal map is the derived counit of the adjunction andthe right-hand vertical map is the equivalence of (2.8). It follows that (cid:15) is an equivalence for thecofree KE L -module C L ( A ), and hence for all bounded KE L -modules. This completes the proof that( Q , R ) restricts to an equivalence between the homotopy categories of bounded E L -comodules andbounded KE L -modules. (cid:3) Remark 2.11.
For an arbitrary operad P in S p , there is an adjunction of the form described inProposition 2.7, between P -comodules and KP -modules. However, this is typically not an equiva-lence, even on bounded objects. The proof of 2.7 relies heavily on [3, 3.67] which in turn crucially depends on the fact that the operad term E L ( n ) is a finite free Σ n -spectrum (i.e. a cell Σ n -spectrumformed from finitely many free cells). The corresponding result does hold for an operad P that sharesthis property. Pirashvili-type equivalence between E L -comodules and functors Γ L → S p . We now turnto the equivalence between E L -comodules and functors Γ L → S p that forms the bottom-left mapin the diagram (2.4). In this case, there is no need to restrict to bounded objects. This is ageneralization of Theorem 3.78 of [3] in the way that work of S(cid:32)lomi´nska [18] generalizes that ofPirashvili [17]. It can be viewed as a covariant, and enriched, version of a theorem of Helmstutler[13].The categories Comod( E L ) and [Γ L , S p ] have projective model structures and in this section webuild a Quillen equivalence between them. This is constructed, Morita-style in a similar manner tothe adjunction of Proposition 2.7 but with a different ‘bimodule’ object that we now introduce. Definition 2.12.
Define a functor B L : E opL × Γ L → T op ∗ by(2.13) B L ( I, J + ) := (cid:95) I → J (cid:94) j ∈ J E L ( I j ) . Notice the similarity between this definition and that of E L in (2.5). The difference is that thecoproduct here is taken over all functions from I to J , not just the surjections. (Crucial here isthat E L (0) is non-trivial.) More precisely, we have an isomorphism(2.14) B L ( I, J + ) ∼ = (cid:95) K ⊆ J E L ( I, K )where a component on the left-hand side corresponding to a function α : I → J is identified with aterm on the right-hand side with K = α ( I ). The E L -module structure on the first variable of B L is chosen so that this is an isomorphism of E L -modules.There is also a close connection between B L and the mapping spaces Γ L ( I + , J + ) for the categoryΓ L . The latter object is a coproduct indexed by all pointed functions α : I + → J + . We can thinkof B L as consisting only of those components corresponding to α with α ( I ) ⊆ J .Here is another way to describe this relationship. For finite sets I, J , consider the cube of basedspaces, indexed by subsets K ⊆ I , given by K (cid:55)→ Γ L ( K + , J + ) . The strict total cofibre of this cube (given, for example, by taking iterated cofibres in each direction)is then homeomorphic to B L , i.e. we have(2.15) tcofib K ⊆ I Γ L ( K + , J + ) ∼ = B L ( I, J + )and we define the functoriality of B L in its second variable in such a way that, for each I , thisis a natural isomorphism of functors Γ L → T op ∗ . (The reader can check that the two structurescommute yielding the required functor B L .The following calculation is crucial. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 11
Lemma 2.16.
For each finite set I , there is a natural weak equivalence, of functors Γ L → T op ∗ : thocofib K ⊆ I Γ L ( K + , − ) ˜ −→ B L ( I, − ) . Proof.
Given (2.15), this amounts to showing that, for each J + ∈ Γ L , the canonical map betweenthe homotopy and strict total cofibres of the cube X ( K ) := { Γ L ( K + , J + ) } is a weak equivalence.A standard condition for this is that, for each K ⊆ I , the mapcolim L (cid:40) K X ( L ) → X ( K )is a cofibration of spaces. In our setting, that map is the inclusion into Γ L ( K + , J + ) of thosecomponents corresponding to pointed functions α : K + → J + for which α ( K ) (cid:42) J . An inclusionof components is a cofibration so this completes the proof. (cid:3) Definition 2.17.
We now build a Quillen equivalence of the form L : Comod( E L ) (cid:29) [Γ L , S p ] : X that is constructed as follows.Viewing B L as a functor E opL → [Γ L , T op ∗ ], we can compose with a (topologically-enriched) cofibrantreplacement functor for [Γ L , T op ∗ ] to obtain ˜ B L : E opL × Γ L → T op ∗ such that ˜ B L (cid:39) B L and withthe property that for each finite set I , ˜ B L ( I, − ) is cofibrant in [Γ L , T op ∗ ].Now define L : Comod( E L ) → [Γ L , S p ] by L ( N ) := N ∧ E L ˜ B L with right adjoint X : [Γ L , S p ] → Comod( E L ) given by X ( G ) := Map Γ L (˜ B L , G ) , a coend and end, respectively, using, respectively, the tensoring and cotensoring of S p over T op ∗ . Proposition 2.18.
The adjunction of Definition 2.17 is a Quillen equivalence between the projec-tive model structures on
Comod( E L ) and [Γ L , S p ] .Proof. This proof is very similar to the proof in [3] that the bottom-left map in (1.1) is part ofa Quillen equivalence. First note that for each finite nonempty set I , the object ˜ B L ( I, − ) is acofibrant object in the projective model structure on [Γ L , S p ]. It follows that the right adjoint X preserves fibrations and trivial fibrations (since these are detected objectwise in both categories)and hence that ( L , X ) is a Quillen adjunction.Our goal is now to show that the derived unit and counit of this adjunction are equivalences.Key to this is the equivalence of Lemma 2.16. Applying the derived mapping space construction (cid:103) Map J + ∈ Γ L ( − , G ( J + )) to this, we get, using Yoneda, an equivalence(2.19) X ( G )( I ) ˜ −→ thofib K ⊆ I G ( K + ) . This describes X ( G )( I ) as the I th cross-effect of G evaluated at ( D L + , . . . , D L + ), hence the notation X .In particular, it follows from this calculation that the right adjoint X preserves homotopy colimits,since the cross-effect can also be recovered from the total homotopy cofibre of the same cube. We canalso use (2.19) to show, by induction, that X reflects equivalences, i.e. if a natural transformation g : G → G (cid:48) induces an equivalence of E L -comodules X ( G ) ˜ −→ X ( G (cid:48) ), then g itself is an equivalence. Now consider the derived unit map N → XL N for a E L -comodule N . Since both X and L commutewith homotopy colimits, we can use the cofibrantly generated model structure on Comod( E L ) toreduce to the case that N is either the source or target of one of the generating cofibrations, i.e. ofthe form N ( r ) = A ∧ E L ( n, r )for some finite spectrum A and positive integer n . In this case, the unit map reduces to A ∧ E L ( n, r ) → Map J + ∈ Γ L (˜ B L ( r, J + ) , A ∧ ˜ B L ( n, J + ))and so, since A is finite, it is sufficient to show that the canonical map(2.20) E L ( n, r ) → Map J + ∈ Γ L (Σ ∞ ˜ B L ( r, J + ) , Σ ∞ ˜ B L ( n, J + ))(determined by the E L -module structure on B L ) is a weak equivalence. This follows from theexistence of the following commutative diagram: E L ( n, r ) Map J + ∈ Γ L (Σ ∞ ˜ B L ( r, J + ) , Σ ∞ ˜ B L ( n, J + ))thofib K ⊆ r (cid:95) I ⊆ K E L ( n, I ) thofib K ⊆ r Σ ∞ B L ( n, K + ) (cid:47) (cid:47) (cid:15) (cid:15) ∼ (cid:15) (cid:15) ∼ (cid:47) (cid:47) ∼ where the right-hand map is the equivalence of (2.19), the bottom map comes from (2.14), and theleft-hand vertical map is induced by the inclusion of E L ( n, r ) into the initial vertex of the given cubeas the term where I = K = r . It is a simple calculation to show that this map is an equivalence.This completes the proof that the derived unit map is an equivalence for all N .Finally, we turn to the derived counit map (cid:15) G : LX G → G . First note that the unit map X G → XLX G is an equivalence by the above, and so the triangle identity implies that X ( (cid:15) G ) : XLX G → X G is an equivalence. But we have already shown that X reflects equivalences, so we deduce that (cid:15) G is an equivalence too. This completes the proof that ( L , X ) is a Quillen equivalence. (cid:3) Proof of Theorem 2.1.
We now return to diagram (2.4). We construct this as follows:(2.21) Mod( KE L ) Coalg( C )Comod( E L ) Comod( Com ) [ T op f ∗ , S p ][Γ L , S p ] [Γ , S p ] (cid:47) (cid:47) U (cid:39) (cid:39) A (cid:55)→ F A (cid:55) (cid:55) Q (cid:47) (cid:47) − ˜ ∧ E L Com (cid:39) (cid:39) L (cid:55) (cid:55) hocolim L − ˜ ∧ E L ˜ B L (cid:47) (cid:47) −∧ Com (cid:94) Σ ∞ X ∧∗ (cid:39) (cid:39) −∧ Com (cid:94) Σ ∞ ( J + ) ∧∗ (cid:47) (cid:47) hLKan (cid:55) (cid:55) hLKan The right-hand part of this diagram is (1.1) without the restrictions to bounded objects or poly-nomial functors. The bottom horizontal map is (enriched) homotopy left Kan extension along thefunctor π : Γ L → Γ that sends the object (cid:87) I D L + of Γ L to the finite pointed set I + , and which for-gets the ‘topological’ information about a morphism. To finish the proof of Theorem 2.1, it is nowsufficient to show that the two quadrilaterals in this diagram commute up to natural equivalence. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 13
For the top-left quadrilateral, take an E L -comodule N . The composite of the middle horizontal anddiagonal arrows applied to N giveshocolim L (cid:48) ( N ˜ ∧ E L Com ˜ ∧ E L (cid:48) ˜ B L (cid:48) which is equivalent to N ˜ ∧ E L (hocolim L (cid:48) Com ˜ ∧ E L (cid:48) ˜ B L (cid:48) ) . Writing
Com = hocolim L (cid:48) E L (cid:48) this reduces to(*) N ˜ ∧ E L (hocolim L (cid:48) ˜ B L (cid:48) ) . The maps B → B → . . . induced by the operad maps E → E → . . . are all equivalences (each term is equivalent to thetrivial bisymmetric sequence ) and so in particular we have an equivalence (of E L -modules in onevariable and C -coalgebras in the other):hocolim L (cid:48) ˜ B L (cid:48) (cid:39) ˜ B L so (*) is equivalent to N ˜ ∧ E L ˜ B L = U ( Q ( N )).For the bottom-left quadrilateral, it is sufficient to show the diagram formed by replacing thehorizontal functors with their right adjoints commutes up to natural equivalence. This diagramhas the form Comod( E L ) Comod( Com )[Γ L , S p ] [Γ , S p ] (cid:15) (cid:15) −∧ E L ˜ B L ∼ (cid:15) (cid:15) ∼ −∧ Com (cid:94) Σ ∞ ( J + ) ∧∗ (cid:111) (cid:111) U (cid:111) (cid:111) res Each direction here takes the form N (cid:55)→ J + (cid:55)→ (cid:95) K ⊆ J N ( K ) which proves the claim. This completes the construction of the diagram (2.21). Theorem 2.1now follows from the fact that Q and L are equivalences ( Q only on those bounded objects thatcorrespond to polynomial functors). (cid:3) Example 2.22.
Taking L = 0 in Theorem 2.1 we see that the following conditions are equivalentfor a polynomial functor F : T op f ∗ → S p : • the C -coalgebra structure on ∂ ∗ ( F ) is ‘trivial’ (in the sense that it is equivalent to thatarising from the symmetric sequence ∂ ∗ ( F ) via the coaugmentation 1 = C → C ); • the Taylor tower of F splits; • F is the left Kan extension from a functor on the subcategory of T op f ∗ consisting of thefinite pointed sets and functions f : I + → J + that are injections away from the basepoint(i.e. for each j ∈ J , f − ( j ) has at most one element). A category of pointed framed manifolds
In this section, we use Theorem 2.1 to find a wider variety of functors whose derivatives are a KE L -module. In particular, we consider the question of when the derivatives of a representable functor,i.e. one of the form Σ ∞ Hom T op ∗ ( X, − ) for a finite based CW-complex X , have the structure of a KE L -module for some given L .We show that this is the case when X is (or is homotopy equivalent to) a ‘pointed framed L -dimensional manifold’ in the sense of Definition 3.1 below. This means that when the basepointof X is removed, the remaining space can be given the structure of a framed smooth manifold ofdimension L , possibly with boundary. In particular, notice that the sphere S L has this property, sowe deduce that the iterated (stable) loop-space functor Σ ∞ Ω L has a KE L -module structure on itsderivatives. Similarly, since S is a pointed framed 1-manifold, the derivatives of the free loop-spacefunctor Σ ∞ L = Σ ∞ Hom T op ∗ ( S , − ) form a KE -module.We also give conditions under which a map f : X → Y between pointed framed manifolds inducesa map between the derivatives of the representable functors that is a morphism of KE L -modules.This happens when f is a ‘pointed framed embedding’, i.e. f is an embedding (when restricted tothe part of X that does not map to the basepoint in Y ) that respects the framing.All the manifolds we consider in this section are smooth, possibly with boundary. A framing on asmooth L -dimensional manifold M is a choice of isomorphism of vector bundles T M ∼ = M × R L . In particular we do not mean ‘stably-framed’, that is, we do not allow for the addition of trivialbundles to
T M get such an isomorphism. Such a structure therefore exists if and only if M isparallelizable.Given a smooth map f : M → N between smooth L -manifolds, each with a framing, we can expressthe derivative Df ( x ) : T x M → T f ( x ) N as an ( L × L ) matrix via the identifications T x M ∼ = R L and T f ( y ) N ∼ = R L that come with the framings. We say that the map f is framed if there is a locallyconstant function λ : M → R + such that Df ( x ) = λ ( x ) Id for all x ∈ M . This means that, on eachcomponent of M , the derivative is a constant positive scalar multiple of the identity matrix. Definition 3.1.
Fix an integer L ≥
0. A pointed framed L -manifold is a based topological space( X, x ) together with the structure of a framed smooth L -dimensional manifold, possibly withboundary, on the topological space X − { x } .Given two pointed framed L -manifolds ( X, x ) and ( Y, y ), a pointed framed embedding f : X → Y is a continuous basepoint-preserving map such that the restriction of f to f − ( Y −{ y } ) is a framedembedding. We write Emb f ∗ ( X, Y ) for the space of pointed framed embeddings X → Y with thecompact-open topology. These spaces form the mapping spaces in a topologically-enriched categoryof pointed framed L -manifolds. Remark 3.2.
Our pointed framed manifolds are a smooth and framed version of the ‘zero-pointedmanifolds’ developed independently, and much more extensively, by Ayala and Francis [4] for theirstudy of factorization homology.We now introduce a condition on a pointed framed manifold (
X, x ) that restricts how the topologyaround the basepoint x is related to the framing on X − { x } . The idea is that x should have abasis of open neighbourhoods that are contractible in a framed sense. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 15
Definition 3.3.
A pointed framed manifold (
V, x ) is f-contractible if there exists a homotopy h : V ∧ [0 , + → V such that h = id V , h is the constant map with value x , and, for each t ∈ [0 , h t : V → V is a pointed framed embedding. Thus V deformation retracts onto x through pointed framedembeddings.A pointed framed manifold ( X, x ) is locally f-contractible if there is a basis of open neighbourhoods { V α } around x such that each ( V α , x ) is framed contractible. Examples 3.4.
Let M be a framed L -dimensional manifold with boundary. Then we can getlocally f-contractible pointed framed manifolds from M in a variety of ways:(1) For any x ∈ M , ( M, x ) is a locally f-contractible pointed framed manifold (with the inducedframing on M − { x } . In this case, a basis of f-contractible neighbourhoods of x is given bythe framed embedded discs with centre x .(2) Adding a disjoint basepoint +, we get a locally f-contractible pointed framed manifold( M + , +).(3) We can add a ‘framed collar’ to the boundary of M by forming M (cid:48) := ( M × { } ) ∪ ∂M ×{ } ( ∂M × [0 , ∂M × [0 ,
1] has the product framing. The quotient M (cid:48) / ( ∂M × { } )is then a locally f-contractible pointed framed manifold that is homotopy equivalent to M/∂M . Examples 3.5.
The sphere S L can be given the structure of a locally f-contractible pointed framed L -manifold by thinking of it as the one-point compactification, either of R L or of the open unitdisc ˚ D L inside R L . In each case, a basis of f-contractible neighbourhoods of the basepoint consistsof the complements of the closed discs centred the origin, with contractions given by Euclideandilations. Definition 3.6.
We now introduce a full subcategory of the category of pointed framed L -manifoldsthat we denote fMfld L ∗ . The objects of fMfld L ∗ are those locally f-contractible pointed framed man-ifolds whose underlying space has the structure of a finite cell complex. In particular, we have aforgetful functor U : fMfld L ∗ → T op f ∗ . Remark 3.7.
Any finite cell complex X is homotopy equivalent to a compact codimension zerosubmanifold of Euclidean space, and hence to a compact framed manifold. Thus any based finitecell complex X is homotopy equivalent to an object of fMfld L ∗ for some L . Similarly, any map f : X → Y between finite cell complexes can be modelled as a pointed framed embedding byembedding the mapping cylinder of f into some Euclidean space. Examples 3.8.
The category fMfld ∗ is equivalent to that denoted Γ in Definition 2.2: the objectsare finite pointed sets, and the morphisms are those basepoint preserving functions that are injectiveaway from the basepoint.For L >
0, the category Γ L is the full subcategory of fMfld L ∗ whose objects are the finite wedgesums (cid:87) n D L + . Interestingly, fMfld L ∗ also contains a full subcategory that is equivalent to Γ opL , namelythat whose objects are the finite wedge sums (cid:87) n S L (with framing coming from that on the openunit disc). The morphisms in this case are the Thom-Pontryagin collapse maps associated to thecorresponding morphisms in Γ L .The following lemma contains the key technical result of this section: a calculation of the homotopytype of the space of pointed framed embeddings from a finite wedge sum (cid:87) n D L + to a locally f-contractible pointed framed manifold X . Lemma 3.9.
Let X be a locally f-contractible pointed framed manifold, and let C ( n, X ) denotethe subspace of X n consisting of n -tuples ( x , . . . , x n ) where x i (cid:54) = x j for i (cid:54) = j , unless x i = x j = x . (Thus C ( n, X ) is the ordinary configuration space of n points in X with the exception thatrepetitions of the basepoint are allowed.) Then there is a natural weak equivalence Θ : Emb f ∗ (cid:32)(cid:95) n D L + , X (cid:33) ˜ −→ C ( n, X ) given by Θ( f ) := ( f (0) , . . . , f n (0)) . Proof.
We use the following criterion of McCord [16, Thm. 6] to show that Θ is a weak equivalence.
Lemma 3.10 (McCord) . Let f : Y → Z be a continuous map and let U be a basis for the topologyon Z such that, for each U ∈ U the map f − ( U ) → U is a weak equivalence. Then f is a weakequivalence. We construct the necessary basis U as follows. Let V denote a basis of f-contractible open neigh-bourhoods of x . The elements of U are then the open subsets of C ( n, X ) of the form U = [ V × · · · × V n ] ∩ C ( n, X )where each V i is either: (i) an embedded open disc in X − { x } (whose closure is also contained in X − { x } ); or (ii) an element of V .We insist that V , . . . , V n are pairwise disjoint with the exception of those that are in V which weinsist are equal. In other words, { V , . . . , V n } is a collection of disjoint embedded open discs in X − { x } , possibly together with an element of V (that can be repeated). The corresponding set U then consists of the n -tuples ( x , . . . , x n ) where x i ∈ V i , and the x i are distinct (though repetitionsof the basepoint are permitted).It is not hard to see that U is a basis for C ( n, X ). Each U ∈ U is contractible: there is adeformation retraction from U to the point ( x , . . . , x n ) where x i is the center of each embeddeddisc V i , and x i = x for those V i that equal V ∈ V . A deformation retraction is given by combininga contraction of each embedded disc to its center with a contraction c of V of the form guaranteedby Definition 3.3.It is now sufficient by Lemma 3.10 to show that each Θ − ( U ) is contractible. We do this in twostages:(1) construct a homotopy between the identity on Θ − ( U ) and a map whose image is containedin a certain subset ∆( U ) ⊆ Θ − ( U );(2) show that ∆( U ) is contractible.Notice that Θ − ( U ) is the set of pointed framed embeddings f : (cid:95) n D L + → X such that f i (0) ∈ V i for each i . We take the subset ∆( U ) to consist of those f such that f i ( D L ) ⊆ V i for each i .Our strategy for part (1) of the proof is simply to ‘shrink’ a given embedding f until its image discsare contained in the sets V i as required. The Tube Lemma ensures that this can be done for some ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 17 nonzero scale factor. More precisely, we set, for f ∈ Θ − ( U ): s ( f ) := sup { t ∈ [0 , | f i ( ty ) ∈ V i for any y ∈ D L and i = 1 , . . . , n } . The number s ( f ) ∈ [0 ,
1] is well-defined since f i (0) ∈ V i for each i and the Tube Lemma impliesthat s ( f ) >
0. We use the construction of s ( f ) to define a homotopy h : Θ − ( U ) ∧ [0 , + → Θ − ( U )by h ( f, t ) i ( y ) := (cid:40) f i ( ty ) if t ≥ s ( f ) / f i (cid:16) s ( f )2 y (cid:17) if t < s ( f ) / . Then h ( f,
1) = f and h ( f, ∈ ∆( U ) for each f ∈ Θ − ( U ). For part (1) of the proof we just need toshow that h is continuous, for which it is sufficient to show that s : Θ − ( U ) → (0 ,
1] is a continuousfunction. To prove this, we have to understand the topology on Θ − ( U ).Recall the definition of the compact-open topology on Emb f ∗ ( (cid:87) n D L + , X ). A basis for this topologyis given by sets of the form O ( K , W ) ∩ · · · ∩ O ( K r , W r )for compact K , . . . , K r ⊆ (cid:87) n D L + and open W , . . . , W r ⊆ X , where O ( K, W ) consists of those f such that f ( K ) ⊆ W .Now suppose that 0 < s ( f ) < (cid:15) > s ( f ) − (cid:15), s ( f ) + (cid:15) ) ⊆ [0 , s ( f ), we have, for all i , f i ([ s ( f ) − (cid:15) ] D L ) ⊆ V i . and also that there is some j and some y ∈ D L such that x (cid:48) := f j ([ s ( f ) + (cid:15)/ y ) / ∈ V j . Suppose that x (cid:48) ∈ ¯ V j . Then the embedding f j must embed some disc around [ s ( f ) + (cid:15)/ y as somedisc around x (cid:48) . Since x (cid:48) ∈ ¯ V j − V j , some point in this disc must be outside of ¯ V j . We can thereforefind some y (cid:48) ∈ D L and some 0 < (cid:15) (cid:48) < (cid:15) such that x (cid:48)(cid:48) := f j ([ s ( f ) + (cid:15) (cid:48) ] y (cid:48) )is contained in an embedded disc D that does not intersect V j .We now observe that f is contained in the open set O ( { [ s ( f ) + (cid:15) (cid:48) ] y (cid:48) } ( j ) , D ) ∩ n (cid:92) i =1 O ([ s ( f ) − (cid:15) ] D L ( i ) , V i ) . Moreover, if g is also in this open set, then s ( f ) − (cid:15) < s ( g ) < s ( f ) + (cid:15). This shows that s is continuous at f .Similarly, if s ( f ) = 1 and 0 < (cid:15) <
1, then there is some open subset of Emb f ∗ ( (cid:87) n D L + , X ) containing f and contained in s − ((1 − (cid:15), s is continuous, and it follows that h is continuous. This completes part (1).Now, for part (2) of the proof, we show that the space ∆( U ) is contractible. Let I = { i ∈ n | V i = V ∈ V} . We then have ∆( U ) ∼ = Emb f ∗ ( (cid:95) I D L + , V ) × (cid:89) i/ ∈ I Emb f ( D L , V i ) where Emb f ( D L , V i ) is the space of framed embeddings D L → V i . Recall that V i is an embeddedopen disc in X − { x } with some centre x i and some finite radius r i >
0. This space of embeddingsis clearly contractible. (For example, it deformation retracts onto the single embedding with centre x i and radius r i / c : V ∧ [0 , + → V is a contraction of V to x such that c t : V → V is a pointed framedembedding for each t , then we define a contraction c (cid:48) of Emb f ∗ ( (cid:87) I D L + , V ) by c (cid:48) t ( f ) := c t ◦ f. Altogether then, we deduce that ∆( U ) is contractible, which completes part (2).Combining a contraction of ∆( U ) with the homotopy h defined above, we obtain a contraction ofΘ − ( U ) as required. This completes the proof of Lemma 3.9. (cid:3) Remark 3.11.
For us, the real significance of Lemma 3.9 is that the configuration spaces C i ( n, X )satisfy a cosheaf property with respect to the covering of X by open subsets that are a disjoint unionof embedded discs and an f-contractible neighbourhood of the basepoint. Lemma 3.9 then impliesthat the embedding spaces Emb f ∗ ( (cid:87) n D L + , X ) satisfy the same property. We use this property inthe proof of Proposition 3.17 below.We now want to construct a KE L -module that models the derivatives of the representable functorΣ ∞ Hom T op ∗ ( X, − ) when X ∈ fMfld L ∗ . It is easier to describe first an E L -module associated to X ,from which we get the required KE L -module via a form of Koszul duality. Definition 3.12.
We define a functor M • : fMfld L ∗ → Mod( E L ) by M X ( n ) := B L ( n, J + ) ˜ ∧ J + ∈ Γ L Emb f ∗ (cid:32)(cid:95) J D L + , X (cid:33) where B L is as in (2.13) and ˜ ∧ denotes the (derived) enriched homotopy coend over the categoryΓ L of Definition 2.2. The symmetric sequence M X inherits an E L -module structure from that onthe first variable of B L , and the functoriality in X arises from that of the embedding space functorEmb f ∗ ( − , − ). Example 3.13.
When X = D L + we have an equivalence of E L -modules M X (cid:39) B L ( ∗ , + ) ∼ = E L . Remark 3.14.
In Example 3.13, we see that M X ( n ) has the homotopy type of a configurationspace of points in X . This is a general phenomenon. It can be shown that there is a natural weakequivalence η X : M X ( n ) ˜ −→ C ∗ ( n, X )where C ∗ ( n, X ) denotes the subspace of X ∧ n consisting of those n -tuples of distinct points in X ,together with the basepoint. The map η X is constructed from the maps B L ( n, J + ) ∧ Emb f ∗ (cid:32)(cid:95) J D L + , X (cid:33) → C ∗ ( n, X ); ( f, g ) (cid:55)→ ( g ( f (0)) , . . . , g ( f n (0)))where f ∈ B L ( n, J + ) is identified with a sequence of embeddings f i : D L → (cid:96) J D L . Example 3.15.
For
L > X = S L we have M S L ( n ) (cid:39) (cid:40) S L for n = 1; ∗ for n > . ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 19
This follows from the equivalence of Remark 3.14. The case n = 1 is immediate and for n > C ∗ ( n, S L ) is contractible. Such a contraction is given by c : C ∗ ( n, S L ) ∧ [0 , + → C ∗ ( n, S L ) by c ([ x , . . . , x n ] , t ) := (cid:40) [ t x , . . . , t x n ] for t > ∗ for t = 0;where we write S L = R L / { x : | x | ≥ } . For any n -tuple x , . . . , x n in S L , at least one of the points x i is not represented by 0 ∈ R L and hence t x i represents the basepoint in S L for sufficiently small t . Thus c is continuous and provides the necessary contraction. Remark 3.16.
For a framed compact L -manifold M (with boundary), the based spaces M + and M/∂M are connected by Atiyah duality. This appears to correspond to a version of Koszul dualitybetween the E L -modules M M + and M M/∂M . (Recall from Example 3.4 that we can build a locallyf-contractible pointed framed manifold that is homotopy equivalent to
M/∂M .) For example,notice from Examples 3.13 and 3.15 that M D L + and M S L are the free and trivial E L -modules onone ‘generator’ respectively. This is a module-level (as opposed to algebra-level) version of anobservation by Ayala and Francis relating Poincar´e and Koszul duality via factorization homology.The key property of the E L -module M X is now given by the following result. Let Com denotethe commutative operad in based spaces: with
Com ( n ) = S for all n . The operad map E L → Com makes
Com into a left E L -module and we can then form the two-sided bar construction B ( M X , E L , Com ). Proposition 3.17.
There is an equivalence of
Com -modules, natural in X ∈ fMfld L ∗ of the form θ X : B ( M X , E L , Com ) ˜ −→ X ∧∗ . Proof.
The map θ X is built from a map M X ◦ Com → X ∧∗ which has components, associated toa surjection α : n (cid:16) k , of the form M X ( k ) → X ∧ n given by the composite M X ( k ) (cid:47) (cid:47) η X C ∗ ( k, X ) → X ∧ k (cid:47) (cid:47) ∆ α X ∧ n where η X is as in Remark 3.14.To prove that θ X is an equivalence, we start with the case X = (cid:87) I D L + , for some finite set I . Wethen have B ( M X , E L , Com )( n ) (cid:39) B ( (cid:95) K ⊆ I E L ( n, K ) , E L , Com ) ∼ = (cid:95) K ⊆ I Com ( n, K ) . We can identify
Com ( n, K ) with the subset of ( I + ) ∧ n consisting of n -tuples in I whose union is K . Taking the sum over all subsets K of I , we get an equivalence B ( M X , E L , Com )( n ) ˜ −→ ( I + ) ∧ n . The map θ X factors this via the equivalence X ∧ n → ( I + ) ∧ n given by contracting each component of X to a single point. It follows that θ X is an equivalencein this case.Next consider the case that X = (cid:87) I D L + ∨ K for some Hausdorff f-contractible pointed framedmanifold K . (So X consists of K together with a disjoint union of discs.) It follows that X is homotopy equivalent to X (cid:48) := (cid:87) I D L + (with the relevant homotopies consisting of pointed framedembeddings). This equivalence induces corresponding equivalences of Com -modules B ( M X , E L , Com )( n ) (cid:39) B ( M X (cid:48) , E L , Com )( n ); X ∧ n (cid:39) X (cid:48)∧ n and it follows by the previous case that θ X is an equivalence.For an arbitrary X ∈ fMfld L ∗ we now let P denote the poset of subsets of X that are of the form (i.e.isomorphic as pointed framed manifolds) (cid:87) I D L + ∨ K for f-contractible K . Consider the followingcommutative diagram:(3.18) hocolim U ∈P B ( M U , E L , Com ) B ( M X , E L , Com )hocolim U ∈P U ∧∗ X ∧∗ (cid:47) (cid:47) (cid:15) (cid:15) ∼ (cid:15) (cid:15) θ X (cid:47) (cid:47) Since the left-hand vertical map is an equivalence by the previous case, it is sufficient to show thatthe horizontal maps are weak equivalences.The bar construction B ( M X , E L , Com ) is built from the embedding functor Emb f ∗ ( (cid:87) n D L + , − ) bytaking various homotopy colimits and so it is sufficient to prove thathocolim U ∈P Emb f ∗ ( (cid:95) n D L + , U ) → Emb f ∗ ( (cid:95) n D L + , X )is an equivalence, for which, by Lemma 3.9, it is enough to show that(3.19) hocolim U ∈P C ( n, U ) → C ( n, X )is an equivalence for any n ≥ Lemma 3.20 (Dugger/Isaksen) . Let U be an open cover of a space X such that each finite in-tersection of elements of U is covered by other elements of U . Then the homotopy colimit of thediagram formed by the elements of U and the inclusions between them is weakly equivalent to X . For U ∈ P , let ˚ U denote the interior of U in X , i.e. the disjoint union of a collection of open discsin X with an f-contractible neighbourhood of the basepoint. We apply Lemma 3.20 to the opencover U := { C ( n, ˚ U ) ⊆ C ( n, X ) | U ∈ P} . This collection covers C ( n, X ) because each n -tuple of points in X is contained in some U ∈ P .Consider a point x = ( x , . . . , x n ) in the finite intersection: C ( n, ˚ U ) ∩ · · · ∩ C ( n, ˚ U r ) . Then each x i not equal to x is the center of some closed disc D i contained in the intersection˚ U ∩ · · · ∩ ˚ U r . Then x is an element of C ( n, ˚ U )where U = (cid:87) i : x i (cid:54) = x ( D i ) + ∨ K for some f-contractible K contained in each U i that does not intersectany D i . (Such a K exists since x has a basis of f-contractible neighbourhoods.) It follows that theopen cover U satisfies the hypotheses of Lemma 3.20 and we deduce that the maphocolim U ∈P C ( n, ˚ U ) → C ( n, X )is a weak equivalence. Finally, note that each inclusion C ( n, ˚ U ) → C ( n, U ) is a weak equivalence,from which we deduce that the map (3.19) is an equivalence. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 21
We now turn to the bottom horizontal map in (3.18). We cannot directly apply Lemma 3.20 in thesame way here because the subsets ˚ U ∧ n ⊆ X ∧ n are typically not open. Instead, we apply 3.20 toshow that the maps hocolim U ∈P U k → X k are equivalences (where U k is the cartesian product of k copies of U ). Combining this with thenatural equivalences X ∧ n (cid:39) thocofib I ⊆ n X I , and using the commutativity of homotopy colimits, we deduce that the bottom horizontal map in(3.18) is an equivalence. This completes the proof that θ X is an equivalence for arbitrary X . (cid:3) Remark 3.21.
Taking X = S L in Proposition 3.17 and recalling from Example 3.15 that M S L istrivial beyond the first term, notice that we get an equivalence of Com -modulesΣ L B ( , E L , Com ) (cid:39) ( S L ) ∧∗ . This provides a different proof of [3, 3.31], which played a key role in the calculation of the comonadthat classifies Taylor towers of functors from based spaces to spectra.
Definition 3.22.
For X ∈ fMfld L ∗ we set M X := Σ ∞ M X . Then M X is an E L -module and Proposition 3.17 implies that we have an equivalence of Com -modules B ( M X , E L , Com ) ˜ −→ Σ ∞ X ∧∗ . The derivatives of the representable functor R X := Σ ∞ Hom T op ∗ ( X, − ) for X ∈ fMfld L ∗ are nowgiven by applying a form of Koszul duality to the E L -module M X . Recall from (2.6) that B L is a bisymmetric sequence that forms an E L -module in one variable and a KE L -module in theother variable. In particular, the (derived) mapping objects Map E L ( M X , B L ) inherit a KE L -modulestructure from that on B L . Proposition 3.23.
For X ∈ fMfld L ∗ , there is an equivalence of C -coalgebras, natural in X : ∂ ∗ ( R X ) (cid:39) Map E L ( M X , B L ) . In other words, the Taylor tower of R X is determined by a KE L -module structure on its derivatives.Proof. First recall from work of the first author [1] that the Taylor tower of R X is given by( P n R X )( Y ) (cid:39) Map
Com ≤ n (Σ ∞ X ∧∗ , Σ ∞ Y ∧∗ ) . By Proposition 3.17, we can rewrite this as( P n R X )( Y ) (cid:39) Map ( E L ) ≤ n ( M X , Σ ∞ Y ∧∗ ) . We now prove that the canonical map(*) Map ( E L ) ≤ n ( M X , E L ) ∧ E L (cid:94) Σ ∞ Y ∧∗ → Map ( E L ) ≤ n ( M X , (cid:94) Σ ∞ Y ∧∗ )is an equivalence. It then follows from Theorem 2.1 that the derivatives of P n R X have a KE L -modulestructure, and that this KE L -module is given byMap ( E L ) ≤ n ( M X , E L ) ∧ E L ˜ B L . We then claim also that there is an equivalence(**) Map ( E L ) ≤ n ( M X , E L ) ∧ E L ˜ B L ˜ −→ Map ( E L ) ≤ n ( M X , B L ) . The right-hand side is equivalent, as a KE L -module, to the n -truncation of Map E L ( M X , B L ) and wededuce therefore that the derivatives of R X are as claimed.It remains to prove (*) and (**), and these follow from a more general result: for any cofibrant E L -module M (cid:48) , the map(3.24) Map ( E L ) ≤ n ( M X , E L ) ∧ E L M (cid:48) → Map ( E L ) ≤ n ( M X , M (cid:48) )is an equivalence. We prove this by induction on n using the diagramMap E L ( M X , ( E L ) = n ) ∧ E L M (cid:48) Map E L ( M X , ( M (cid:48) ) = n )Map E L ( M X , ( E L ) ≤ n ) ∧ E L M (cid:48) Map E L ( M X , ( M (cid:48) ) ≤ n )Map E L ( M X , ( E L ) ≤ ( n − ) ∧ E L M (cid:48) Map E L ( M X , ( M (cid:48) ) ≤ ( n − ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) in which the vertical maps form fibre sequences. It is sufficient then to show that the top horizontalmap here is an equivalence for all n . We can rewrite this map as(3.25) Map( B ( M X , E L , )( n ) , E L ( n, − )) Σ n ∧ E L M (cid:48) → Map( B ( M X , E L , )( n ) , M (cid:48) ( n )) Σ n . The key observation now is that the Σ n -spectrum B ( M X , E L , )( n ) is built from finitely-many freeΣ n -cells. To see this we note that by Proposition 3.17 we have B ( M X , E L , )( n ) (cid:39) B (Σ ∞ X ∧∗ , Com , )( n ) (cid:39) Σ ∞ X ∧ n / ∆ n X where ∆ n X denotes the ‘fat diagonal’ inside X ∧ n . Since X is homotopy equivalent to a basedfinite complex, this Σ n -spectrum is equivalent to the suspension spectrum of a finite complex offree Σ n -cells. It follows that in (3.25) we can replace the homotopy fixed points with homotopyorbits. These now commute with the coend and we see that the resulting map is an equivalence.This completes the proof. (cid:3) Remark 3.26.
We stress that the KE L -module structure on ∂ ∗ ( R X ) for a based space ( X, x )depends on a choice of smooth structure and framing on X − { x } . Different choices correspondto potentially non-equivalent KE L -modules that yield equivalent C -coalgebra structures on thosederivatives. Definition 3.27.
Let G : fMfld L ∗ → S p be a pointed simplicially-enriched functor. We define d ( G ) := G ( Y ) ∧ Y ∈ fMfld L ∗ Map E L ( M Y , B L )with KE L -module structure on d ( G ) induced by that on B L . Theorem 3.28.
Let F : T op f ∗ → S p be a functor that is the enriched (homotopy) left Kan extensionof a pointed simplicially-enriched functor G : fMfld L ∗ → S p along the forgetful functor fMfld L ∗ → T op f ∗ . Then the derivatives of F have a KE L -module structure that classifies the Taylor tower.Conversely, if F : T op f ∗ → S p is polynomial and the derivatives of F have a KE L -module structure,then F is the Kan extension of such a G .Proof. Suppose that F is the homotopy left Kan extension of G : fMfld L ∗ → S p , i.e. we have F ( Y ) (cid:39) G ( Y ) ∧ Y ∈ fMfld L ∗ Σ ∞ Hom T op ∗ ( Y, − ) . ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 23
Taking derivatives, which commutes with homotopy colimits for spectrum-valued functors, we getan equivalence of C -coalgebras ∂ ∗ ( F ) (cid:39) G ( Y ) ∧ Y ∈ fMfld L ∗ ∂ ∗ ( R Y ) (cid:39) G ( Y ) ∧ Y ∈ fMfld L ∗ Map E L ( M Y , B L ) = d ( G )which has a KE L -module structure as required.The converse follows from Theorem 2.1 since Γ L can be identified with a full subcategory of fMfld L ∗ . (cid:3) Example 3.29.
Using the description of M S L in Example 3.15, we see that the Taylor tower ofthe representable functor R S L = Σ ∞ Ω L ( − ) is determined by the KE L -moduleΣ − L Map E L ( , B L ) . Example 3.30.
Let M be any compact parallelizable L -dimensional manifold possibly with bound-ary. Then the derivatives of the functors Σ ∞ Hom T op ∗ ( M + , − ) and Σ ∞ Hom T op ∗ ( M/∂M, − ) have a KE L -module structure. If we give M a basepoint, the same is true of Σ ∞ Hom T op ∗ ( M, − ).We finish this section with a conjecture about one way in which the existence of a KE L -module and KE L − -module structures on the derivatives of functors might be related. Conjecture 3.31.
Let
Σ : T op ∗ → T op ∗ denote the usual reduced suspension functor. Suppose thederivatives of F : T op f ∗ → S p have a KE L -module structure. Then the derivatives of F Σ have a KE L − -module structure. Evidence for this conjecture is provided by the stable splitting of mapping spaces of B¨odigheimer[5] and others. These results imply, for example, that if M is a compact framed L -manifold, thenthe Taylor tower of the functor Σ ∞ Hom T op ∗ ( M + , Σ L − )splits, i.e. has a KE -module structure on its derivatives. This result would follow by applyingConjecture 3.31 L times to Example 3.30. We therefore view Conjecture 3.31 as a refinement ofthese splitting results to encompass the intermediate functors Σ ∞ Hom T op ∗ ( M + , Σ m − ) for m < L .4. The Taylor tower of Waldhausen’s algebraic K-theory of spaces functor
Let A : T op ∗ → S p denote Waldhausen’s algebraic K -theory of spaces functor and let ˜ A be thecorresponding reduced functor so that A ( X ) (cid:39) A ( ∗ ) × ˜ A ( X ) . The derivatives of ˜ A (at ∗ ) were calculated by Goodwillie and can be written as ∂ n ( ˜ A ) (cid:39) Σ − n Σ ∞ (Σ n /C n ) + where C n is the cyclic group of order n sitting inside the symmetric group Σ n .Our goal is to show that the Taylor tower of ˜ A (and hence that of A ) is determined by a KE -module structure on these derivatives. Our strategy is to use the arithmetic square to break ˜ A intoits p -complete and rational parts, which we deal with separately.Consider first the p -completion of ˜ A which we write ˜ A p . We investigate this using the relationshipbetween A ( X ) and the topological cyclic homology of X , denoted T C ( X ). One of the main resultsof [6] is that the difference between A ( X ) and T C ( X ) is locally constant. In particular, this means that the corresponding reduced theories agree in a neighbourhood of the one-point space and sothe Taylor towers agree. There is therefore an equivalence of C -coalgebras ∂ ∗ ( ˜ A p ) (cid:39) ∂ ∗ ( ˜ T C p ).The p -completion of T C ( X ) is given by the following homotopy pullback square (see, for example,[15]):(4.1) ˜ T C ( X ) p Σ (cid:0) Σ ∞ Hom T op ∗ ( S , X ) p (cid:1) hS Σ ∞ Hom T op ∗ ( S , X ) p Σ ∞ Hom T op ∗ ( S , X ) p (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) T r (cid:47) (cid:47) − ∆ p where the right-hand vertical map is the transfer associated to the S -action on the free loop space,and the bottom map is given by the difference between the identity map and the map induced bythe p th power map ∆ p : S → S .Taking derivatives, we get a corresponding homotopy pullback of C -coalgebras(4.2) ∂ ∗ ( ˜ T C p ) ∂ ∗ (Σ[ L p ] hS ) ∂ ∗ ( L p ) ∂ ∗ ( L p ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) − ∆ p where we are writing L ( X ) := Σ ∞ Hom T op ∗ ( S , X ) for the suspension spectrum of the free loopspace on X . Our goal now is to construct a model for the bottom and right-hand maps in thisdiagram in the category of KE -modules.When X is a finite CW-complex, the homotopy groups of L ( X ) are finitely generated and so wehave L p ( X ) (cid:39) S p ∧ L ( X )where S p denotes the p -complete sphere spectrum. It follows that L p is the left Kan extension to T op f ∗ of the functor G : fMfld ∗ → S p ; G ( X ) = S p ∧ Emb f ∗ ( S , X )and so, by Theorem 3.28, the derivatives of L p are given by the KE -module S p ∧ ∂ ∗ ( L ) = ∂ ∗ ( L ) p . By a similar argument the derivatives of Σ[ L p ] hS are given by the KE -module Σ[ ∂ ∗ ( L ) p ] hS .The action of S on L is determined by the action of S on itself via translations which are framedembeddings. The induced action on S therefore commutes with the KE -module structure and soinduces a transfer map Σ[ ∂ ∗ ( L ) p ] hS → ∂ ∗ ( L ) p which models the right-hand vertical map in (4.2).On the other hand, the bottom map in (4.2) is a problem. The p th power map on S is not aframed embedding (or even an embedding) so the induced map∆ ∗ p : ∂ ∗ ( L ) → ∂ ∗ ( L ) ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 25 does not commute with the KE -module structure. We can solve this problem, however, by mod-elling the p th power map as a framed embedding between solid tori, i.e. as a map in fMfld ∗ . Thisallows us to construct a map of KE -modules that models the bottom horizontal map in (4.2). Definition 4.3.
Let T be the solid torus obtained from the cylinder D × [0 ,
1] via the identifications( x, y, ∼ ( x, y,
1) for ( x, y ) ∈ D . We give T (cid:48) the standard framing arising from that on the cylinderas a subset of R .Now recall that we have fixed a prime p . Let T (cid:48) be the subset of T given by T (cid:48) := (cid:40) ( x, y, z ) | (cid:18) x − cos(2 πz )2 p (cid:19) + (cid:18) y − sin(2 πz )2 p (cid:19) ≤ p (cid:41) with the induced framing. A diagram of T (cid:48) in the case p = 3 is shown in Figure 4.4. Notice that T (cid:48) is a thickened helix with one twist for each revolution around the torus T . Figure 4.4. T (cid:48) as a subset of T for p = 3The inclusion i : T (cid:48) → T is a framed embedding and a homotopy equivalence that commutes withthe obvious collapse maps T → S and T (cid:48) → S (that project onto the z -coordinate). In particular,the map i ∗ : ∂ ∗ (Σ ∞ Hom T op ∗ ( T + , − )) → ∂ ∗ (Σ ∞ Hom T op ∗ ( T (cid:48) + , − ))can be used to model the identity map on ∂ ∗ ( L ).We define ∆ p : T (cid:48) → T by ∆ p ( x, y, z ) := ( px, py, pz )where pz is interpreted modulo 1. Notice that ∆ p models the p th power map on S . The choice ofthickness and radius for the helix T (cid:48) also ensure that ∆ p is a framed embedding. A diagram of thisembedding for p = 3 is given in Figure 4.5. It follows that ∆ p induces a map of KE -modules∆ ∗ p : ∂ ∗ (Σ ∞ Hom T op ∗ ( T + , − )) → ∂ ∗ (Σ ∞ Hom T op ∗ ( T (cid:48) + , − )) . Figure 4.5. T (cid:48) embedded in T via ∆ p for p = 3 Definition 4.6.
We now construct a homotopy pullback square of KE -modules of the form(4.7) M p Σ[ ∂ ∗ (Σ ∞ Hom T op ∗ ( T + , − )) p ] hS ∂ ∗ (Σ ∞ Hom T op ∗ ( T + , − )) p ∂ ∗ (Σ ∞ Hom T op ∗ ( T + , − )) p ∂ ∗ (Σ ∞ Hom T op ∗ ( T (cid:48) + , − )) p (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) T r (cid:15) (cid:15) ∼ i ∗ (cid:47) (cid:47) i ∗ − ∆ ∗ p where the top right-hand map is the transfer associated to the S -action on T given by action onthe z -coordinate, and the bottom horizontal map models the difference between i ∗ and ∆ ∗ p in thestable homotopy category of KE -modules. This defines the KE -module M p .The following result then follows from a comparison between diagrams (4.7) and (4.2). We usehere the fact that the forgetful functor from KE -modules to C -coalgebras preserves homotopypullbacks. Proposition 4.8.
There is an equivalence of C -coalgebras ∂ ∗ ( ˜ A p ) (cid:39) M p . We now turn to the rationalization of ˜ A which we denote by ˜ A Q . There is a rational equivalence˜ A ( X ) (cid:39) Σ L ( X ) hS and so, by the argument used above for p -completion, the derivatives of the rationalization ∂ ∗ ( ˜ A Q )are given by the KE -module M (cid:48) Q := Σ[ ∂ ∗ ( L ) hS ] Q which, via the operad map KE → KE , we also view as a KE -module. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 27
In order to form the arithmetic square for ˜ A , we need also a model for the derivatives of the functor (cid:34)(cid:89) p ˜ A p (cid:35) Q . On the one hand, since rationalization commutes with taking derivatives, Proposition 4.8 impliesthat there is an equivalence of C -coalgebras ∂ ∗ (cid:34)(cid:89) p ˜ A p (cid:35) Q (cid:39) ˆ M Q := (cid:34)(cid:89) p M p (cid:35) Q . On the other hand, an argument similar to that of the previous paragraph gives us an equivalenceof C -coalgebras ∂ ∗ (cid:34)(cid:89) p ˜ A p (cid:35) Q (cid:39) ˆ M (cid:48) Q := (cid:34)(cid:89) p Σ[ ∂ ∗ ( L ) p ] hS (cid:35) Q . We now have a diagram of the form(4.9) (cid:89) p M p ˆ M Q M (cid:48) Q ˆ M (cid:48) Q ∂ ∗ (cid:32)(cid:34)(cid:89) p ˜ A p (cid:35)(cid:33) (cid:15) (cid:15) (cid:15) (cid:15) ∼ (cid:47) (cid:47) (cid:63) (cid:63) (cid:47) (cid:47) ∼ where the top vertical map (given by rationalization) and the left-hand horizontal map (inducedby p -completion for each p ) are maps of KE -modules.In order to form the pullback in the category of KE -modules, we need to show that there is anequivalence of KE -modules ˆ M (cid:48) Q ˜ −→ ˆ M Q (the dotted arrow above) that lifts the given equivalenceof C -coalgebras. We do this by showing that any morphism of C -coalgebras of this form lifts toa morphism of KE -modules (which is then necessarily an equivalence since these are detected onthe underlying symmetric sequences).Let (cid:103) Map KE ( − , − ) and (cid:103) Map C ( − , − ) denote the derived mapping spectra for KE -modules and C -coalgebras respectively. Lemma 4.10.
The map π (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ) → π (cid:103) Map C ( ˆ M (cid:48) Q , ˆ M Q ) , induced by the map C KE → C of comonads, is surjective. Proof.
In [2] we showed that the comonad C encodes the structure for a ‘divided power’ moduleover the operad KCom that is Koszul dual to the commutative operad. In particular, there is amap of comonads C → C KCom that is built from norm maps of the formMap(
KCom ( n ) ∧ . . . , A ( n )) h Σ n ×···× Σ nr (cid:47) (cid:47) N Map(
KCom ( n ) ∧ . . . , A ( n )) h Σ n ×···× Σ nr . These norm maps are equivalences when A ( n ) is a rational spectrum, and, in that case, the termsof the symmetric sequence C ( A ) are again rational. Since the module ˆ M Q is formed from rationalspectra, it follows that there is an equivalence (cid:103) Map C ( ˆ M (cid:48) Q , ˆ M Q ) ˜ −→ (cid:103) Map
KCom ( ˆ M (cid:48) Q , ˆ M Q )induced by the map C → C KCom . It is therefore sufficient to show that the map(*) π (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ) → π (cid:103) Map
KCom ( ˆ M (cid:48) Q , ˆ M Q )is a surjection, where we can think of this as induced by the map of operads KCom → KE that isdual to the standard map E → Com . In other words, it is sufficient to show that any (derived)morphism of
KCom -modules lifts, up to homotopy, to a morphism of KE -modules.To prove this, we construct a square of the following form:(4.11) π (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ) π (cid:103) Map
KCom ( ˆ M (cid:48) Q , ˆ M Q )Hom H ∗ KE ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) Hom H ∗ KCom ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) . (cid:47) (cid:47) (*) (cid:15) (cid:15) (cid:15) (cid:15) (A) (cid:15) (cid:15) ∼ = (B) (cid:47) (cid:47) (C) ∼ = The terms in the bottom row are the homomorphism of modules over the operads of graded Q -vector spaces given by the rational homology of the operads KE and KCom , and of the modulesˆ M (cid:48) Q and ˆ M Q . The map labelled (C) is induced by the operad map. The maps labelled (A) and (B)are edge homomorphisms associated with the Bousfield-Kan spectral sequences used to calculatedthe homology of the mapping spectra in the top row, as we explain below. Given that the diagramis commutative, it will then be sufficient to show that (A) is a surjection, and that (B) and (C) areisomorphisms.We start by analyzing the Bousfield-Kan spectral sequence for calculating the homotopy groups of (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ). The E term of this spectral sequence takes the form E − s,t ∼ = π t − s Map Σ ( nondeg [ ˆ M (cid:48) Q ◦ KE ◦ · · · ◦ KE (cid:124) (cid:123)(cid:122) (cid:125) s factors ] , ˆ M Q )where nondeg means that we take the ‘nondegenerate’ terms in the given iterated compositionproduct, i.e. where each of the factors KE contributes something from a term higher than thefirst. Claim 4.12.
The E term of the spectral sequence is given by E − s,t ∼ = Ext s,tH ∗ KE ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) . Proof.
In the formula for the E -term Map Σ ( − , − ) denotes the mapping spectrum for symmetricsequences, which we can expand out (assuming cofibrant models for KE and ˆ M (cid:48) Q ) as E − s,t ∼ = (cid:89) n π t − s Map( nondeg [ ˆ M (cid:48) Q ◦ KE ◦ · · · ◦ KE (cid:124) (cid:123)(cid:122) (cid:125) s factors ]( n ) , ˆ M Q ( n )) h Σ n . ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 29
For a rational spectrum X with action of a finite group G , we have π ∗ ( X hG ) ∼ = π ∗ ( X hG ) ∼ = [ π ∗ X ] G ∼ = [ π ∗ X ] G and so we can write E − s,t ∼ = (cid:89) n π t − s Map( nondeg [ ˆ M (cid:48) Q ◦ KE ◦ · · · ◦ KE (cid:124) (cid:123)(cid:122) (cid:125) s factors ]( n ) , ˆ M Q ( n )) Σ n . The homotopy groups of a rational spectrum are equivalent to its rational homology groups andso, using the K¨unneth Theorem, we get E − s,t ∼ = (cid:89) n Hom( nondeg [ H ∗ ˆ M (cid:48) Q ◦ H ∗ KE ◦ · · · ◦ H ∗ KE (cid:124) (cid:123)(cid:122) (cid:125) s factors ]( n ) , H ∗ ˆ M Q ( n )) Σ n t − s . We can now see that ( E , d ) is precisely the complex used to calculate Ext in the category ofmodules over the operad H ∗ KE of graded Q -vector spaces. (cid:3) Claim 4.13.
The Bousfield-Kan spectral sequence for the homotopy groups of (cid:103)
Map KE ( ˆ M (cid:48) Q , ˆ M Q ) collapses at E and so the edge homomorphism π (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ) → E , = Hom H ∗ KE ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) is a surjection.Proof. First note that, since ˆ M (cid:48) Q and ˆ M Q are rational, the spectral sequence for calculating thehomotopy groups of (cid:103) Map KE ( ˆ M (cid:48) Q , ˆ M Q ) is isomorphic to that for calculating the homotopy groupsof (cid:103) Map KE ∧ H Q ( ˆ M (cid:48) Q , ˆ M Q ) where we have replaced KE with its rationalization.We now use rational formality of the operad KE . Recall that Kontsevich proved in [14] that theoperads E L are formal over the real numbers, and that Guill´en Santos et al. proved in [12] thatformality over R descends to formality over Q for operads with no zero term. This means thatthere is an equivalence of operads E L ∧ H Q (cid:39) H E L where the right-hand side denotes the operad formed by the generalized Eilenberg-MacLane spectraassociated to the rational homology groups of E L . Taking Koszul duals commutes with rational-ization and so we have an equivalence KE L ∧ H Q (cid:39) K ( H E L ) . Koszul duality for an operad of Eilenberg-MacLane spectra reduces to Koszul duality of the under-lying operad of graded abelian groups. Getzler and Jones [10] proved that the homology of E L is a‘Koszul’ operad in the sense of Ginzburg and Kapranov [11] from which it follows that K ( H E L ) (cid:39) H KE L . Altogether we have an equivalence KE L ∧ H Q (cid:39) H KE L , so, in particular, KE is rationally formal.The modules ˆ M (cid:48) Q and ˆ M Q are already Eilenberg-MacLane spectra (since the derivatives of ˜ A arewedges of copies of sphere spectra) and so the spectral sequence in question is equivalent to thatused to calculate the homotopy groups of the mapping spectrum (cid:103) Map H KE ( H ˆ M (cid:48) Q , H ˆ M Q ) , or, equivalently, that of the corresponding object in the world of rational chain complexes (cid:103) Map H ∗ KE ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) . But the differentials in the underlying chain complexes here are all trivial so this spectral sequencecollapses at E as claimed. (cid:3) This shows that the map (A) in diagram (4.11) is a surjection. We now turn to the correspondingspectral sequence for the homotopy groups of (cid:103)
Map
KCom ( ˆ M (cid:48) Q , ˆ M Q ). Claim 4.14.
The E -term of this spectral sequence is given by E − s,t ∼ = Ext s,tH ∗ KCom ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) and this is zero for t (cid:54) = 0 . The spectral sequence therefore collapses and the edge homomorphism isan isomorphism π (cid:103) Map
KCom ( ˆ M (cid:48) Q , ˆ M Q ) ∼ = E , ∼ = Hom H ∗ KCom ( H ∗ ˆ M (cid:48) Q , H ∗ ˆ M Q ) . Proof.
The same argument as for KE gives the form of the E -term. But the rational homologies H ∗ KCom ( n ), H ∗ ˆ M (cid:48) Q ( n ) and H ∗ ˆ M Q ( n ) are all concentrated in a single degree (1 − n ). It is then easyto see directly that E − s,t = 0 for t (cid:54) = 0. (cid:3) This shows that the map (B) in diagram (4.11) is an isomorphism, and the diagram commutes bynaturality of the construction of the spectral sequence.It remains to check that the map (C) is an isomorphism. For this, we notice that, for degreereasons, the action of the operad H ∗ KE on the modules H ∗ ˆ M (cid:48) Q and H ∗ ˆ M Q is determined fully bywhat happens on the terms H − n ( KE ( n )). Since the maps H − n ( KCom ( n )) → H − n ( KE ( n ))are isomorphisms (each side is isomorphic to Lie ( n )), we can see directly that a morphism of H ∗ KE -modules of the form H ∗ ˆ M (cid:48) Q → H ∗ ˆ M Q carries precisely the same data as a morphism of H ∗ KCom -modules. Thus the map (C) is anisomorphism as required. This completes the proof of Lemma 4.10. (cid:3)
We then build a homotopy pullback square of KE -modules of the form(4.15) M (cid:89) p M p M (cid:48) Q ˆ M (cid:48) Q ˆ M Q (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) ∼ and obtain the following result. Theorem 4.16.
There is an equivalence of C -coalgebras ∂ ∗ ( ˜ A ) (cid:39) M . In other words, the derivatives of the functor ˜ A are a KE -module. ANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS 31
Proof.
The arithmetic square for ˜ A gives us a homotopy pullback of C -coalgebras ∂ ∗ ˜ A ∂ ∗ (cid:32)(cid:89) p ˜ A p (cid:33) ∂ ∗ ˜ A Q ∂ ∗ (cid:32)(cid:89) p ˜ A p (cid:33) Q (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) Comparison with (4.15) gives us the required equivalence. (cid:3)
Corollary 4.17.
The Taylor tower of A has the following descriptions: (1) there is an E -comodule N n (Koszul dual to the KE -module ∂ ≤ n ( ˜ A )) such that P n A ( X ) (cid:39) A ( ∗ ) × [ N n ∧ E Σ ∞ X ∧∗ ] with the Taylor tower map P n A → P n − A induced by a map of E -comodules N n → N n − ; (2) there is a functor G n : fMfld ∗ → S p such that P n A is the enriched homotopy left Kanextension of G n along the forgetful functor fMfld ∗ → T op f ∗ with the Taylor tower map P n A → P n − A induced by a natural transformation G n → G n − . Remark 4.18.
We do not know if our result for the Taylor tower of A is the best possible. It isconceivable that the derivatives of ˜ A have, in fact, a KE or KE -module structure. The Taylortower of A (Σ X ) is known to split by work of B¨okstedt et al. [6], so, given Conjecture 3.31, onemight guess that the latter is true. References
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