aa r X i v : . [ m a t h . A T ] M a r On Cohen-Jones Isomorphismin String Topology
Syunji Moriya ∗ Abstract
The loop product is an operation in string topology. Cohen and Jones [15, 16] gave ahomotopy theoretic realization of the loop product as a classical ring spectrum LM − TM for a manifold M . Using this, they presented a proof of the statement that the loopproduct is isomorphic to the Gerstenhaber cup product on the Hochschild cohomology HH ∗ ( C ∗ ( M ) ; C ∗ ( M )) for simply connected M . However, some parts of their proof istechnically difficult to justify. The main aim of the present paper is to give detailedmodification to a geometric part of their proof. To do so, we set up an ”up to higherhomotopy” version of McClure-Smith’s cosimplicial product. We prove a structuredversion of Cohen-Jones isomorphism in the category of symmetric spectra. Contents A ∞ -symmetric ring spectrum in string topology 15 e B on LM − τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Conditions which e ǫ and ψ satisfy . . . . . . . . . . . . . . . . . . . . . 183.1.3 Construction of e ǫ and ψ . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Comparison with Cohen-Jones spectrum . . . . . . . . . . . . . . . . . . . . . 24 LM − τ . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Intermediate cosimplicial object . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.1 Point-set description of hom-spectrum . . . . . . . . . . . . . . . . . . 26 ∗ Corresponding address: Department of Mathematics and Information Sciences, Osaka Prefecture Uni-versity, Sakai, 599-8531, Japan E-mail adress: [email protected] ohen-Jones isomorphism A ∞ -structure . . . . . . . . . . 284.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4.2 Cofacial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4.3 Operad CK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4.4 Monad CK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.5 A ∞ -operad CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Action of CK on IM • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5.2 Formula of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5.3 Conditions which e ψ , z , and ω satisfy . . . . . . . . . . . . . . . . . . . 414.5.4 Construction of e ψ , z , and ω . . . . . . . . . . . . . . . . . . . . . . . . 444.5.5 Construction of b Υ and e Υ . . . . . . . . . . . . . . . . . . . . . . . . . 484.6 Comparison of A ∞ -structures on Tot and g Tot . . . . . . . . . . . . . . . . . . 504.7 Proof of Theorem 1.0.1 (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Throughout this paper, M denotes a smooth closed manifold of a finite dimension d . Let LM denote the free loop space of M , i.e., LM is the space of all continuous maps from thecircle S to the manifold M with the compact-open topology.String topology was initiated by Chas and Sullivan [10]. It is a study of certain alge-braic structures on the homology of LM , which can be seen as a generalization of Goldman’sLie algebra for a Riemann surface. These structures are studied in many ways e.g., re-lation to counting problem of closed geodesics (see Goresky-Hingston [27]), generalizationto Gorenstein spaces (see F´elix-Thomas [25], Kuribayashi-Menichi-Naito[35, 38], and Naito[36]). Another interesting subject is the relationship between string topology operations andintrinsic operations on Hochschild cohomology of the cochain due to Gerstenhaber [2] andJones [6]. Cohen and Jones [15, 16] gave a proof of the claim that there exists an isomorphismbetween the loop product, a most basic string topology operation, and the Gerstenhaber cupproduct on the Hochschild cohomology of the singular cochain algebra C ∗ ( M ) over a field k (cid:0) H ∗− d ( LM ; k ) , the loop product (cid:1) ∼ = (cid:0) HH ∗ ( C ∗ ( M ); C ∗ ( M )) , the cup product (cid:1) , which we call the Cohen-Jones isomorphism . In the case of characteristic zero, F´elix andThomas [22] showed an isomorphism of BV-algebras. (In contrast, Menichi [28] showed non-existence of such an isomorphism in characteristic 2.)Though the idea of proof of Cohen-Jones isomorphism in [15, 16] is very interesting,the author of the present paper encountered technical difficulties with their argument. Thedifficulties can be divided to two parts. One is the comparison between the loop product anda natural product on the topological Hochschild cohomology. The other is the comparisonbetween topological and chain level Hochschild cohomologies. The aim of the present paperis to resolve the former part, which we consider the greater difficulty. We shall state themain theorem. In [15] the loop product was realized as a product on a Thom spectrum LM − T M in the stable homotopy category. By this product, we regard LM − T M as a classicalassociative ring spectrum, i.e., a monoid in the stable homotopy category, and call it the
Cohen-Jones ring spectrum . The main result of the present paper is the following theorem,which is regarded as a symmetric spectrum version of the Cohen-Jones isomorphism. ohen-Jones isomorphism Theorem 1.0.1.
There exists a non-unital A ∞ -symmetric ring spectrum LM − τ satisfyingthe following two conditions.(1) (Theorem 3.2.1) LM − τ is isomorphic to the Cohen-Jones ring spectrum LM − T M as aclassical non-unital associative ring spectrum.(2) Suppose M is simply connected. Let Q be a fibrant cofibrant replacement of the func-tion spectrum F ( M ) of M in a suitable model category (see Proposition 2.5.3). Then, LM − τ and the topological Hochschild cohomology THC( Q ; Q ) is weak equivalent asnon-unital A ∞ -symmetric ring spectra.For terminologies, see subsection 1.1 and for topological Hochschild cohomology see 2.5.The function spectrum F ( M ) is the spectrum of maps from the manifold with a disjoint basepoint M + to the sphere spectrum. In other words, it is (a structured version of) the Spanier-Whitehead dual of M + . F ( M ) is regarded as an analogue of the cochain algebra C ∗ ( M ).The equivalence in (2) or the isomorphism of Cohen-Jones is more or less a folklore (see alsoKlein [19]), but it is important to have a detailed proof for further study of the relationshipbetween string topology and Hochschild cohomology. For works related to (1), Poirier andRounds [34] constructed a chain map which encodes string topology TQFT operations, andIrie [40] constructed a chain-level BV-structure including the loop product over reals andgive applications to symplectic topology, based on Fukaya’s idea [18]We shall look at the outline of the proof of the Cohen-Jones isomorphism in [15, 16] toexplain the main task in the present paper. The authors of [15] use two cosimplicial object( L M ) ∗ , CH • M . ( L M ) ∗ is the cosimplicial spectrum given by ( L M ) n = ( M × n + ) ∧ M − T M for each cosimplicialdegree n , where M − T M = Σ − N T h ( ν ) is the Thom spectrum for the normal bundle ν of afixed embedding M → R N . There is a weak equivalence LM − T M ≃ Tot( L M ) ∗ , where Totdenotes the totalization, see subsection 1.1. (In the present paper, we denote the cosimplicialdegree by the superscript • but here, for ( L M ) ∗ we use the notation of [15].) CH • M is thecosimplicial cochain complex given by CH nM = Hom ( C ∗ ( M ) ⊗ n , C ∗ ( M )). The (normalized)total complex of CH • M is the Hochschild complex CH ∗ ( C ∗ ( M ) , C ∗ ( M )). Though CH • M doesnot appear in [15], it is used implicitly (see [16]). They use products on these cosimplicialobjects ( L M ) p ∧ ( L M ) q → ( L M ) p + q , CH pM ⊗ CH qM → CH p + qM . The first one is defined by using a natural product on M − T M and is claimed to induce theloop product on Tot( L M ) ∗ ≃ LM − T M , and the second one is the product which inducesthe Gerstenhaber cup product on the Hochschild complex. They define a degreewise quasi-isomorphism C ∗ (( L M ) ∗ ) ≃ CH • M of cosimplicial chain complexes, which preserves products,using the Atiyah duality M − T M ≃ F ( M ) and a quasi-isomorphism C ∗ ( F ( M )) ≃ C ∗ ( M ).(Here, C ∗ is an unclear chain functor for symmetric spectra.) Using this quasi-isomorphism,they essentially state that there is a zig-zag of quasi-isomorphisms C ∗ ( LM − T M ) ≃ C ∗ (Tot( L M ) ∗ ) ≃ Tot C ∗ (( L M ) ∗ ) ≃ Tot CH • M = CH ∗ ( C ∗ ( M ); C ∗ ( M ))which sends the loop product to the cup product. (Here Tot denotes different two notions,spectrum-level totalization and total complex.)In the construction of the quasi-isomorphism C ∗ (( L M ) ∗ ) ≃ CH • M , the topological Hochschildcohomology for F ( M ) is used. The topological Hochschild is not well-defined in the classicalstable homotopy category since it is defined by using the symmetry of the monoidal structureand the totalization of a cosimplicial object. It is well-defined in the category of symmetricspectra [12, 13] since that category has a structure of a monoidal model category. In [16],Cohen gives details of constructions of [15] in the category of symmetric spectra, especiallya very effective realization of the Thom spectrum M − T M which is also useful for other ap-plication. Actually, the author of the present paper crucially applied it to a model for a knot ohen-Jones isomorphism d , d n on ( L M ) n are not well-defined. In [16], thecoface operators are defined by using morphisms∆ r : M − τ ( e ) → M − τ ( e ) ∧ ( ν ǫ ( e ) + ) , ∆ l : M − τ ( e ) → ( ν ǫ ( e ) + ) ∧ M − τ ( e )(see the paragraph under Theorem 6 of [16]), where M − τ ( e ) is a model of M − T M whichhas a structure of unital commutative symmetric ring spectrum. These morphisms are notwell-defined morphisms of symmetric spectra since they do not commute with the action ofthe sphere spectrum (see Remark 3.1.1 for an account). The author tried to modify this usinganother model M − τ of M − T M introduced by Cohen in [16]. While M − τ is non -unital, themorphisms analogous to ∆ r and ∆ l are well-defined morphisms of symmetric spectra for thismodel. Unfortunately, this modification caused another problem. In order for a product ona cosimplicial object to induce a product on the totalization, it must satisfy some condition.The only sufficient condition which the author knows is the condition of McClure-Smith [17](for A ∞ -structure), which states some compatibility between coface and degeneracy operatorsand the product. For example, CH • M satisfies this condition. The problem is that the non-unital version of Cohen’s product on ( L M ) ∗ does not satisfy the condition of McClure-Smith(see Remark 3.1.1).In the present paper, for a transparent proof, we give all involved constructions in thecategory of symmetric spectra. Our constructions are based on the non-unital model ofCohen. We define a product on a model of LM − T M which encodes the loop product inthe category of symmetric spectra as in (1) of Theorem 1.0.1, and a product on a modifiedversion of ( L M ) ∗ ( L • in the notion of the present paper) which satisfies a slight generalizationof the McClure-Smith’s condition. The generalization is given as an action of a monad e K ona cosimplicial object. This product is different from the above non-unital version of Cohen’sproduct. We prove the product induced by the e K -action on L • is isomorphic to the producton LM − τ . Then we construct a zig-zag of weak equivalences L • ← IM • → THC • ( A ′ , B )where A ′ and B are suitable (fibrant or cofibrant) models of F ( M ) and THC • is the cosimpli-cial object the totalization of which is the topological Hochschild cohomology. As the actionof e K is a generalization of the McClure and Smith’s product, THC • has a natural action of e K . The author tried to connect L • and THC • ( A ′ , B ) by a zig-zag of weak equivalences whichpreserve the action of e K but it turned out to be difficult. To avoid this difficulty, we need ageneralization of the McClure-Smith’s product, laxer than an action of e K . We give such ageneralization as an action of another monad CK . The constructions of the monad CK andthe zig-zag of weak equivalences preserving the action of CK are main tasks in the presentpaper.The other main task is to establish an equivalence of two multiplicative objects definedby McClure-Smith [17]. Recall from [17] the notion of homotopy totalization g Tot, which isthe homotopy limit over the category of standard simplices ∆, see subsection 1.1 for thedefinition. g Tot has a better homotopy invariance than Tot and we need both of them. Weconstruct an explicit isomorphism LM − τ ∼ = Tot( L • ) while a CK -action induces an ( A ∞ -)product on g Tot (not on Tot). In [17], a notion of Ξ n -algebra structure on a cosimplicialobject was introduced and it was proved that a Ξ n -structure induces E n -operad actions bothon Tot and on g Tot. Here, an E n -operad is an operad weakly equivalent to the little n -cubesoperad. The Ξ -structure is the McClure-Smith product mentioned above. We prove thefollowing theorem: ohen-Jones isomorphism Theorem 1.0.2 (Theorem 4.6.15) . Let X • be a cosimplicial object over the category oftopological spaces or symmetric spectra, and suppose X • is equipped with a Ξ n -structureand suppose a canonical morphism f ∗ : Tot( X • ) → g Tot( X • ) (see subsection 1.1) is a weakequivalence. Then, the E n -actions on Tot( X • ) and g Tot( X • ) which are induced from theΞ n -structure are equivalent.We use the case of n = 1 of this theorem to prove Theorem 1.0.1 (2). As the both E n -actions have many applications [14, 20, 17, 39, 37], this theorem will be useful in othercontext. Theorem 1.0.2 is not so trivial as it looks since the E n -operad actions on Tot and g Tot are realized by different operads and there is no obvious morphism between them. Thekey observation is that the two involved operads are naturally regarded as endomorphismoperads on two different objects contained in a colored operad.There still exists a gap between Theorem 1.0.1 and the Cohen-Jones isomorphism. Let H ∗ (THC( Q ; Q )) = π ∗ (THC( Q ; Q ) ∧ Hk ) with Hk is the Eilenberg-MacLane spectrum for abase field k . If there exists an isomorphism of algebras H ∗ (THC( Q ; Q )) ∼ = HH ∗ ( C ∗ ( M ) , C ∗ ( M )) · · · · · · ( ∗ ) , where Q is the object in Theorem 1.0.1, we can complete the proof of Cohen-Jones isomor-phism, combining the isomorphism ( ∗ ) with Theorem 1.0.1. The isomorphism ( ∗ ) is plausiblesince the function spectrum is an analogy of the singular cochain (the fibrant cofibrant re-placement is necessary for the topological Hochschild to have the right homotopy type).Nevertheless, the construction of ( ∗ ) is non-trivial problem since it is related to comparisonof symmetry of monoidal structures of symmetric spectra and chain complexes. A similarproblem appeared in an earlier paper of Jones [6], which is modified by Unghretti [42]. Sincethe construction of the isomorphism ( ∗ ) is a quite general homotopical algebraic problem, wewill resolve it in another paper.An outline of this paper is as follows: In section 2, we recall basic definitions and knownresults we use later. Nothing is essentially new. We review the Cohen-Jones ring spectrumand (a slightly different version of) the realization of the Atiyah duality in symmetric spectradue to [16]. We also recall a description of the Stasheff’s associahedral operad by trees andthe McClure-Smith’s product for A ∞ -structures and introduce a slight generalization of it,which is applicable to the A ∞ -structure on LM − τ . We also recall the definition of topologicalHochschild cohomology.In section 3 we define the non-unital A ∞ -symmetric ring spectra LM − τ and prove thepart (1) of Theorem 1.0.1.In section 4, we prove the part (2) of Theorem 1.0.1. We construct a chain of equivalencesof non-unital A ∞ -symmetric ring spectra: LM − τ (A) ∼ = Tot( L • ) (B) ≃ g Tot( L • ) (C) ≃ g Tot( IM • ) (D) ≃ g Tot(THC • ( A ′ , B )) (E) ≃ Tot(THC • ( A ′ , B )) = THC( A ′ , B ) (F) ≃ THC(
Q, Q )For the definitions of A ′ , B and Q , see Proposition 2.5.4, and the equivalence (F) is provedin the same proposition. THC denotes the topological Hochschild cohomology. In subsection4.2, we define a cosimplical symmetric spectrum L • whose totalization is isomorphic to LM − τ and prove the A ∞ -structure on LM − τ comes from a (slight generalization of) McClure-Smithproduct on L • , which implies the isomorphism (A). Equivalences (C) and (D) are inducedfrom morphisms between cosimplical symmetric spectra L • p ←− IM • ¯ q −→ THC • ( A ′ , B ). Insubsection 4.3, we define IM • and the two morphisms p , ¯ q . In subsection 4.4, we introducethe monad CK , and in subsection 4.5, we define an action of CK on IM • . We also need toestablish the equivalences of A ∞ -structures on Tot and g Tot (B) and (E). We deal with themain part of this problem in subsection 4.6. In the final subsection 4.7, we put the results of ohen-Jones isomorphism
Acknowledgements : The author is most grateful to Masana Harada for reading thefirst draft of this paper and giving valuable comments. The author is partially supported byJSPS KAKENHI Grant Number 26800037 and 17K14192. • T OP denotes the category of all (unpointed) topological spaces and continuous maps. CG denotes the full subcategory of T OP consisting of all compactly generated spacesin the sense of [11, Definition 2.4.21] • Our notion of a symmetric spectrum is that of Mandell-May-Schwede-Shipley[13] andthe category of symmetric spectra is denoted by SP . For a symmetric spectrum werefer to the numbering of the underlying sequence as the level. • In this paper, C denotes either of CG or SP . We regard C as a closed symmetricmonoidal category tensored and cotensored over CG . The monoidal product (resp.external tensor) is denoted by ⊗ (resp. ˆ ⊗ ). If C = CG , both ⊗ and ˆ ⊗ are given bythe cartesian product. If C = SP , ⊗ is equal to the product ∧ S defined in [13], andˆ ⊗ is given by ( X ˆ ⊗ K ) l = X l ∧ ( K + ) for X ∈ SP , K ∈ CG , where K + is the basedspace made by adding a disjoint base point to K , and ∧ is the usual smash productsof pointed spaces. The unit of SP is denoted by S and called the sphere spectrum.We denote by Map the internal hom, which is adjoint to ⊗ , and by ( − ) K the cotensor,which is adjoint to ˆ ⊗ K for each K ∈ CG . The function spectrum F ( M ) is defined as S M . (If C = CG , ( − ) K = Map( K, − ) but we use the both notations to ease notations.)Even when the category C is specified, we sometimes use the notation ⊗ instead of × or ∧ S for simplicity. • Our notion of a model category is that of [11]. In this paper, we mainly deal with thefollowing model categories. – We endow CG the standard model structure, see [11, Theorem 2.4.25] – For a cofibrantly generated model category M , we denote the category of cosimpli-cial objects over M by M ∆ . We endow two model structures on M ∆ . One is thetermwise model structure whose fibrations and weak equivalences are termwiseones. We call cofibrations in this model structure projective cofibrations. Theother is the Reedy model structure, see [11, Theorem 5.2.5] for the definition. Weabbreviate cosimplicial symmetric spectrum as cs-spectrum . – We use two model structures on SP . One is the level model structure and theother is the stable model structure, see [13] for the definition. Recall that amorphism f : X ∗ → Y ∗ is called a π ∗ -isomorphism if it induces an isomorphismbetween stable homotopy groups. If a morphism is a π ∗ -isomorphism, it is a weakequivalence in the stable model structure but the converse is false in general.To emphasize the model structure we consider, we sometimes prefix names of morphismswith the name of model structure. For example, we say a morphism f : X •∗ → Y •∗ between cs-spectra is a termwise stable fibration if each morphism f n : X n ∗ → Y n ∗ is afibration in the stable model structure. For the compatibility of model structures withsymmetric monoidal and tensor structures, see [11, 13]. • The homotopy category of SP with respect to the stable model structure is equivalentto the classical stable homotopy category as symmetric monoidal categories. We meanby a classical (associative) ring spectrum an associative monoid in this category. ohen-Jones isomorphism • Operad means non-symmetric (or non-
Σ) operad (see [7, 33]) and the object at arity n of an operad O is denoted by O ( n ). We mainly consider operads in CG or in the categoryof posets (with the cartesian monoidal structure) and operads in CG is called topologicaloperads. Our notion of a A ∞ -operad is the non-unital version i.e., a topological operad O is an A ∞ -operad if O (0) is empty and for each n ≥ i ≥ π i ( O ( n )) ∼ = ∗ . • Recall the notion of action of O or O -algebras (or algebras over O ) for an operad O and the notions of monads and algebras over a monad (see [7] where the case ofsymmetric operad are considered but the case of non-symmetric operad are completelyanalogous except for just not taking coinvariants for the symmetric groups). In thispaper, an A ∞ -sturucture means an action of an A ∞ -operad. For a monad or operad M in the category C , we denote by ALG M ( C ) the category of algebras over M . There is acanonical way to construct a monad from an operad (see [7]). With this construction,algebras over an operad are the same as algebras over the corresponding monad. Let X , Y be two objects on which two monads (or operads) M , N act respectively (insome category). A morphism f : X → Y is said to be compatible with a morphism F : M → N of monads (or operads) if the following square is commutative: M ( X ) F ( f ) / / (cid:15) (cid:15) N ( Y ) (cid:15) (cid:15) X f / / Y where the vertical morphisms are the actions of monads (or operads). • Let F : CG → SP be the functor given by K S ˆ ⊗ K . For a topological operad O we define an operad F ( O ) in SP by F ( O )( n ) = F ( O ( n )) with the naturally inducedcomposition. By abusing notation we mean by an O -algebra in SP an F ( O )-algebra.For an A ∞ -operad O , we call an O -algebra in SP a non-unital A ∞ -symmetric ringspectrum (or nu- A ∞ -ring spectrum in short). Let X and Y be two symmetric spectraon which two A ∞ -operads O , P act on respectively. A stable equivalence f : X → Y ofsymmetric spectra is called a (weak) equivalence of nu- A ∞ -ring spectra if there exists amorphism of operads g : O → P which is compatible with f . Two nu- A ∞ -ring spectraare said to be (weakly) equivalent if they can be connected by a zig-zag of equivalencesof nu- A ∞ -ring spectra. • Recall the notion of totalization Tot( X • ) and its variant g Tot( X • ) for a cosimplicialobject X • over C from [17]. For a cosimplicial space K • , an object ( X • ) K • is definedas the subobject of Q n ( X n ) K n consisting of elements consistent with cosimplicial op-erators (see Definition 18.3.2 of [24] for details, in which the corresponding notation ishom ∆ ( K, X )). Let ∆ • denote the cosimplicial space of standard simplices. We fix aprojective cofibrant replacement of ∆ • denoted by ˜∆ • and a (termwise) weak equiva-lence f : ˜∆ • → ∆ • . Tot( X • ) is as usual, defined as ( X • ) ∆ • and g Tot( X • ) as ( X • ) ˜∆ • . Amorphism f ∗ : Tot( X • ) → g Tot( X • ) is naturally induced by f . g Tot has better homotopyinvariance than Tot. In practice, g Tot is invariant under termwise equivalences betweentermwise fibrant objects while Tot is invariant only under those between Reedy fibrantobjects (see Corollary 18.4.4 and Theorem 18.6.6 of [24]).
In this subsection, we recall the definition of the Cohen-Jones ring spectrum, so we deal withthe classical spectra. ohen-Jones isomorphism e : M → R k be a smooth embedding. For ǫ >
0, we denote by ν ǫ ( e ) the opensubset of R k consisting of points whose Euclidean distance from e ( M ) are smaller than ǫ .Let L e denote the minimum of 1 and the least upper bound of ǫ > π e : ν ǫ ( e ) → e ( M ) satisfying the following conditions:1. For any x ∈ ν ǫ ( e ) and any y ∈ M , | π e ( x ) − x | ≤ | e ( y ) − x | and equality holds if andonly if r ( x ) = e ( y ). Here, | − | denotes the usual Euclid norm in R k .2. For any y ∈ M , π − e ( { e ( y ) } ) = B ǫ ( e ( y )) ∩ ( e ( y ) + ( T y M ) ⊥ ). Here B ǫ ( e ( y )) is the openball with center e ( y ) and radius ǫ .3. The closure ¯ ν ǫ ( e ) of ν ǫ ( e ) is a smooth submanifold of R k with boundary.(Such a retraction exists for a sufficiently small ǫ > π e satisfying the above three conditions is unique. Wealso consider π e as a map to M by identifying M and e ( M ) so we have a disk bundle π e : ¯ ν ǫ ( e ) → M over M .In the rest of this paper, we fix an embedding e : M → R k and for a linear injectivemap φ : R k → R k , we abbreviate ν ǫ ( φ ◦ e ) and π φ ◦ e as ν ǫ ( φ ) and π φ , respectively.In this subsection, we fix a linear injective map φ : R k → R N with N >> d . Let ǫ > ev ∗ ν ǫ ( φ ) be the pullback of the disk bundle ν ǫ ( φ ) bythe evaluation ev : LM → M at a fixed base point of S . We define a (classical) spectrum LM − T M (an object of the homotopy category Ho( SP )) by LM − T M = Σ − N T h ( ev ∗ ν ǫ ( φ )) . Here,
T h ( − ) denote the Thom space. Let E : ev ∗ ν ǫ ( φ ) × ev ∗ ν ǫ ( φ ) −→ ν ǫ ( φ ) × ν ǫ ( φ )be the bundle map induced from the evaluation, and ev ∞ : LM × M LM −→ M be theevaluation at the base point, considering LM × M LM as the space of two loops which havecommon base point.We denote by φ × φ : R k → R N the linear map given by v ( φ ( v ) , φ ( v )). ¯ ν ǫ ( φ × φ ) is a subspace of ¯ ν ǫ ( φ ) × ¯ ν ǫ ( φ ). To define a product on LM − T M , we need the followinglemma whose proof is an easy excercise of differential topology.
Lemma 2.1.1.
Under the above notations, there exists a homeomorphism α : E − ( ν ǫ ( φ × φ )) ∼ = ev ∗∞ ν ǫ ( φ × φ ) which makes the following diagram commute: LM × M LM ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:15) (cid:15) E − ( ν ǫ ( φ × φ )) α / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ev ∗∞ ν ǫ ( φ × φ ) (cid:15) (cid:15) ν ǫ ( φ × φ ) , where the top vertical and slanting arrows are induced by the embedding φ × φ : M → ¯ ν ǫ ( φ × φ ) and the bottom arrows are natural bundle maps. Furthermore, a homeomorphismwhich makes the above diagram commutative is unique up to isotopies.We define a product on LM − T M as follows. LM − T M ∧ LM − T M ∼ = Σ − N T h ( ev ∗ ν ǫ ( φ ) × ev ∗ ν ǫ ( φ )) → Σ − N ( E − ( ν ǫ ( φ × φ ) /E − ∂ν ǫ ( φ × φ ))(collapse the outside of ν ǫ ( φ × φ ) ⊂ ν ǫ ( φ ) × ν ǫ ( φ )) ∼ = Σ − N ( T h ( ev ∗∞ ν ǫ ( φ × φ )) (by Lemma 2.1.1) → Σ − N ( T h ( ev ∗ ν ǫ ( φ × φ )) (concatenation) ∼ = Σ − N (Σ N T h ( ev ∗ ν ǫ ( φ )) ∼ = Σ − N ( T h ( ev ∗ ν ǫ ( φ )) ∼ = LM − T M . ohen-Jones isomorphism Definition 2.1.2.
We call the classical ring spectrum LM − T M equipped with the aboveproduct the
Cohen-Jones ring spectrum . In this subsection, we exhibit a realization of Atiyah duality in the category of symmetricspectra essentially due to Cohen [16]. We warn the reader that a symmetric spectrum M − τ defined here is slightly different from the object of the same notation in [16].For each k ≥
0, we put V k = ( φ : R k → R k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ is a linear map such that ∃ c ≥ ∀ v ∈ R k | φ ( v ) | = c | v | ) . Of course, if k < k , V k is empty. V k is topologized as a subspace of the Euclidian space oflinear maps (see subsection 2.1 for notations).We define a sequence of unpointed spaces {M k } k ≥ by M k = { ( φ, ǫ, v ) | φ ∈ V k , < ǫ < L e / , v ∈ ¯ ν ǫ ( φ ) } and put e V k = V k × (0 , L e / M k is considered as a disk bundle over e V k × M with theprojection ( φ, ǫ, v ) ( φ, ǫ, π φ ( v )). We define a pointed space M − τk as the Thom space asso-ciated to M k . The sequence M − τ = { M − τk } k ≥ is equipped with a structure of symmetricspectrum as follows: The action of Σ k is induced by the permutation of the components of R k . The action of S is given by ( φ, ǫ, v ) (0 × φ, ǫ, ( t, v )) for t ∈ S = R ∪ {∞} . As e V k is k/ − M − τ is isomorphic to the Thom spectra M − T M in [15] as objects of thehomotopy category of SP .To connect M − τ and the function spectrum F ( M ) by π ∗ -isomorphisms, we define twosymmetric spectra Γ( M ), Γ ′ ( M ) and three π ∗ -isomorphisms κ , κ , and ρ fitting into thefollowing diagram: M − τ ρ −→ Γ( M ) κ ←− Γ ′ ( M ) κ −→ F ( M ) . We first define Γ( M ). For ( φ, ǫ ) ∈ e V k , let B φ,ǫ be the trivial k -sphere bundle over M whosefiber at x ∈ M is ¯ B ǫ ( φ ( x )) /∂ ¯ B ǫ ( φ ( x )), where ¯ B ǫ ( φ ( x )) ⊂ R k is the closed ball with radius ǫ and center φ ( x ). Let Γ φ,ǫ be the space of sections M → B φ,ǫ . We put˜Γ k ( M ) = { ( φ, ǫ, s ) | ( φ, ǫ ) ∈ e V k , s ∈ Γ φ,ǫ } . We give Γ φ,ǫ the compact-open topology and ˜Γ k ( M ) the topology as a fiber bundle over e V k .The space Γ k ( M ) at level k is obtained from ˜Γ k ( M ) by collapsing the subspace { ( φ, ǫ, σ ∞ ) | ( φ, ǫ ) ∈ e V k } to one point, where σ ∞ is the section consisting of the points represented by theboundaries. The actions of S and Σ k on the sequence Γ( M ) = { Γ k ( M ) } k is defined exactlyanalogously to those on M − τ .The definition of Γ ′ ( M ) is as follows: We put Γ ′ k ( M ) = { ( φ, ǫ, f ) | ( φ, ǫ ) ∈ e V k , f ∈ F ( M ) k } / { ( φ, ǫ, ∗ ) } , where ∗ is the base point of F ( M ) k , and the actions of S and Σ k issimilar to those on Γ( M ).The morphism κ is given by the canonical projection and κ is the morphism inducedby collapsing map S k → ¯ B ǫ ( φ ( x )) /∂ ¯ B ǫ ( φ ( x )). Finally, the morphism ρ is given by ρ ( h φ, ǫ, v i ) = h φ, ǫ, s v i , s v ( y ) = (cid:26) v if | v − φ ( y ) | < ǫ base point if | v − φ ( y ) | ≥ ǫ for h φ, ǫ, v i ∈ M − τk .The following theorem is a realization of the Atiyah duality in symmetric spectra andeasily follows from the original Atiyah duality ([1, 16]) and the fact that e V k is k/ − Theorem 2.2.1.
The morphisms κ , κ and ρ defined above are π ∗ -isomorphisms. ohen-Jones isomorphism In this section we review the Stasheff’s associahedral operad. We define it as the geometricrealization of an operad in the category of posets. This definition is well-known, see Sinha[29, 4.4] or Fiedrowicz-Gubkin-Vogt [30] for example. (In [30] the authors use “parenthizedwords” instead of trees which we use here.) We will use the description of the associahedraby trees, explained below, to construct a A ∞ -structure which governs the loop product. Wegive a proof of the consistency of our description and the original Stasheff’s definition in somedetail since we use a similar argument in more complicated situation in subsections 4.4.2 and4.4.3. Definition 2.3.1. A tree is a finite connected acyclic graph. (We do not distinguish thesource or target from two endpoints of an edge at this point.) For an integer n ≥
2, an embedded n -tree is a pair ( T, f ) of a tree T and a continuous injective map f from thegeometric realization of T to the plane R × [0 ,
1] such that f ( T ) ∩ R × { } consists of aunique vertex called the root , which is at least bivalent, and f ( T ) ∩ R × { } consists of n univalent vertices called the leaves , and all vertices different from the root and leavesare at least trivalent. An isotopy between two embedded n -trees ( T , f ) and ( T , f ) iscontinuous family of homeomorphisms { g t : R × [0 , → R × [0 , | ≤ t ≤ } such that g = id R × [0 , and g maps f ( T ) homeomorphically onto f ( T ). An n -tree is an isotopyclass of embedded n -tree T . We label the leaves with the numbers 1 , . . . , n according to theusual order on R × { } = R . The vertex of an edge which is farther from the root is called its source , and the other vertex is called its target . For two numbers i, j with 1 ≤ i < j ≤ n , the( i, j ) -join is the first vertex at which the root paths from the i -th and j -th leaves join. Herethe root path is the unique shortest path to the root. The ( i, j ) -bunch is the vertex which isthe ( k, l )-join for a pair ( k, l ) if and only if i ≤ k < l ≤ j . For a n -tree T the characteristicset CH( T ) is the set of pairs ( i, j ) such that 1 ≤ i < j ≤ n and T has the ( i, j )-bunch.The set of all n -trees is denoted by T ( n ). T (0) and T (1) are defined as the empty setand one point set respectively. We define a partial order on T ( n ) by declaring T ≤ T ′ if T is obtained from T ′ by succesive contractions of internal edges (edges whose sources are notleaves). We give the collection T = { T ( n ) } n a structure of operad over posets as follows.Let T ∈ T ( n ) and T ∈ T ( n ) and 1 ≤ i ≤ n . The n + n − T ◦ i T is obtainedby identifying the i -th leaf of T with the root of T . When we consider T and T as asubtree of T ◦ i T , The root of T ◦ i T is the root of T and j -th leaf of T ◦ i T is j -th leafof T if j ≤ i − j − i + 1-th leaf of T if i ≤ j ≤ i + n − j − n + 1-th leaf of T if i + n ≤ j ≤ n + n −
1. Finally, we define a topological operad K by K ( n ) = | T ( n ) | withthe induced operad structure.Note that for a given ( i, j ) a n -tree may not have the ( i, j )-bunch but any vertex whichis not a leaf is the ( i, j )-bunch for some ( i, j ). Clearly the function T ( n ) ∋ T CH( T ) ⊂{ ( i, j ) | ≤ i < j ≤ n } is injective, and T ≤ T ′ if and only if CH( T ) ⊃ CH( T ′ ).Let P be a poset and p ∈ P an elment. We denote by h p i the subposet { q ∈ P | q ≤ p } . The codimension of p is the muximum of numbers N such that a chain p < p < · · · < p N ∈ P exists. The following is the fundamental property of T . Verification is trivial. Lemma 2.3.2. (1) For each n ≥ T ( n ) has the muximum. We denote it by T ( n ). T isgenerated by the set of maximum elements of arity ≥ n , n ≥
1, and 1 ≤ i ≤ n , the composition ( − ◦ i − ) : T ( n ) × T ( n ) −→ T ( n + n −
1) induces a bijection from T ( n ) × T ( n ) onto h T ( n ) ◦ i T ( n ) i .(3) An element T of T ( n ) is of codimension one if and only if it is equal to a composition oftwo maximum elements (of arity ≥ T is of codimension two if and only if it is equalto a composition of three maximum elements (of arity ≥ T and T be two different elements of T ( n ) of codimension one. If h T i ∩ h T i is notempty, there exists an element T of codimension two such that h T i ∩ h T i = h T i .We shall show the operad K is isomorphic to the Stasheff’s associahedral operad usingthese properties. ohen-Jones isomorphism T ( n ) , be the subposet of T ( n ) consisting of elements of codimension one or two.We define a diagram B n : T ( n ) , −→ CG as follows: An element of codimension one is uniquely presented as T ◦ i T with T , T muximum trees of arity ≥ B n ( T ◦ i T ) = K ( n ) ×K ( n ),where n t is the arity of T t . Similary, an element of codimension two is uniquelypresented as ( S ◦ j S ) ◦ k S with S , S , S muximul trees of arity ≥
2, and j ≤ k . We put B n (( S ◦ j S ) ◦ k S ) = K ( m ) × K ( m ) × K ( m ), where m t is the arity of S t .On morphisms, suppose ( S ◦ j S ) ◦ k S ≤ T ◦ i T . By (2) of Lemma2.3.2, only one ofthe following three cases occurs.1. S ◦ j S ≤ T , S = T , k = i ,2. S = T , S ◦ k − j +1 S ≤ T , j = i ,3. S ◦ k − m +1 S ≤ T , S = T , i = j .Using these relations, we define the map B n (( S ◦ j S ) ◦ k S ) → B n ( T ◦ i T ). For example,in the first case, we define the map as the following map( − ◦ j − ) × id : K ( m ) × K ( m ) × K ( m ) → K ( n ) × K ( n )and similarly for the rest cases.A natural transformation B n ⇒ K ( n ) : T ( n ) , −→ CG is defined by using the composi-tion of K (the map B n ( T ◦ i T ) → K ( n ) is ( − ◦ i − )). Here, K ( n ) is considered as the constantdiagram over T ( n ) , . So we obtain the induced map θ n : colim T ( n ) , B n −→ K ( n ). Theimage of θ n is ∂ K ( n ), the subcomplex spanned by all n -trees of codimension one.A point of K ( n ) is presented as t T + · · · + t k T k with T < · · · < T k ∈ T ( n ), t + · · · + t k = 1and t i ≥
0. Using this presentation, we define a map˜ θ n : Cone (colim B n ) −→ K ( n )by ˜ θ n ( t · u ) = tθ n ( u ) + (1 − t ) T ( n ). Here, Cone ( X ) = [0 , × X/ { } × X and t · u is thepoint represented by ( t, u ). Note that the construction Cone (colim B n ) coincides with thedefinition of the associahedra given by Stasheff [3]. Proposition 2.3.3.
Under the above notations, the maps θ n : colim B n → ∂ K ( n ) and˜ θ n : Cone (colim B n ) → K ( n ) are homeomorphisms. Proof.
To ease the notations, let K ( T ) denote the subspace |h T i| ⊂ K ( n ). By (2) of Lemma2.3.2, the map ( − ◦ i − ) : K ( n ) × K ( n ) −→ K ( T ( n ) ◦ i T ( n )) is a homeomorphism for each n , n and i . Since K ( T ) ∩ K ( T ) = ∪ T ≤ T ,T K ( T ), (3) and (4) of the same lemma imply θ n is a homeomorphism onto ∪ T Let X • be a cosimplicial object over C . A McClure-Smith product (MSproduct, for short) on X • is a family of morphisms { µ p,q : X p ⊗ X q → X p + q | p, q ≥ } which satisfy the following conditions: d i ( x · y ) = (cid:26) ( d i x ) · y if 0 ≤ i ≤ px · ( d i − p y ) if p + 1 ≤ i ≤ p + q + 1( d p +1 x ) · y = x · ( d y ) s i ( x · y ) = (cid:26) ( s i x ) · y if 0 ≤ i ≤ p − x · s i − p y if p ≤ i ≤ p + qx · ( y · z ) = ( x · y ) · z ohen-Jones isomorphism p = deg x, q = deg y . Here, we denote µ a,b ( z ⊗ w ) by z · w , and we interpret theseequations as equations of morphisms if C = SP .We define a (non-symmetric) monoidal structure (cid:3) on C ∆ which is closely related toMS-product. For X • , Y • ∈ C ∆ , we put( X • (cid:3) Y • ) r = G p + q = r X p ⊗ Y q / ∼ , d p +1 x ⊗ y ∼ x ⊗ d y. The cosimplicial operators are defined similary to the above formulae of MS-conditions. Wecall a semigroup (an object with associative product but without unit) with respect to themonoidal product (cid:3) a (cid:3) -object. The following is clear. Proposition 2.4.2. Let X • be a cosimplicial object over C . MS-products on X • and struc-tures of a (cid:3) -object on X • are in one to one correspondence.We denote by B the co-endmorphism operad of the cosimplicial space ∆ • , i.e., B ( n ) =Map CG ∆ (∆ • , (∆ • ) (cid:3) n ) with a natural composition product. The following is proved in [17] Proposition 2.4.3. The operad B is an A ∞ -operad.In fact, B ( n ) is homeomorphic to the space of weakly order preserving surjections fromthe interval [0 , 1] to itself. For later use, we shall describe such a homeomorphism explicitly.We take a presentation of the standard topological simplex ∆ n as∆ n = { ( t , . . . , t n ) | ≤ t ≤ · · · ≤ t n ≤ } . We define a morphism of cosimplicial spaces ζ n : ∆ • (cid:3) · · · (cid:3) ∆ • → ∆ • ( n − (cid:3) ) by[( t , . . . , t p ) , . . . , ( t n, , . . . t n,p n )] (cid:18) t n , . . . , t p n , . . . , n − t n, n , . . . , n − t n,p n n (cid:19) The following is proved in [17]. Lemma 2.4.4 (Lemma 3.6 of[17]) . ζ n is an isomorphism of cosimplicial spaces.Thus we obtain an identification B ( n ) ( ζ n ) ∗ ∼ = Map(∆ • , ∆ • ) ∼ = { u : [0 , → [0 , | u is a weakly monotone surjection } The second homeomorphism is given by the natural projection Map(∆ • , ∆ • ) → Map(∆ , ∆ ).For an element f ∈ B ( n ), we denote by f the corresponding weakly monotone surjection.We shall describe the composition product of B using this identification. For f ∈ B ( n ) and g ∈ B ( m ), f ◦ i g ( t ) = nm + n − f ( t ) if t ∈ f − [0 , i − n ] m + n − [ i − mg ( nf ( t ) − i + 1) ] if t ∈ f − [ i − n , in ] m + n − [ m − nf ( t ) ] if t ∈ f − [ in , Proposition 2.4.5 (Theorem 3.1 of [17]) . For an (cid:3) -object X • over C , Tot( X • ) has a naturalaction of the operad B . In other words, Tot induces a functor from the category of (cid:3) -objectsto the category of B -algebras. ohen-Jones isomorphism We will use a straightforward generalization of MS-product described as follows.In general, the monoidal product (cid:3) is not symmetric but if one of variables is a constantcosimplicial object, there exist obvious natural isomorphisms c ( X ) (cid:3) c ( Y ) ∼ = c ( X ⊗ Y ) , c ( X ) (cid:3) Z • ∼ = Z • (cid:3) c ( X ) ∼ = X ⊗ Z • . for X , Y ∈ C and Z • ∈ C ∆ . Here c ( − ) is the constant cosimplical object, and X ⊗ Z • denotesthe cosimplicial object defined by ( X ⊗ Z • ) n = X ⊗ Z n .Recall that there is a canonical procedure which associate a monad to an operad in asymmetric monoidal category. We shall define a monad e K over C ∆ by a similar way. As afunctor e K : C ∆ −→ C ∆ , e K ( X • ) = G n ≥ K ( n ) ⊗ ( X • ) (cid:3) n , and structure morphisms e K ◦ e K ⇒ e K and id C ∆ ⇒ e K is defined by the same formula as in thesymmetric monoidal case by using the above commuting isomorphisms. Definition 2.4.6. A topological operad e B is defined as follows: As a topological space, e B ( n ) = Map CG ∆ (∆ • , K ( n ) ⊗ (∆ • ) (cid:3) n )for each n . For an element ( f, g , . . . , g n ) ∈ B ′ ( n ) × B ′ ( m ) × · · · × B ′ ( m n ) the compositionproduct f ◦ ( g , . . . , g n ) is given by∆ • f −→ K ( n ) ⊗ (∆ • ) (cid:3) n g (cid:3) ··· (cid:3) g n −−−−−−−→ K ( n ) ⊗ (cid:0) K ( m ) ⊗ (∆ • ) (cid:3) m (cid:1) (cid:3) · · · (cid:3) (cid:0) K ( m n ) ⊗ (∆ • ) (cid:3) m n (cid:1) ∼ = K ( n ) ⊗ K ( m ) ⊗ · · · ⊗ K ( m n ) ⊗ (∆ • ) (cid:3) m + ··· + m n −◦− −−−→ K ( m + · · · + m n ) ⊗ (∆ • ) (cid:3) m + ··· + m n . Proposition 2.4.7. (1) As operads, e B ∼ = B × K . In particular, e B is an A ∞ -operad.(2) For a e K -algebra X • , Tot( X • ) has a natural action of the operad e B . In other words, Totinduces a functor ALG e K ( C ∆ ) → ALG e B ( C ). Proof. (1) is clear. the isomorphism is induced by the natural projection to B and evaluationat ∆ . For (2), the action of e B on Tot( X • ) for a e K -algebra X • is given by e B ( n ) ˆ ⊗ Tot( X • ) ⊗ n → e B ( n ) ˆ ⊗ Map C ∆ ((∆ • ) (cid:3) n , ( X • ) (cid:3) n ) → e B ( n ) ˆ ⊗ Map C ∆ ( K ( n ) ⊗ (∆ • ) (cid:3) n , K ( n ) ˆ ⊗ ( X • ) (cid:3) n ) → Map C ∆ (∆ • , K ( n ) ˆ ⊗ ( X • ) (cid:3) n ) → Tot( X • ) . Here, the first morphism is induced by the monoidal structure (cid:3) , the second by the tensorwith the identity on K ( n ), and the third by the action of e K . (Here we use the notation Mapinstead of ( − ) ( − ) for simplicity.)Finally, we state a similar claim for g Tot (see subsection 1.1). Let e B ′ be an operad obtainedby replacing ∆ • with ˜∆ • in the definition of e B (see subsection 1.1). Proposition 2.4.8. e B ′ is an A ∞ -operad, and g Tot induces a functor ALG e K ( C ∆ ) → ALG e B ′ ( C ). ohen-Jones isomorphism Definition 2.5.1. Let A be a monoid in SP (i.e., an associative symmetric ring spectrum).A A − A -bimodule in the category SP is as usual, an object M of SP with a morphism A ⊗ M ⊗ A → M called a (two-sided) action of A , which satisfies the usual associativityand unity axioms. A morphism of A − A -bi-modules is a morphism in SP compatible withactions of A . Let M be a A − A -bimodule over SP . We define a cs-spectrum THC • ( A, M ) byTHC p ( A, M ) = Map SP ( A ⊗ p , M ) . The coface operator d : THC p ( A, M ) → THC p +1 ( A, M )is defined as the adjoint of the following compositionMap SP ( A ⊗ p , M ) ⊗ A ⊗ p +1 ∼ = → A ⊗ Map SP ( A ⊗ p , M ) ⊗ A ⊗ p → A ⊗ M → M, where the first arrow is the transposition of the first component of A ⊗ p +1 , the second arrowis induced by the evaluation Map( X, Y ) ⊗ X → Y , and the third arrow is the left actionof A . The last coface operator d p is defined similarly by using the right action. The othercoface operators d i (1 ≤ i ≤ p − 1) and the codegeneracy operators s j : THC p ( A, M ) → THC p − ( A, M ) (0 ≤ j ≤ p − 1) is defined as the pullback by the following morphisms A ⊗ p +1 = A ⊗ i − ⊗ ( A ⊗ A ) ⊗ A ⊗ p − i → A ⊗ i − ⊗ A ⊗ A ⊗ p − i = A ⊗ p A ⊗ p − = A ⊗ j ⊗ S ⊗ A p − j − → A ⊗ j ⊗ A ⊗ A ⊗ p − j − = A ⊗ p induced by the product and unit morphism of A , respectively. We call the totalizationTot(THC • ( A, M )) the topological Hochschild cohomology of A with coefficients in M anddenote it by THC( A, M ).Let A be a monoid in SP . Note that the category of A − A bimodules has naturalstructure of (non-symmetric) monoidal category with its monoidal product given by usualtensor over A of left and right A -modules. We consider semigroups in this monoidal category,which we call non-unital A -algebras . Lemma 2.5.2. Let A be a monoid in SP and B a non-unital A -algebra. THC • ( A, B ) hasa natural structure of a (cid:3) -object. In particular, THC( A, B ) has an induced structure of B -algebra (see Proposition 2.4.5). Proof. The associative product is given as the adjoint ofMap( A p , B ) ⊗ Map( A q , B ) ⊗ A p + q ∼ = Map( A p , B ) ⊗ A p ⊗ Map( A q , B ) ⊗ A q evaluation −−−−−−−→ B ⊗ B → B Note that a (cid:3) -object is regarded as a e K -algebra via the unique morphism from K to theassociative operad. Hence we may regard the cs-spectrum THC • ( A, B ) as a e K -algebra usingthe (cid:3) -object structure in Lemma 2.5.2.In the rest of this subsection, we prepare some technical results used later . The part oneof the following is proved in [13] and the proof of part two is similar. Proposition 2.5.3. (1) The category of monoids in SP has the model category structurewhere a morphism is weak equivalence or fibration if and only if so it is as a morphismof SP with the stable model structure.(2) Let A be a monoid in SP . The category of non-unital A -algebras has a model categorystructure where a morphism is a weak equivalence or fibration if and only if so it is as amorphism of SP with the stable model structure.Note that the function spectrum F ( M ) has a structure of monoid induced by that of S . We define a structure of non-unital F ( M )-algebra on Γ( M ) as follows (see subsection2.2 for the definition of Γ( M )): For h φ , ǫ , s i , h φ , ǫ , s i ∈ Γ( M ), we put h φ , ǫ , s i ·h φ , ǫ , s i = h φ × φ , min { ǫ , ǫ } , s ∧ s i , where s ∧ s denotes the section taking x ∈ M to the point represented by ( s ( x ) , s ( x )). For f ∈ F ( M ) and h φ, ǫ, s i ∈ Γ( M ), we put f · ( φ, ǫ, s ) = (0 × φ, ǫ, f ∧ s ) where f ∧ s is understood similarly to s ∧ s . ohen-Jones isomorphism Proposition 2.5.4. Let A ′ → F ( M ) be a cofibrant replacement of F ( M ) as a monoid in SP . We consider Γ( M ) as a non-unital A ′ -algebra by pulling back the structure of F ( M )-algebra defined above. Let Γ( M ) → B be a fibrant replacement of Γ( M ) as A ′ -algebra and Q be any fibrant cofibrant replacement of F ( M ) (as a monoid in SP ). Then THC( Q, Q ) andTHC( A ′ , B ) is weak equivalent as nu- A ∞ -ring spectra. Here, all replacements are taken withrespect to the model structures in Proposition 2.5.3. Proof. It is easy to see the weak equivalence class of THC( Q, Q ) as a B -algebra is independentof a fibrant-cofibrant replacement Q . So all we have to do is to prove the claim for oneparticular Q . Take a trivial cofibration i : A ′ → Q with Q fibrant in the category of monoids.Clearly, Q is also cofibrant. We have a zig-zag of weak equivalences of non-unital A ′ -algebrasas follows: Q ← A ′ → F ( M ) κ ←− Γ ′ ( M ) κ −→ Γ( M ) → B where κ , κ , and Γ ′ ( M ) is defined in subsection 2.2, and the non-unital A ′ -algebra structureon Γ ′ ( M ) is defined completely analogously to Γ( M ). Using Proposition 2.5.3, we can replacethis chain by the following zig-zag of weak equivalences: Q ← B ′ → B where B ′ is a fibrant object. From this and by homotopy invariance properties of ⊗ (andMap) in [13], we obtain a zig-zag of termwise level equivalences of (cid:3) -objects:THC • ( Q, Q ) → THC • ( A ′ , Q ) ← THC • ( A ′ , B ′ ) → THC • ( A ′ , B )where the first map is induced by pullback by A ′ → Q . Again by properties of ⊗ in [13], anycs-spectra in this zig-zag are Reedy fibrant, so the application of Tot produces a zig-zag oflevel equvalences between THC( Q, Q ) and THC( A ′ , B ) as B -algebras. A ∞ -symmetric ring spectrum in string topol-ogy In this section we refine the Cohen-Jones ring spectra recalled in subsection 2.1 to a nu- A ∞ -ring spectrum. It is realized as an action of the operad e B defined in sub-subsection 2.4.1on a symmetric spectrum LM − τ which is isomorphic to LM − T M in the homotopy categoryHo( SP ). We begin by defining the symmetric spectrum LM − τ . Definition 3.0.1. Recall from subsection 2.2 the definitions of e V k , M k , and M − τ . Let ev : e V k × LM → e V k × M denote the product of the identity and the evaluation at 0 ∈ S for each k ≥ 0. We define a space LM − τk as the Thom space associated to the pullback diskbundle ev ∗ M k . We endow the sequence LM − τ = { LM − τk } k ≥ with a structure of symmetricspectrum exactly analogous to that of M − τ . Convention. In the rest of paper, the bold letter v (or v i ) denotes an element of M k ,and the components of v (or v i ) are denoted by( φ, ǫ, v ) ( or ( φ i , ǫ i , v i ) )where φ ∈ V k , ǫ ∈ (0 , L e / v ∈ ¯ ν ǫ ( φ ) ( or φ i ∈ V k , ǫ i ∈ (0 , L e / v i ∈ ¯ ν ǫ i ( φ i ) ).Similarly, the bold letter c ( or c i ) denotes an element of ev ∗ M k . The components of c ( or c i ) are denoted by ( φ, ǫ, c, v ) ( or ( φ i , ǫ i , c i , v i ) )where ( φ, ǫ, v ) ∈ M k and c ∈ LM ( or ( φ i , ǫ i , v i ) ∈ M k and c i ∈ LM ).For a sequence of spaces { X k } k ≥ and an integer n ≥ X [ k , . . . , k n ] = X k × · · · × X k n . ohen-Jones isomorphism e B on LM − τ The action of e B on LM − τ which we will construct is denoted by Φ = { Φ n } n ≥ where Φ n isa morphism Φ n : e B ( n ) ˆ ⊗ ( LM − τ ) ⊗ n −→ LM − τ . We first give an outline of the construction. We will define a continuous function e ǫ = e ǫ k ,...,k n : K ( n ) × e V [ k , . . . , k n ] −→ (0 , L e / k . . . k n ≥ 0. This function gives the upper boundof the distance between base points of loops which the loop product do not collapse loopsto the base point We shall explain this more precisely. Let D nk ,...,k n denote the subset of K ( n ) × M [ k , . . . , k n ] consisting of elements ( u ; v , . . . , v n ) which satisfy the condition d (( v , . . . v n ) , φ ( M ) ) ≤ e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) , where φ is the linear map φ × · · · × φ n : R k → R k + ··· + k n and d ( − , − ) is the Eucliddistance. We consider the space D nk ,...,k n as a “ fiberwise tubular neighborhood of M ”in the sense that the inverse image of a point ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) by the projection p : D nk ,...,k n → K ( n ) × e V [ k , . . . , k n ] is a tubular neighberhood of φ ( M ) in R k + ··· + k n . Forelements u ∈ K ( n ) and c , . . . , c n ∈ ev ∗ M , if the vector ( v , . . . , v n ) belongs to the tubularneighborhoods which fibers over the element ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )), we create a new loopby concatenating c , . . . , c n after perturbing them so that their base points (the images of0 ∈ S ) coincide, and otherwise we collapse them to the base point of LM − τ .To make precise the procedure“concatenation after perturbation”, we will define a con-tinuous map e Φ k ,...,k n : B ( n ) × ev ∗ D nk ,...,k n −→ ev ∗ M k + ··· + k n Here, ev ∗ D k ,...,k n is defined by the following pullback diagram ev ∗ D k ,...,k n / / (cid:15) (cid:15) D k ,...,k n (cid:15) (cid:15) K ( n ) × ev ∗ M [ k , . . . , k n ] / / K ( n ) × M [ k , . . . , k n ] , where the right vertical arrow is the inclusion and the bottom horizontal arrow is the obviousbundle map induced by ev . Assuming the existence of e Φ, the action Φ n : e B ( n ) ˆ ⊗ ( LM − τ ) ⊗ n −→ LM − τ is defined using e Φ and a collapsing map as follows: If the values of e ǫ are sufficientlysmall, by Proposition2.4.7 (1), we may regard the space B ( n ) × ev ∗ D k ,...,k n as a subspace of e B ( n ) × ev ∗ M [ k , . . . , k n ] by B ( n ) × ev ∗ D k ,...,k n ⊂ B ( n ) × K ( n ) × ev ∗ M [ k , . . . , k n ] = e B ( n ) × ev ∗ M [ k , . . . , k n ] . Using this inclusion, the morphism Φ n is defined by the following composition( e B ( n ) + ) ∧ LM − τk ∧ LM − τk ∧ · · · ∧ LM − τk n ∼ = ( e B ( n ) × ev ∗ M [ k , . . . , k n ]) / e B ( n ) × ∂ ( ev ∗ M [ k , . . . , k n ]) collapse −−−−−→ ( B ( n ) × ev ∗ D k ,...,k n ) / ( B ( n ) × ∂ev ∗ D k ,...,k n ) e Φ −→ ( ev ∗ M k + ··· + k n ) / ( ∂ev ∗ M k + ··· + k n ) = LM − τk + ··· + k n Here, e B ( n ) + denotes e B ( n ) with disjoint base point, ∂ev ∗ D k ,...,k n , ∂ ( ev ∗ M [ k , . . . , k n ]),and ∂ev ∗ M k + ··· + k n (the total space of ) the boundary sphere bundles of ev ∗ D k ,...,k n , ohen-Jones isomorphism ev ∗ M [ k , . . . , k n ], and ev ∗ M k + ··· + k n respectively.For elements ( f, u ) ∈ e B ( n ) = B ( n ) × K ( n ), c i ∈ ev ∗ M , e Φ( f, u ; c , . . . , c n ) will be of thefollowing form ( φ × · · · × φ n , e ǫ, e c, ( v , . . . , v n ) ) (1)for some loop e c ( e ǫ = e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n ))). To define e c , we will need a continuous map ψ = ( ψ , . . . , ψ n ) : D k ,...,k n −→ Map( M, M ) × n . for each k , . . . , k n . This map encodes the perturbation procedure. Actually, ψ satisfy theequality ψ ( d )( π φ ( v )) = ψ ( d )( π φ ( v )) = · · · = ψ n ( d )( π φ n ( v n )) = π φ ×···× φ n ( v , . . . , v n )for any d = ( u ; v , . . . , v n ) ∈ D . So if ( u ; c , . . . , c n ) ∈ ev ∗ D , as c i (0) = π φ i ( v i ), the n loops ψ ( d ) ◦ c , . . . , ψ n ( d ) ◦ c n has common base point π φ ×···× φ n ( v , . . . , v n ) and we canconcatenate these loops at the base point. Here, d is the image of ( u ; c , . . . , c n ) by theprojection ev ∗ D → D . e c is defined as the loop created by concatenation of the loops rescaledwith the rate determined by an element f ∈ B ( n ). Precisely speaking, we put e c ( t ) = ψ i ( d ) ◦ c i ( nf ( t ) − i + 1) for t ∈ f − (cid:20) i − n , in (cid:21) , ≤ i ≤ n. (2)(Here we identify f with a weakly monotone surjection on [0 , e B = B × K of Proposition 2.4.7 (1), we may say K controlls the purterbation and B the rescaling.Thus, the construction of the action of e B on LM − τ is reduced to the construction ofmaps e ǫ and ψ . In sub-subsection 3.1.2, we state conditions which e ǫ and ψ satisfy and provethese conditions ensure the action Φ is well-defined. In sub-subsection 3.1.3, we constructthese maps. The construction is a standard inductive one using the cone presentation of theassociahedra given in subsection 2.3. An instructive explanation of essense of the inductiveconstruction is found in Adams[4]. Remark 3.1.1. Cohen described a multiplicative structure on a cs-spectrum which shouldinduce the loop product on the totalization in [16]. But it is unclear whether this structureinduces a product on its totalization and this is why we do not use it ( and this is a motivationfor writing the present paper). To complete the exposition, we explain this point moreprecisely. In the rest of this remark, we follow the notations in [16]. Cohen gives two modelsof M − T M both of which are slightly different from one given in the present paper. One isunital and the other is non-unital. To define a cs-spectrum, he uses the unital model denotedby M − τ ( e ) and define a map ∆ r : M − τ ( e ) −→ M − τ ( e ) ∧ ( ν ǫ ( e ) + ) to define a coface mapbut this map is not a morphism of symmetric spectra as it does not commute with the actionof the sphere spectrum. In fact, ∆ r takes a point ( φ, x ) ∈ M − τ ( e ) to the point ( φ, x ) ∧ x where x ∈ ν ǫ ( e ) is the point determined by the unique decomposition x = φ ( x ) + x with x ∈ Im ( φ ) ⊥ , but x for ( φ × ∆ m , ( x, t )) is in general different from x for ( φ, x ). (One caneasily find a counter-example. See [16] for notations.)If we replace the unital model with the non-unital one denoted by M − τ , a map definedsimilarly to ∆ r becomes a well-defined morphism of symmetric spectra and we obtain awell-defined cosimplicial object. But even in this case, a kind of product on the cosimplicialobjects defined as in [16] does not satisfy the conditions of McClure-Smith because thefollowing square does not commute for a reason similarly to the unital case M − τ ∧ M − τ µ / / id ∧ ∆ r (cid:15) (cid:15) M − τ ∆ r (cid:15) (cid:15) M − τ ∧ M − τ ∧ ( M + ) µ ∧ id / / M − τ ∧ ( M + ) . ohen-Jones isomorphism e ǫ and ψ satisfy In this subsection, we list the conditions which e ǫ and ψ satisfy and prove the conditions en-sure the formulae of the previous section give a well-defined operad action ( see Proposition3.1.2). The construction is given in next subsection. Conditions on e ǫ . e ǫ satisfies the following five conditions ( ǫ -1)- ( ǫ -5) for any n, n , n ≥ u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) ∈ K ( n ) × e V [ k , . . . , k n ].( ǫ -1) e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) = e ǫ ( u ; ( φ , ǫ ) , . . . , (0 × φ i , ǫ i ) , . . . , ( φ n , ǫ n )), where 0 × φ i ( x ) =(0 , φ i ( x )) ∈ R k i +1 .( ǫ -2) For any permutations σ , . . . , σ n , e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) = e ǫ ( u ; ( σ φ , ǫ ) , . . . , ( σ n φ n , ǫ n )),where σ i φ i is the composition of φ i with the permutation of the component of R k i as-sociated to σ i .( ǫ -3) For an element ( u ◦ i u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) such that u s ∈ K ( n s ) ( s = 0 , 1) and n + n − n , e ǫ ( u ◦ i u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n ))= e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ i − , ǫ i − ) , ( φ ′ , ǫ ′ ) , ( φ i + n , ǫ i + n ) , . . . , ( φ n , ǫ n ) ),where ( φ ′ , ǫ ′ ) = ( φ i × · · · × φ i + n − , e ǫ ( u ; ( φ i , ǫ i ) , . . . , ( φ i + n − , ǫ i + n − )) ).( ǫ -4) e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) ≤ | φ ×···× φn | min { ǫ , . . . , ǫ n } .( ǫ -5) If u ∈ ∂ K ( n ), e ǫ ( tu ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) ≤ e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) for any t ∈ [0 , e ǫ well-defined at eachinductive step. Conditions on ψ . To state conditions which ψ satisfies, we need three maps: α i : K ( n ) × K ( m ) × M [ k , . . . , k n + m − ] −→ K ( n ) × M [ k , . . . , k i + · · · + k i + m − , . . . , k n + m − ] β i : K ( n ) × K ( m ) × M [ k , . . . , k n + m − ] −→ K ( n + m − × M [ k , . . . , k n + m − ] γ i : K ( n ) × K ( m ) × M [ k , . . . , k n + m − ] −→ K ( m ) × M [ k i , . . . , k i + m − ]defined by α i ( u , u ; v , . . . , v n + m − ) = ( u ; v , . . . , v i − , v ( i ) , v i + m , . . . , v n + m − ) .β i = ( − ◦ i − ) × id γ i = proj × proj i,...,i + m − , where v ( i ) = ( φ i × · · · × φ i + m − , e ǫ ( u ; ( φ i , ǫ i ) , . . . , ( φ i + m − , ǫ i + m − )) , ( v i , . . . , v i + m − )), and proj and proj i,...,i + m − are the usual projections to the components indicated by subscripts. ψ satisfies the following five conditions ( ψ -1)- ( ψ -5).( ψ -1) For t ∈ R and ( u ; v , . . . , v n ) ∈ D n , if ( u ; v , . . . , t × v i , . . . , v n ) ∈ D n , ψ ( u ; . . . , v i , . . . ) = ψ ( u ; . . . , t × v i , . . . ). Here t × v i denotes the element (0 × φ i , ǫ i , ( t, v i )).( ψ -2) For σ i ∈ Σ k i , ψ ( u, σ · v , . . . , σ n · v n ) = ψ ( u, v , . . . , v n ). Here, the k -th symmetricgroup acts on M k by permutations of components. ohen-Jones isomorphism ψ -3) The following diagram commutes for each n, m , and i : β − i D n + m − k ,...,k n + m − α i × γ i / / β i (cid:15) (cid:15) D n...,k i + ··· + k i + m − ,... × D mk i ,...,k i + m − ψ × ψ (cid:15) (cid:15) Map( M, M ) n × Map( M, M ) mComp i (cid:15) (cid:15) D n + m − k ,...,k n + m − ψ / / Map( M, M ) n + m − . Here, Comp i is defined as follows: Comp i ( g , . . . , g n ; f , . . . , f m ) = ( g , . . . , g i ◦ f , . . . , g i ◦ f m , g i +1 , . . . , g n ) . ( ψ -4) For ( u, v , . . . , v n ) ∈ D n , ψ i ( u, v , . . . , v n )( π φ i ( v i )) = π φ ×···× φ n ( v , . . . , v n ).( ψ -5) | ψ ( u, v , . . . , v n )( y ) − y | ≤ n d ( v, φ ( M )) where v = ( v , . . . , v n ), and φ = φ × · · ·× φ n ,and d ( − , − ) denotes the Euclid distance.Note that by the conditions ( ǫ -3)and ( ǫ -4), we have α i ( β − i D n + m − k ,...,k n + m − ) ⊂ D nk ,...,k i + ··· + k i + m − ,k i + m ,...,k n + m − γ i ( β − i D n + m − k ,...,k n + m − ) ⊂ D mk i ,...,k i + m − . By these inclusions, the top horizontal map of ( ψ -3) is well-defined.The role of first three conditions are similar to those on e ǫ . Proposition 3.1.2. If e ǫ and ψ satisfy the conditions ( ǫ -1), ( ǫ -2), ( ǫ -3), ( ψ -1), ( ψ -2), ( ψ -3),and ( ψ -4), the formulae (1) and (2) in sub-subsection 3.1.1 define an action of e B on LM − τ . Proof. By ( ψ -4), e c is a well-defined loop (see sub-subsection 3.1.1). By conditions ( ǫ -1)and ( ψ -1) (resp.( ǫ -2) and ( ψ -2)) the map Φ : ( e B ( n ) + ) ∧ LM − τk ∧ LM − τk ∧ · · · ∧ LM − τk n −→ LM − τk + ··· + k n commutes with the action of S (resp. Σ ∗ ) so defines a morphism of symmetricspectra. The rest thing is compatibility of Φ and composition of e B (= B × K ). Let ( f, u ) ∈B ( n ) × K ( n ) and ( g, w ) ∈ B ( m ) × K ( m ) be two elements. We must show e c ( f ◦ i g, u ◦ i w ; c , . . . , c n ) = e c ( f, u ; c , . . . , c i − , c ( i ) , c i + m , . . . , c n + m − ) , where c ( i ) = ( φ i × · × φ i + m − , ǫ ( i ) , e c ( g, w ; c i , . . . , c i + m − ) , ( v i , . . . , v i + m − )) and we regard e c as a map B ( n ′ ) × ev ∗ D → LM . The compatibility with composition obviously follows fromthis equality and the condition ( ǫ -3). To prove the equality, note that t ∈ ( f ◦ i g ) − (cid:20) k − n + m − , kn + m − (cid:21) ⇐⇒ t ∈ f − [ k − n , kn ] if k ≤ i − nf ( t ) − i + 1 ∈ g − [ k − im , k − i +1 m ] if i ≤ k ≤ i + m − f − [ k − mn , k − m +1 n ] if i + m ≤ k ≤ n + m − i ≤ k ≤ i + m − 1. Proofs of other cases are similar.By the above condition on t , t ∈ f − [ i − n , in ] so the right hand side of the above equation isequal to ψ i ( u ) ◦ c ( i ) ( nf ( t ) − i + 1) = ψ i ( u ) ◦ ψ k − i +1 ( w )( c k [ mg ( nf ( t ) − i + 1) − ( k − i + 1) + 1])= ψ k ( u ◦ i w )( c k (( n + m − f ◦ i g ( t ) − k + 1)) (by ( ψ -3))= the right hand side.Here, for example, ψ i ( u ) is abbreviation of ψ i ( u ; v , . . . , v ( i ) , . . . v n + m − ). ohen-Jones isomorphism e ǫ and ψ We first describe construction of e ǫ and ψ , then verify well-definedness of the construction. Construction of e ǫ . We construct e ǫ by induction on the arity n . When n = 2, K (2) isone point set { pt } . We put e ǫ ( pt ; ( φ , ǫ ) , ( φ , ǫ )) = 110 | φ × φ | min { ǫ , ǫ } . Suppose e ǫ is constructed up to n − ǫ -1) -( ǫ -5) as long as they makesense. Let B V : T ( n ) , −→ CG be the diagram given by B V ( T ) = B n ( T ) × e V [ k , . . . , k n ](see subsection 2.3). Let ∂ K ( n ) denote the subspace of K ( n ) consisting of all faces of codi-mension one. To define e ǫ on ∂ K ( n ) × e V [ k , . . . , k n ], we define a natural transformationˆ ǫ : B V → (0 , L e / , L e / 16) denotes the constant functor taking the value on theopen interval. For an element T of codimension one, we define ˆ ǫ T by the above equation ofconditon ( ǫ -3). In other words, if T = T ◦ i T , and the arity of T t is n t , we putˆ ǫ T ( u, w ; ( φ , ǫ ) , . . . , ( φ n + n − , ǫ n + n − ))= e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ i − , ǫ i − ) , ( φ ′ , ǫ ′ ) , ( φ i + n , ǫ i + n ) , . . . , ( φ n + n − , ǫ n + n − )) . If T is of codimension two, we take an element T ′ of codimension one such that T ≤ T ′ , anddefine ˆ ǫ T to be the composition B V ( T ) → B V ( T ′ ) ˆ ǫ T ′ −−→ (0 , L e / ǫ is verified below. We define e ǫ on ∂ K ( n ) × e V [ k , . . . , k n ] as the composition ∂ K ( n ) × e V [ k , . . . , k n ] ∼ = colim T ( n ) , B V ˆ ǫ −→ (0 , L e / , where the homeomorphism ∼ = is induced from the homeomorphism θ n : colim B n ∼ = ∂ K ( n ).Recall that any point of K ( n ) is of the form tu for some t ∈ [0 , 1] and u ∈ ∂ K ( n ). Using e ǫ | ∂ K ( n ) × e V [ k ,...,k n ] , we put e ǫ ( tu ; ( φ , ǫ ) , . . . , ( φ n , ǫ n ))= (1 − t ) min w ∈ ∂K ( n ) { e ǫ ( w ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) } + t e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) . Construction of ψ . We also construct ψ by induction on the arity n . When n = 2, weput ψ i ( pt ; v , v )( y ) = π φ ( b ψ i ( pt ; v , v )( y )) , where b ψ i ( pt ; v , v )( y ) = v + φ ( y − π φ i ( v i )) . Here, φ = φ × φ , v = ( v , v ), and sum and scalar multiplication are taken in R k + k .Suppose ψ is constructed up to arity n − ψ -1) -( ψ -5) aslong as they make sense. First, we shall construct ψ on δ D n = { ( u, v , . . . , v n ) ∈ D n | u ∈ ∂ K ( n ) } . We define a diagram B D : T ( n ) , −→ CG by B D ( T ◦ i T ) = β − i D n , B D (( S ◦ j S ) ◦ k S ) = β − j,k D n . ohen-Jones isomorphism sss ψ i ( tu )( y ) ψ i ( u )( y ) b ψ i ( tu )( y ) t − t v + φ ( y − π φ i ( v i )) φ ( M ) R k + ··· + k n ψ ( ∂ K (4))( y )Figure 1: construction of ψ for n = 4 ( ψ i ( u )( y ) is the abbreviation of ψ i ( u ; v , . . . , v )( y ), ψ i ( tu )( y ) and b ψ i ( tu )( y ) are similar abbreviations.)Here, T ◦ i T denotes an element of codimension one and β i is the map given in the statementof conditions on ψ . ( S ◦ j S ) ◦ k S denotes an element of codimension two and β j,k is definedas follows: β j,k :=(( − ◦ j − ) ◦ k − ) × id : K ( m ) × K ( m ) × K ( m ) × M [ k , . . . , k m + m + m − ] −→ K ( m + m + m − × M [ k , . . . , k m + m + m − ]where m t is the arity of S t . On morphisms, B D is defined by exactly same way as B n .Clearly θ n in 2.3 induces a homeomorphism colim B D ∼ = δ D n . Similarly to the constructionof e ǫ , we shall define a natural transformation ψ : B D −→ Map( M, M ) n . Let T be an elementof T ( n ) , . If T is of codimension one, ψ T is the composition of the top right angle in thecondition ( ψ -3). If T is of codimension two, we take an element T ′ ≥ T of codimension one,and define ψ T to be the composition B D ( T ) ⊂ B D ( T ′ ) ψ T ′ → Map( M, M ) n . Well-definedness of ψ is verified bellow. Define ψ on δ D n as the composition δ D n ∼ = colim B D ψ −→ Map( M, M ) n Let tu denote the element of K ( n ) with t ∈ [0 , 1] and u ∈ ∂ K ( n ). Using ψ | δ D n , we put ψ i ( tu ; v , . . . , v n )( y ) = π φ ( b ψ i ( tu ; v , . . . , v n )( y ))where b ψ i ( tu ; v , . . . , v n )( y ) = (1 − t )[ v + φ ( y − π φ i ( v i ))] + tφ ( ψ i ( u, v , . . . , v n )( y )) . Here, φ = φ × · · · × φ n , v = ( v , . . . , v n ), and sum and scalar multiplication are takenin R k + ··· + k n , see Figure 1. (We will verify the image of b ψ i belongs to the domain of π φ below.) Verifications on construction of e ǫ and ψ . (On e ǫ ) We shall verify the well-definednessof the natural transformation ˆ ǫ . An element T of codimension two is presented as ( S ◦ j S ) ◦ k S for unique 5-tuple ( S , S , S , j, k ) such that j ≤ k . Let T ′ = T ◦ i T be an element of ohen-Jones isomorphism T ≤ T ′ . We must show the map ˆ ǫ T is independent of a choiceof T ′ . We shall consider the case of k ≤ j + m − m t is the arity of S t ). In this casethere are exactly two possibilities: (i) S ◦ j S < T , S = T and k = i , or (ii) S = T , S ◦ k − j +1 S < T , and j = i . The values ˆ ǫ T at ( u , u , u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) correspondingto (i) and (ii) are e ǫ ( u ◦ j u , ( φ , ǫ ) , . . . , ( φ k × · · · × φ k + m − , e ǫ ( u ; ( φ k , ǫ k ) , . . . , ( φ k + m − , ǫ k + m − )) , . . . ) and e ǫ ( u ; . . . , ( φ j × · · · × φ j + m + m − , e ǫ ( u ◦ k − j +1 u , ( φ j , ǫ j ) , . . . , ( φ j + m + m − , ǫ j + m + m − )) , . . . )respectively . As m t ≤ n − 2, using the inductive hypothesis concerning ( ǫ -3), we can obviouslyverify these two values are equal. We can also verify the case of k > j + m similarly.The condition ( ǫ -3) is satisfied by definition, and verification of the rest conditions isstraightforward.(On ψ ) We shall verify the natural transformation ψ is well-defined. Let T = ( S ◦ k S ) ◦ j S be an element of codimension two. We must verify the map ψ T is independent ofa choice of an element T ′ = T ◦ i T of codimension one with T ′ > T . We shall consider thecase of k ≤ j + m − 1. The maps ψ T corresponding to the above two choices (i),(ii) in theverification on e ǫ are compositions β − j,k D n β j × id −−−−→ β − k D n α k × γ k −−−−→ D m + m − × D m Comp k ◦ ( ψ ) −−−−−−−−→ Map( M, M ) n and β − j,k D n id × β k − j +1 −−−−−−−→ β − j D n α j × γ j −−−−→ D m × D m + m − Comp j ◦ ( ψ ) −−−−−−−−→ Map( M, M ) n , respectively. These two maps fit into the top-right and bottom-left corners of the followingdiagram: β − j,k D n β j × id / / α ′ × γ ′ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ id × β k − j +1 (cid:15) (cid:15) α ′′ × γ ′′ ❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆ β − k D n α k × γ k / / (A) D m × D m ψ (cid:15) (cid:15) β − j D m × D m β j × id ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ α × γ j × id (cid:15) (cid:15) (B) β − j D nα j × γ j (cid:15) (cid:15) (C) D m × β − k D m id × β k } } ④④④④④④④④④④④④④④④④④④④④ id × α × γ k − j +1 / / D m × D m × D m ψ (cid:15) (cid:15) (E) Map m × Map m × Map m id × C k − j +1 (cid:15) (cid:15) C j × id / / Map m × Map m C k (cid:15) (cid:15) D m × D m (D) ψ / / Map m × Map m C j / / (F) Map n Here, • Map, m st , and C l are abbreviations of Map( M, M ), m s + m t − 1, and Comp l , • the maps α ′ , γ ′ , α ′′ , and γ ′′ are defined by the following formulae: α ′ ( u , u , u , v , . . . v n ) =( u , u , v , . . . , v ( k ) , . . . , v n ), v ( k ) = ( φ ( k ) , ǫ ( k ) , v ( k ) ), φ ( k ) = φ k × · · · × φ k + m − , . . . , γ ′ ( u , u , u ; . . . ) = ( u , v k , . . . v k + m − ) α ′′ ( u , u , u , v , . . . , v n ) = ( u , v . . . , v ( j ) , . . . v n ), and γ ′′ ( u , u , u , v , . . . , v n ) =( u , u , v j , . . . , v j + m + m − ).Commutativity of this diagram is verified as follows: Commutativity of the triangles (A),(C) and square (B) follows from the following obvious equalities. α ◦ α ′ = α ′′ , γ ′ = γ k − j +1 ◦ γ ′′ , γ j ◦ α ′ = α ◦ γ ′′ . ohen-Jones isomorphism ψ -3) as m s , m st ≤ n − 2, and commutativity of (F) is clear. Thus, we have verified the map ψ T is independent of choices of T ′ for the case k ≤ j + m − 1. For the case k ≥ j + m , wecan verify the well-definedness of ψ T by a similar way.To show ψ i is well-defined on whole D n , we must show values of b ψ i belong to the domainof π φ . In other words, we must show d ( b ψ i ( tu ; v , . . . , v n )( y ) , φ ( M )) < | φ |· L e . for any element( tu ; v , . . . , v n ) ∈ D n , and y ∈ M . By straightforward calculation, we have | b ψ i ( tu )( y ) − φ ( y ) | ≤ (1 − t ) | v − π φ i ( v i ) | + t | φ | · | ψ i ( u )( y ) − y | . Here, φ = φ × · · · × φ n , v = ( v , . . . , v n ), and for example, b ψ i ( tu )( y ) is abbreviation of b ψ i ( tu ; v , . . . , v n )( y ). We can write u = u ◦ j u for some u , u with n = ar u ≥ n = ar u ≥ 2. By inductive hypothesis, we have | ψ i ( u )( y ) − y | ≤ | ψ i ( u )( ψ i ( u )( y )) − ψ i ( u )( y ) + ψ i ( u )( y ) − y |≤ (6 n + 6 n ) d ( v, φ ( M )) . ( i , i are suitable integers.) We also have | v − φπ φ i ( v i ) | ≤ | v − φπ φ ( v ) | + | φπ φ ( v ) − φπ φ i ( v i ) | = d ( v, φ ( M )) + | φ | · | π φ ( v ) − π φ i ( v i ) |≤ d ( v, φ ( M )) + | φ || φ i | ( | φ i π φ ( v ) − v i | + | v i − φ i π φ i ( v i ) | ) ≤ (1 + 2 | φ | ) d ( v, φ ( M )) ( ∵ | φ i ( π φ ( v )) − v i | ≤ | φ ( π φ ( v )) − v | )So we have d ( b ψ i ( tu ; v , . . . , v n )( y ) , φ ( M )) ≤ | b ψ i ( tu )( y ) − φ ( y ) |≤ max { n + n ) | φ | , | φ |} · d ( v, φ ( M )) ≤ max { n + n ) | φ | , | φ |} | φ | min { ǫ , . . . , ǫ n } (by ( ǫ -4)) ≤ | φ | L e . ( ψ -3) is satisfied by definition, and verification of ( ψ -1), ( ψ -2), ( ψ -4) is trivial. We shall verifythe condition ( ψ -5). We have | b ψ i ( tu )( y ) − φ ( ψ i ( u )( y )) | ≤ (1 − t ) | v − φπ φ i ( v i ) | + (1 − t ) | φ | · | ψ i ( u )( y ) − y | . So with the above equalities, we have | ψ i ( tu )( y ) − y | ≤ | φ | ( | b ψ i ( tu )( y ) − ψ ( y ) | + | b ψ i ( tu )( y ) − φ ( ψ i ( u )( y )) | ) ≤ | φ | (2(1 − t ) | v − φπ φ i ( v i ) | + | φ | · | ψ i ( u )( y ) − y | ) ≤ | φ | (2(1 + 2 | φ | ) + 6( n + n )) d ( v, φ ( M )) ≤ (6 + 6 n + 6 n ) · d ( v, φ ( M )) . As n , n ≥ n = n + n − n − (6 + 6 n + 6 n ) = − n + 12 n · n + 12 n − n − n − n − n ) − ≥ . Thus, we have completed the construction of an action of e B on LM − τ . ohen-Jones isomorphism Note that the classical stable homotopy category is equivalent to the homotopy categoryof symmetric spectra Ho( SP ) as symmetric monoidal category, so the Cohen-Jones ringspectrum LM − T M is considered as a monoid in Ho( SP ). Theorem 3.2.1. If we consider LM − τ as a semigroup in the homotopy category Ho( SP ),it is isomorphic to the Cohen-Jones ring spectrum. (A semigroup in a monoidal category isan object equipped with an associative product (but without a unit)) Proof. Define a map α : E − ( ν ǫ ( φ × φ )) −→ ev ∗∞ ν ǫ ( φ × φ ) as follows: (For the definitionsof E and ev ∞ , see subsection 2.1) α (( c , v ) , ( c , v )) = (( e c , e c ) , ( v , v )) , e c i ( t ) = ψ i (( φ , v ) , ( φ , v ))( c i ( t )) ( i = 1 , . Here, ψ i is the one constructed in sub-subsection 3.1.3 and note that the formula of ψ i doesnot depends on ǫ so we omit it here. We shall prove α is homeomorphism for sufficientlysmall ǫ . First we show α is injective. Let (( c , v ) , ( c , v )) and (( c ′ , v ′ ) , ( c ′ , v ′ )) be twoelements of E − ( ν ǫ ( φ )) ( φ = φ × φ ). If α (( c , v ) , ( c , v )) = α (( c ′ , v ′ ) , ( c ′ , v ′ )), we have( v , v ) = ( v ′ , v ′ ) and b ψ i (( φ , v ) , ( φ , v ))( c i ( t )) and b ψ i (( φ , v ′ ) , ( φ , v ′ ))( c ′ i ( t )) belong tothe same fiber of the tubular neighborhood of φ ( M ) for each t ∈ [0 , i = 1 , ψ in sub-subsection 3.1.3). It follows that φ ( c i ( t ) − c ′ i ( t )) ∈ T φ ( e c i ( t )) ( φ ( M )) ⊥ . If ǫ is sufficiently small, three points c i ( t ) , c ′ i ( t ) and e c i ( t ) belong to sufficiently small openset of M which is approximated by the affine space φ ( e c i ( t )) + T φ ( e c i ( T )) φ ( M ), so the aboverelation can not occur if c i ( t ) = c ′ i ( t ). This implies α is injective.Let (¯ c , ¯ c , ( v , v )) be an element of ev ∗∞ ν ǫ ( φ ). We shall show there exists a uniqueelement c i ( t ) ∈ B ǫ ( φ (¯ c i ( t )) ∩ φ ( M ) such that b ψ i (( φ , v ) , ( φ , v ))( c i ( t )) belongs to the fiberover φ (¯ c i ( t )) of the tubular neighborhood of φ ( M ) for each i and t , where B ǫ ( φ (¯ c i ( t ))) isthe neighborhood of φ (¯ c i ( t )) with radius 10 ǫ . If ǫ is sufficiently small, the two points ( v , v )and φ ( π φ ( v i )) are sufficiently close, and the set B ǫ ( φ (¯ c i ( t ))) ∩ φ ( M ) is approximated bythe affine space φ (¯ c i ( t )) + T φ (¯ c i ( t )) φ ( M ). So the intersection[ φ ( π φ ( v i )) − ( v , v ) + φ (¯ c i ( t )) + T φ (¯ c i ( t )) φ ( M ) ⊥ ] ∩ φ ( M ) ∩ B ǫ ( φ (¯ c i ( t )))consists of exactly one point. We denote this point by φ ( c i ( t )), which is easily seen to satisfythe above condition. It is easy to see c i ( t ) continuously depends on t . Thus we obtain anelement (( c , v ) , ( c , v )) ∈ E − ( ν ǫ ( φ × φ )) with α (( c , v ) , ( c , v )) = (¯ c , ¯ c , ( v , v )).It is also easy to see c i continuously depends on ¯ c i . So the left inverse (¯ c , ¯ c , ( v , v )) (( c , v ) , ( c , v )) is continuous and as α is injective, this is the continuous inverse of α .This homeomorphism α makes the diagram in Lemma 2.1.1 commutative so by the samelemma, we can use this homeomorphism in the definition of the Cohen-Jones ring spec-trum. An isomorphism LM − T M ∼ = LM − τ in the homotopy category of spectra is givenby T h ( ev ∗ ν ( φ )) → LM − τN , ( c, v ) ( φ , ǫ , c, v ) . So what we have to prove is homotopycommutativity of the following diagram: T h ( ev ∗ ν ( φ ) × ev ∗ ν ( φ )) / / (cid:15) (cid:15) LM − τN ∧ LM − τN (cid:15) (cid:15) S N T h ( ev ∗ ν ( φ )) / / LM − τ N But we can easily construct a homotopy for the commutativity using a linear isotopy R N × I → R N covering an isotopy R N × I → R N between the diagonal and 0 × id. ohen-Jones isomorphism To prove Theorem 1.0.1 (2), it is enough to prove LM − τ and THC( A ′ , B ) are weak equivalentas nu- A ∞ -ring spectra (see Proposition 2.5.4 for the definition of A ′ and B ). We will provethis claim by constructing a zig-zag of termwise weak equivalences between two cosimplicialobjects whose totalizations are isomorphic to LM − τ and THC( A ′ , B ). In subsection 4.2, wedefine a cs-spectrum L • with an action of the monad e K such that the totalization Tot L • is isomorphic to LM − τ as e B -algebras (see sub-subsection 2.4.1 for e K and e B ). We want toconnect two cs-spectra L • and THC • ( A ′ , B ) by a zig-zag of termwise stable equivalenceswhich preserve actions of e K , but this turns out difficult (see sub-subsection 4.4.1 for anexplanation). To avoid this difficulty, we define a monad CK over cosimplicial objects whichis considered as a ”up to homotoy coherency version” of e K and connect L • and THC • ( A ′ , B )by a zig-zag of termwise stable equivalences which preserve actions of CK as follows: L • p ←− IM • ¯ q −→ THC • ( A ′ , B ) , where the actions of CK on the left and right are induced by a natural morphism U : CK → e K of monads, see subsections 4.3 and 4.5. In subsubsection 4.4.5, we prove an action of CK induces an action of an A ∞ -operad on g Tot, but we cannot prove it does on Tot. So we mustprove A ∞ -structures on Tot and g Tot both induced by an action of e K are equivalent, andthis is done in subsections 4.6 and 4.7. The rest thing we have to prove is that the twomorphisms IM • → L • and IM • → THC • ( A ′ , B ) induce weak equivalences on g Tot. for thefirst morphism, this is clear as it is a termwise levelwise weak equivalence. For the lattermorphism, we need more care and prove it in subsection 4.7 (note that g Tot is not alwayshomotopy invariant for symmetric spectra as the model category SP is not fibrant). LM − τ We shall define a cs-spectrum L • such that Tot( L • ) is isomorphic to LM − τ . We put L p = ( M × p ) ˆ ⊗ M − τ (= ( M × p + ) ∧ M − τ ) . (See subsection 2.2 for M − τ .) If 1 ≤ i ≤ p , the coface morphism d i : L p → L p +1 isdefined as the morphism repeating the i -th component of M × p , and d and d p +1 are de-fined as the ones taking an element ( x , . . . , x p , h φ, ǫ, v i ) to ( π φ ( v ) , x , . . . , x p , h φ, ǫ, v i ) and( x , . . . , x p , π φ ( v ) , h φ, ǫ, v i ) respectively. The codegeneracy morphism s i : L p → L p − is theone skipping the i + 1-th component for each i = 0 , . . . , p − p = { ( t , . . . , t p ) ∈ R p | ≤ t ≤ · · · ≤ t p ≤ } be the standard p -simplex for p ≥ ϕ p : ∆ p ˆ ⊗ LM − τ → L p by [( t , . . . , t p ) , h φ, ǫ, c, v i ] [ c ( t ) , . . . c ( t p ) , φ, ǫ, v ] . As the collection { ϕ p } p ≥ forms a morphism between cosimplicial spectra, by adjointness weobtain a morphism ϕ : LM − τ → Tot( L • ) ∈ SP . The following can be proved similarly tothe corresponding statement for the usual cosimplicial model of a free loop space. Proposition 4.2.1. Under the above notations, ϕ : Tot L • → LM − τ is an isomorphism in SP .We shall define an action of e K on L • . We define a morphism Ψ : e K ( L • ) −→ L • byΨ( u ; ( x , . . . , x p , h φ , ǫ , v i ) , . . . ( x n, , . . . , x n,p n , h φ n , ǫ n , v n i )= ( ψ ( x ) , . . . , ψ ( x p ) , . . . , ψ n ( x n, ) , . . . , ψ n ( x n,p n ) , h φ × · · · × φ n , e ǫ, ( v , . . . , v n ) i ) , where ψ i and e ǫ mean ψ i ( u ; ( φ , ǫ , v ) , . . . , ( φ n , ǫ n , v n )) and e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) respec-tively, see subsection 3.1 for ψ and e ǫ . ohen-Jones isomorphism Proposition 4.2.2. Ψ is an action of e K . Proof. Ψ actually factors through the quotient e K ( L • ) because ψ i ( π φ i v i ) = π φ ( v , . . . , v n ) foreach i (see the condition ( ψ -4) in sub-section 3.1.2). Verification of compatibility of Ψ withcoface and codegeneracy morphisms is a routine work. (Compatibility with the first and lastcoface morphisms also follows from the identity ψ i ( π φ i v i ) = π φ ( v , . . . , v n ). ) Compatibilitywith the product of the monad follows from the condition ( ψ -3) stated in sub-subsection3.1.2. Proposition 4.2.3. The isomorphism ϕ : LM − τ → Tot L • in Proposition 4.2.1 is compatiblewith the actions of e B . Here, the action on LM − τ is the one defined in subsection 3.1 andthe action on Tot L • is the one induced by Ψ via Proposition 2.4.7. Proof. We have to prove the following diagram is commutative e B ( n ) ˆ ⊗ ( LM − τ ) ⊗ n / / (cid:15) (cid:15) LM − τ (cid:15) (cid:15) e B ( n ) ˆ ⊗ (Tot L • ) ⊗ n / / Tot L • Let ( f, u ) ∈ e B ( n ) = B ( n ) × K ( n ), c , . . . , c n ∈ LM − τ , and ( t , . . . t k ) ∈ ∆ k be elements. Let[( t , . . . , t ,k ) , . . . , ( t n, , . . . , t n,k n )] ∈ (∆ • (cid:3) · · · (cid:3) ∆ • ) k be a representative of f ( t , . . . t k ) . The image of (( f, u ) ; c , . . . , c n ) by the left-bottom cornercomposition takes ( t , . . . , t k ) to ψ ( c ( t )) , . . . , ψ ( c ( t ,k )) , . . . , ψ n ( c n ( t n, )) , . . . , ψ n ( c n ( t n,k n )) , φ ×· · ·× φ n , e ǫ, ( v , . . . , v n )) . Since ( i − t i,l ) /n = f ( t L ), where L = P i − j =1 k j + l , for each 1 ≤ i ≤ n and 1 ≤ l ≤ k i , bythe definition of ˜ c in sub-subsection 3.1.1, the above value is equal to the image by right-topcomposition. In this subsection, we define a cs-spectrum IM • which intermediates L • and THC • ( A ′ , B )(see subsection 4.1). In the rest of this paper, we fix A ′ an B as in Proposition 2.5.4 To write down constructions efficiently in the following, we shall describe the internal homobject Map in SP at the point-set level. This description is used in later (sub)sectionsimplicitly. Of course, the contents of this subsection is not new.Let X , . . . , X n and Y be symmetric spectra. The k -th space of the spectrum Map( X ⊗· · · ⊗ X n , Y ) consisting of sequences of continuous maps f l : G l + ··· + l n = l ( X ) l × · · · × ( X n ) l n → Y l + k , l ≥ x , . . . , x n ) ∈ ( X ) l × · · · × ( X n ) l n , if at least one of x i is the base point, f l ( x , . . . , x n ) is also the base point. ohen-Jones isomorphism f l induces a map on the corresponding smashproduct. Then the following diagram commutes S ∧ ( X ) l ∧ · · · ∧ ( X n ) l n T / / f l (cid:15) (cid:15) ( X ) l ∧ · · · ∧ S ∧ ( X i ) l i ∧ · · · ∧ ( X n ) l n (cid:15) (cid:15) S ∧ Y l + k (cid:15) (cid:15) ( X ) l ∧ · · · ∧ ( X i ) l i +1 ∧ · · · ∧ ( X n ) l n f l +1 (cid:15) (cid:15) Y l + k +1 σ · / / Y l + k +1 Here, T is the obvious transposition, and the permutation σ ∈ Σ l + k +1 is the cyclicpermutation of the first l + · · · + l i − + k +1 letters which takes 1 to l + · · · + l i − + k +1(the rest arrows are induced by the action of the sphere spectrum).3. When we regard Σ l × · · · × Σ l n ⊂ Σ l ⊂ Σ k + l (permutations on the last l -letters), themap f l is Σ l × · · · × Σ l n -equivariant. Put T • = THC • ( F ( M ) , Γ( M ))(see subsection 2.2 for F ( M ) and Γ( M )). We define a morphism e ρ p : L p → T p by e ρ p ( x , . . . , x p , h φ, ǫ, v i )( f , . . . , f p ) = ρ ( h φ, ǫ, v i ) · f ( x ) · · · f p ( x p ) . (See subsection 2.2 for ρ and ( − · − ) denotes the action of the sphere spectrum.) A troubleis that the collection { e ρ p } p does not commutes with coface morphisms. To remedy this, weintroduce an intermediate cs-spectrum IM • .Let p : ( T p ) [0 , → T p be the evaluation at 0 and L pk × T pk ( T pk ) [0 , be the pullback of thediagram L pk e ρ p −→ T pk p ←− ( T pk ) [0 , . We define a subspace IM pk ⊂ L pk × T pk ( T pk ) [0 , as follows. A pair ( λ, h ) of λ ∈ L pk and h ∈ ( T pk ) [0 , belongs to IM pk if and only if it satisfiesthe following conditions:1. If λ is the base point, h is also the base point, i.e., the constant path at the base point.2. If λ is not the basepoint, λ is represented uniquely by an element ( x , . . . , x p , φ, ǫ, v ) ∈ M × p × M k . Then for each l ≥ t ∈ [0 , f ∈ F ( M ) × pl , h t,l ( f ) ∈ Γ( M ) k + l is represented by an element of a form (0 × φ, ǫ, s t ) where the section s t satisfies thecondition that | s t,l ( y ) − φ ( y ) | ≥ | v − φ ( y ) | for each y ∈ M and t ∈ [0 , { IM pk } k ≥ forms a symmetric spectrum, denoted by IM p , whose structuremaps are the restrictions of those of L p × T p ( T p ) [0 , . We shall define the coface morphisms.Fix a map H x,t : M → M depending on x ∈ M and t ∈ [0 , 2] continuosly such that • H x, takes B / L e ( x ) ∩ M to the one point set { x } , and • H x,t is the identity on the whole M if 1 ≤ t ≤ λ, h ) ∈ IM pk , ( λ i , h i ) = d i ( λ, h ) is defined as follows: λ i = d i λ , the coface of L . If λ is the base point, h i is the base point, otherwise λ is represented by a unique element( x , . . . , x n , ( φ, ǫ, v )) ∈ M × n × M k ) and we put h it,l ( f , . . . , f n +1 ) = σ l ,k · H ∗ π φ v,t f · h t,l ( f , . . . , f n +1 ) ( i = 0) h t,l ( f , . . . , f i · f i +1 , . . . , f n +1 ) (1 ≤ i ≤ n ) h t,l ( f , . . . , f n ) · H ∗ π φ v,t f n +1 ( i = n + 1) . ohen-Jones isomorphism l is the level of f and σ l ,k is a permutation which transposes the first l letters andthe next k letters (this is the natural permutation appearing when one transpose elements oflevel k and l ). If | v − φ ( y ) | < ǫ , we have | π φ ( v ) − y | = 1 | φ | | φπ φ ( v ) − φ ( y ) | ≤ | φ | | φπ φ ( v ) − v | + | v − φ ( y ) | < ǫ < L e / , which implies H ∗ π φ v f ( y ) = f ( π φ ( v )). This implies the above formulae of the first and lastcoface maps actually defines maps between pullbacks. The above two defining conditions on IM • ensure that these cofaces are continuous and thus well-defined morphisms of symmetricspectra. The codegeneracy operators on IM • is the one induced by those on L • and T • in acomponent-wise manner. The verification of cosimplicial identity is trivial as it is equivalentto verify the identity for THC • ( F ( M ) , H ∗ π φ ( v ) ,t Γ( M )), and we have completed the definitionof IM • . Definition 4.3.1. We define a morphism p : IM • → L • as the projection to the first factorof the pullback and q : IM • → T • as the evaluation at 2 ∈ [0 , φ : A ′ → F ( M ) and ϕ : Γ( M ) → B , and let ϕ ∗ ψ ∗ : T • → THC • ( A ′ , B ) denote the morphisminduced by the pushforward by ϕ and the pullback φ ⊗ n : ( A ′ ) ⊗ n → F ( M ) ⊗ n at the n -thterm for each n ≥ 0, and ¯ q : IM • → THC • ( A ′ , B ) the composition ϕ ∗ ψ ∗ ◦ q .The two defining conditions on IM • clearly imply the following lemma Lemma 4.3.2. The morphism p is a termwise level equivalence. A ∞ -structure In this subsection, we introduce a generalization of McClure-Smith product reviewed insubsection 2.4. This is somewhat complicated so we first explain the motivation. It is idealthat we could construct an action of e K on IM • such that the two morphisms IM • → L • and IM • → T • are compatible with the action. A most natural approach is as follows. Take amap e ψ : D n × [0 , −→ Map( M, M ) n which satisfies some compatibility conditions analogous to those on ψ and the equalities e ψ = ψ , e ψ = id × n , then we define an action of e K on IM • by˜Ψ : ( u ; ( λ , h ) , . . . , ( λ n , h n )) (Ψ( u ; λ , . . . , λ n ) , ˜ h )˜ h t ( f , . . . , f p + ··· + p n ) = σ · ( h ,t ( e ψ ∗ ,t f , . . . e ψ ∗ ,t f p ) · · · h n,t ( e ψ ∗ n,t f p ≤ n − +1 , . . . , e ψ ∗ n,t f p ≤ n )) . where Ψ is defined in subsection 4.2, and σ is a suitable permutation determined by thelevel of h i ’s and f j ’s, and e ψ i,t actually represents the value of e ψ i,t at the element of D n representing ( u ; λ , . . . , λ n ) (and we use similar abbreviation in the following). The equality e ψ i, = ψ i ensures that the image of ˜Ψ is contained in the fiber product L • × T • ( T • ) [0 , andthat the projection IM • → L • is compatible with the monad action, and e ψ i, = id × n doesthe compatibility of the evaluation IM • → T • at 2 with the action. Consider the case of arity2. If the action is well-defined, ˜Ψ must satisfies the equation d ( ˜Ψ(( λ , h ) , ( λ , h ))) = ˜Ψ( d ( λ , h ) , ( λ , h )) . Unwinding this equation, we see the equation H π φ ( v ) ,t = e ψ t, ◦ H π φ ( v ) ,t ,where we write λ i = ( x i , . . . , x ip i ; h φ i , ǫ i , v i i ), φ = φ × φ and v = ( v , v ), must be satisfiedfor each t ∈ [0 , 2] on the neighborhood { y ∈ M | | v − φ ( y ) | < e ǫ } , but it is difficult to define e ψ ohen-Jones isomorphism d x ) y ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d d ( xy ) ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ( d p +1 x ) y ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d xd y ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ h Figure 2: two components of T c (2) ( p = | x | , ◦ represents the presence of a mark on thevertex or edge)to satisfy this equation. For example, we can choose H y,t so that H y,t takes a neighborhoodof y to { y } for each t ∈ [0 , H y,t , it is impossible to define e ψ to satisfy theabove equality as a single-valued map. On the contrary, we can choose H y,t so that for any t > H y,t is a homeomorphism on M . In this case, by the above equality e ψ t is independentof u ∈ K ( n ) for t > ψ defined insubsection 3.1.Thus, it is difficult to construct an action of e K on IM • . To avoid this difficulty, weintroduce an ’up to homotopy coherency version’ of the McClure-Smith product. Practicallyspeaking, we relax the equalities d ( x · y ) = ( d x ) · y, d p + q +1 ( x · y ) = x · d q +1 y, ( d p +1 x ) · y = x · d y ( | x | = p, | y | = q )by homotopy. We first define an operad which parametrize coherency homotopies concerningthese equations, and then we define a monad over cosimplicial objects by using the operad,finally we prove an action of this monad induces an A ∞ -structure on g Tot. In this sub-subsection, we introduce an operad T c over posets. An element of T c ( n ), whichis called a cofacial n -tree , is an n -tree (see Definition 2.3.1) with some marks attached to itsvertices and edges. We consider three kinds of marks d , d , and h i . ( i = 1 , . . . , n − d represents the first coface map and d the last one. h i represents a homotopy x · · · ( dx i ) · x i +1 · · · x n ≃ x · · · x i · ( dx i +1 ) · · · x n . For example, a correspondence between cofacial trees and variables or homotopies are pre-sented in Figures 2 and 3.We impose the following rules on the attached marks. • d and d can be attached to any vertex. • d (resp. d ) can be attached to an edge e if and only if it is the leftmost (resp. rightmost)one among edges which have the same target as e . • The only one vertex which h i can be attached to is the ( i, i + 1)-join. h i cannot beattached to any edge. • If a mark can be attached to a vertex or edge, any number of copies of the mark canbe attached to it. The number of copies attached is called the multiplicity.For example, trees with marks presented in Figures 2 and 3 satisfy the above rules. Anotherexample which satisfies the rules is the following: ohen-Jones isomorphism ★★★★★★★★★❇❇❇❇❇❇❇❇ ❝❝❝❝❝❝❝❝❝✂✂✂✂✂✂✂✂ (( d x ) y ) z ( d x )( yz ) d ( x ( yz )) ( d ( xy )) zd (( xy ) z ) ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ★★★★★★★★★❇❇❇❇❇❇❇❇ ❝❝❝❝❝❝❝❝❝✂✂✂✂✂✂✂✂ (( d p +1 x ) y ) z ( d p +1 x )( yz ) xd ( yz ) ( xd y ) zx (( d y ) z ) ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ h ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ h ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ h Figure 3: two components of T c (3) ohen-Jones isomorphism ❅❅❅ (cid:0)(cid:0)(cid:0)❅❅ ❜ d h ❜ d ❜ d ❜ h ❜ d ❜ dd Here, each superscript represents the multiplicity. For example, the multiplicity of d on theroot is 4 and that of h is 5. On the other hand, the following three trees with marks do not satisfy the above rules: ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ h We introduce a partial order on T c ( n ). We define a relation ≤ on T c ( n ) for each n byiteration of the following three kind of operations.(o-1) Suppose a cofacial n -tree T has an edge where some marks are attached. We shift someof the marks on the edge to one of its endpoints and denote the result by T ′ . Then T ′ ≤ T . (Examples of this operation are presented on the left line segment of Figure2. If T is the tree at the middle of the segment, T ′ is each of those at the endpoints ofthe segment.)(o-2) Suppose T has the mark h i . We remove a part of copies of h i and add the same numberof d (resp. d ) to the source of the edge which has the ( i, i + 1)-join as its target and ison the root path from i + 1-th leaf (resp. i -th leaf), and denote the result by T ′ . Then T ′ ≤ T . (Examples are on the right line segment of Figure 2. If T is the tree at themiddle of the segment, T ′ is each of those at the endpoints of the segment.)(o-3) Suppose T ′ has an internal edge e which has neither d nor d . We contract e to a vertex(so two endpoints of e are identified with the vertex), and if e has the mark d or d ,we shift all copies of them to one of edges incoming to the source of e (such an edge isuniquely determined by the rules of attaching marks), and we carry the marks not on e over to the new tree but it may not satisfy the rule in attaching marks. Concretelyspeaking, d (resp. d ) may be on an edge which is not rightmost (resp. leftmost). In thiscase, say the mark is on the root path from the i -th leaf, then we remove all copies ofthe mark and add the same number of h i − (resp. h i ) to the created vertex. We denotethe result by T . Then T ′ ≤ T . (Examples are the relation between the trees T ′ at themiddle of the edges (( d x ) y ) z − ( d ( xy )) z , ( d ( xy )) z − d (( xy ) z ), d ( x ( yz )) − ( d x )( yz ),(( d p +1 x ) y ) z − ( xd y ) z , x (( d y ) z ) − xd ( yz ), xd ( yz ) − ( d p +1 x )( yz ) in Figure 3 and thetree T at the center of the pentagon to which the edge belongs. ) Example 4.4.1. In Figures 2 and 3 each cofacial tree at the middle of each edge is largerthan cofacial trees at its endpoints, and the one at the center of each pentagon is the largestamong those in the same pentagon.We shall prepare some notations. Definition 4.4.2. For a cofacial n -tree T we define a number m i ( T ) for each i = 0 , . . . , n as follows. m ( T ) (resp. m n ( T )) is the number of d ’s on the root path of the first (resp.last) leaf (counted with multiplicity). In the case 0 < i < n − 1, consider the shortest pathconnecting the i -th and i + 1-th leaves. Starting from the i -th leaf, we count the number of d ’s on the path until we arrive at the ( i, i + 1)-join. Next we count the number of h i ’s onthe join and then we count the number of d ’s from the join to the i + 1-th leaf. (We do notcount d and d on the ( i, i + 1)-join.) m i ( T ) is the total number. (See Figure 4.) ohen-Jones isomorphism Lemma 4.4.3. If T ≤ T ′ in T c ( n ), m i ( T ) = m i ( T ′ ) for each i = 0 , . . . , n . Lemma 4.4.4. For n ≥ 2, the relation ≤ on T c ( n ) is a partial order. Each connectedcomponent of T c ( n ) has the maximum of the following form : ❝ h m · · · h m n − n − ❅❅❅❅ ❏❏❏❏❏ ✡✡✡✡✡(cid:0)(cid:0)(cid:0)(cid:0) ❝ d m ❝ d m n · · · whose underlying tree is the maximum of T ( n ). We denote this cofacial tree by T ( m , . . . , m n ).(A connected component of a poset is an equivalence class of the relation generated by ≤ .) Proof. Clearly ≤ satisfies the transitivity and reflexivity lows. We shall show the anti-symmetry low. Let T and T ′ be two cofacial n -tree with T ≤ T ′ . Let CH( T ) denote thecharacteristic set of the underlying n -tree of T (see Definition 2.3.1). As the set of verticesof T is naturally bijective to the set CH( T ) ∪ { , . . . , n } , we can identify the set of verticesof T ′ with a subset of vertices of T by the inclusion CH( T ) ⊃ CH( T ′ ). The anti-symmetrylow immediately follows from the following observations.1. The multiplicity of d and d at each vertex of T ′ is smaller than or equal to the multi-plicity at the corresponding vertex of T ,2. The multiplicity of h i on T ′ is larger than or equal to that on T for i = 1 , . . . , n − d and d at each incoming edge of each vertex of T ′ is larger than orequal to the multiplicity at each incoming edge of the corresponding vertex of T .For any cofacial n -tree T we can take a cofacial n -tree T ′ such that T ≤ T ′ and the multiplicityof d and d on any vertex of T ′ are zero (using the operation (o-1)). Then we contract allinternal edges of T ′ and obtain an element of a form T ( m , . . . , m n ) with T ′ ≤ T ( m , . . . , m n ).This and Lemma 4.4.3 imply T ( m , . . . , m n ) is the maximum of the connected componentcontaining T (and m i = m i ( T )).We denote by T c ( m , . . . , m n ) the component including T ( m , . . . , m n ). By definition T c (1) is a descrete poset consisting of formal symbols d m d m with m , m ≥ 0. We denotethe one point set { d m d m } by T c ( m , m ).Similarly to T , The collection T c = { T c ( n ) } n ≥ has a structure of an operad in thecategory of posets. Let T ∈ T c ( n ) and T ∈ T c ( m ) be two cofacial trees. The underlyingtree of the composition T ◦ i T is the composition of the underlying tree: U ( T ◦ i T ) = U ( T ) ◦ i U ( T ). Let T ′ be an n -tree with marks obtained from T by replacing h j with h j + m − for each j = i, . . . , n − T ′ be an m -tree with marks obtained from T byreplacing h j with h j + i − for each j = 1 , . . . , m − 1. The marks on the vertex which connects T and T is the union of the marks on i -th leaf of T ′ and on the root of T ′ (with takingmultiplicity into account). The marks on other vertices or edges are equal to the marks onthe corresponding vertices or edges of T ′ or T ′ . For example, ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ h ❜ d ◦ ❅❅❅ (cid:0)(cid:0)(cid:0) ❜ h = ❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❜ dh ❜ h The following property of T c is analogous to the property of T given in Lemma.2.3.2 ohen-Jones isomorphism Lemma 4.4.5. (1) The map ( − ◦ i − ) : T c ( m , . . . , m n ) × T c ( m ′ , . . . , m ′ n ) −→ T c ( n + n − 1) is a monomorphism of sets.(2) For three cofacial trees T , T , and T ′ with T ′ ≤ T ◦ i T ∈ T c ( n ) there exist two elements T ′ and T ′ such that T ′ ≤ T , T ′ ≤ T and T ′ = T ′ ◦ i T ′ .(3) Let n ≥ 2. Any element of T c ( n ) of codimension 1 is a composition of two irreducibleelements. Conversely, a composition of two irreducible elements is of codimension 1. If acofacial tree T is presented as T ◦ i T with T , T irreducible, the data ( T , T , i ) is unique.Here a cofacial tree T is irreducible if T = T ◦ i T implies T = 1 or T = 1.(4) Let n ≥ 2. Any element of T c ( n ) of codimension 2 is a composition of three irreducibleelements. Conversely, a composition of three irreducible elements is of codimension 2. Sup-pose a cofacial tree T is presented as ( S ◦ j S ) ◦ k S with S , S , and S irreducible and j ≤ k . Then(a) if S = S and ar( S ) = ar( S ) = 1 and j = k , there are exactly two choices of the data( S , S , S , j, k ) and if we take one, the other is ( S , S , S , j, k ),(b) if S = S and ar( S ) = ar( S ) = 1, there are exactly two choices of the data( S , S , S , j, k ) and if we take one, the other is ( S , S , S , j, k ),(c) otherwise the data ( S , S , S , j, k ) is unique.(5) If T and T ∈ T c ( n ) are two different elements of codimension one such that h T i∩h T i 6 = ∅ , there exists a (unique) T of codimension 2 such that h T i ∩ h T i = h T i . Proof. (1) is clear. (2) We consider the case of ar( T ) ≥ 2. The case ar( T ) = 1 is similarand easier. We put ar( T ) = n . Both of T ◦ i T and T ′ have the ( i, i + n − T ′ ≤ T ◦ i T do not shift marks through the( i, i + n − µ , ˇ µ , and m ′ l (0 ≤ l ≤ n ) by1. ˆ µ (resp. ˇ µ ) being the multiplicity of d (resp. d ) on the ( i, i + n − m ′ (resp. m ′ n ) = the number of d ’s (resp d ’s) on the path in T ′ from the i -th (resp. i + n − i, i + n − ≤ l ≤ n − m ′ l = m i + l ( T ′ ),then we have m ′ ≥ m ( T ) ≥ m ′ − ˆ µ, m ′ n ≥ m n ( T ) ≥ m ′ n − ˇ µ, m ′ i = m i ( T )(1 ≤ i ≤ n ) . So we can take T ′ and T ′ such that T ′ = T ′ ◦ i T ′ and m i ( T ′ ) = m i ( T ), m i ( T ′ ) = m i ( T ).The operations which realize T ′ ≤ T ◦ i T can be separated to operations on T ′ and T ′ andthese two realize T ′ ≤ T and T ′ ≤ T . (3) and (4) are clear.We shall prove (5). Let T and T ′ be two cofacial n -trees of codimension 1. Suppose T = T ◦ i T and T ′ = T ′ ◦ i ′ T ′ such that T k and T ′ k are irreducible for k = 1 , 2. Forsimplicity, we assume n = ar( T ) ≥ n ′ = ar( T ′ ) ≥ i ≤ i ′ . By the condition h T i ∩ h T ′ i 6 = ∅ , one of the following two cases occurs. (i) i ′ + n ′ − ≤ i + n − 1, (ii) i + n − ≤ i ′ . In the former case, the same condition andLemma 4.4.3 imply m i ′ − i ( T ) ≥ m ( T ′ ) , m i ′ − i + l ( T ) = m l ( T ′ ) (1 ≤ l ≤ n ′ − ,m i ′ − i + n ′ ( T ) ≥ m n ′ ( T ′ ) . By these inequalities, we can put T ′′ := T ( m ( T ) , . . . , m i ′ − i ( T ) − m ( T ′ ) , m i ′ − i + n ′ ( T ) − m n ′ ( T ′ ) , . . . , m n ( T )) ∈ T c ( n − n ′ +1) , and ( T ◦ i T ′′ ) ◦ i ′ T ′ is the element of codimension 2 such that h T i ∩ h T ′ i = h ( T ◦ i T ′′ ) ◦ i ′ T ′ i .The proof of the latter case (ii) is similar. ohen-Jones isomorphism T c d ji : T c ( m , . . . , m i , . . . , m n ) −→ T c ( m , . . . , m i + 1 , . . . , m n ) ,s ji : T c ( m , . . . , m i , . . . , m n ) −→ T c ( m , . . . , m i − , . . . , m n )which will be used to define the monad CK . ( i = 0 , . . . , n . For d ji , j = 0 , . . . , m if i = 0,and j = 1 , . . . , m n + 1 if i = n , otherwise j = 1 , . . . , m i . For s ji , j = 0 , . . . , m i − 1. )We first define d ji ’s. We first consider the case of i = 0. In the case of j = 0, d is themap which increases the multiplicity of d on the root by one (and does not change the othermarks). In the case of j > 0, consider the root path of the first leaf. We count the numberof d on the path, starting from the root (taking the multiplicity into account). We define d j as the map which increases the multiplicity of d on the vertex or edge at which j -th d is, byone.In the case 0 < i < n − 1, We count the number of marks on the path connecting the i -thand i + 1-th leaves as in Definition 4.4.2 (see also Figure 4). ¯ d ji is the map which increasesthe multiplicity of the j -th mark by one.In the case i = n , consider the root path the last leaf. We count the number of d on thepath from the last leaf. For 1 ≤ j ≤ m n , d ji is the map which increases the multiplicity ofthe j -th d . For j = m n + 1, d is the map which increases the multiplicity of d on the root.For each j = 0 , . . . , n , To define s ji , we use the path used in defining d ji . s ji is the mapwhich decreases the j + 1-th mark. Example 4.4.6. Let T be an element of T c (1 , , 1) as follows. ❜ dh ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) d ( T ) , d ( T ) , d ( T ) , d ( T ) , d ( T ) , d ( T ) are equal to ❜ ddh ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ dh ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ dh ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ dh ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ d h ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ d h ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) ,respectively. s ( T ) , s ( T ) , s ( T ) , s ( T ) are equal to ❜ dh ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ dh ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ d ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) , ❜ h ❜ d ❜ d ❅❅❅ (cid:0)(cid:0)(cid:0) .The proof of the following lemma is routine. Lemma 4.4.7. The maps s ji and d ji satisfy the following identities: d ji d ki = d ki d j − i ( k < j ) s ji d ki = d ki s j − i ( k < j )= id ( k = j, j + 1)= d k − i s ji ( k > j + 1) s ji s ki = s ki s ji ( k > j ) d ji d kl = d kl d ji , s ji d kl = d kl s ji , s ji s kl = s kl s ji ( i = l ) . (Note that the first five identities have the same form as the cosimplicial identities.) ohen-Jones isomorphism ❅❅❅❅❅❅❅❅■ d j m ( T ) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠ d j m ( T ) ❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)❅❅❅❅■ d j m ( T ) ❅❅❅(cid:0)(cid:0)(cid:0)✒ d j m ( T ) ❅❅❅❅❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅ Figure 4: paths used to define m i ( T ) and { d ji } CK Definition 4.4.8. We define a topological operad CK by CK ( n ) = | T c ( n ) | with the inducedcomposition. Let CK ( m , . . . , m n ) denote the connected component of CK ( n ) correspondingto T c ( m , . . . , m n ) and d ji and s ji denote the maps on CK ( m , . . . , m n ) induced by d ji and s ji defined in sub-subsection 4.4.2 via the realization functor. U : CK → K denotes themorphism of operads induced by the morphism : T c → T forgetting marks.We shall give a description of CK analogous to the description of K given in subsection2.3. Let T c ( m , . . . , m n ) , be the subposet of T c ( m , . . . , m n ) consisting of elements ofcodimension one or two. We define a diagram B c = B cm ,...,m n : T c ( m , . . . , m n ) , −→ CG as follows (compare with the definition of B n in subsection 2.3). As an element of codimensionone is uniquely presented as T ◦ i T by Lemma 4.4.5, we put B c ( T ◦ i T ) = CK ( T ) ×CK ( T ).An element of codimension two is of the form ( S ◦ j S ) ◦ k S . If this presentation is not unique,we fix a presentation once and for all, and put B c (( S ◦ j S ) ◦ k S ) = CK ( S ) ×CK ( S ) ×CK ( S ).Suppose ( S ◦ j S ) ◦ k S ≤ T ◦ i T . For example, in the case (a) of Lemma 4.4.5,(4), by (2)of the same lemma, one of the following cases occurs.1. S ◦ j S ≤ T , S = T ,2. S ◦ j S ≤ T , S = T .In each case, we define a map B c (( S ◦ j S ) ◦ k S ) → B c ( T ◦ i T ) using the composition of CK .In the other cases (b),(c) of Lemma 4.4.5, we define a map B c (( S ◦ j S ) ◦ k S ) → B c ( T ◦ i T )similarly.A natural transformation B c −→ CK ( m , . . . , m n ) is defined using the composition of CK .Here, CK ( m , . . . , m n ) denotes the corresponding constant diagram. Thus we obtain a map θ c : colim B c −→ CK ( m , . . . , m n ) . The image of θ c is ∂ CK ( m , . . . , m n ), the subcomplexspanned by all cofacial n -trees of codimension one. A point of CK ( m , . . . , m n ) can bepresented as t T + · · · t k T k with T < · · · < T k ∈ T c ( m , . . . , m n ) and t , . . . , t k ≥ t + · · · + t k = 1, . Using this presentation, we define a map˜ θ c : Cone (colim B c ) −→ CK ( m , . . . , m n )by ˜ θ c ( t · u ) = tθ c ( u ) + (1 − t ) T ( m , . . . , m n ). Proposition 4.4.9. With the above notations, the maps θ c : colim B c → ∂ CK ( m , . . . , m n )and ˜ θ c : Cone (colim B c ) → CK ( m , . . . , m n ) are homeomorphisms. Proof. In view of Lemma.4.4.5, the proof is completely analogous to the proof of Proposition2.3.3. ohen-Jones isomorphism CK Let C ∆ be the category of cosimplicial objects over C (see subsection 1.1). In this sub-subsection, we define a monad CK : C ∆ −→ C ∆ . We first define a functor CK n : ( C ∆ ) × n −→ C ∆ . Let X , . . . , X n be objects of C ∆ . The objectof cosimplicial degree l of CK n ( X , . . . , X n ) is defined as follows. CK n ( X , . . . , X n ) l = G CK ( m , . . . , m n ) ˆ ⊗ X p ⊗ · · · ⊗ X p n n / ∼ . Here, the sequence of non-negative integers m , . . . , m n , p , . . . , p n ranges over sequencessatisfying m + · · · + m n + p + · · · + p n = l , and in the point-set expression the equivalencerelation ∼ is generated by the following relations( u ◦ i d ; x , · · · , x i , · · · , x n ) ∼ ( u ; x , · · · , d x i , · · · , x n ) , ( u ◦ i d ; x , · · · , x i , · · · , x n ) ∼ ( u ; x , · · · , d p i +1 x i , · · · , x n )where u ∈ CK ( m , . . . , m n ) , x j ∈ X p j j , and 1 ≤ i ≤ n , and d and d denote the elements d d and d d of CK (1) respectively.We define the coface and codegeneracy morphisms on {CK n ( X , . . . , X n ) l } l by using thecorresponding maps of CK and X i alternately. For example, for 0 ≤ j ≤ m , d j is definedusing d j of CK , and for m + 1 ≤ j ≤ m + p , d j is defined using d j − m of X , and for m + p + 1 ≤ j ≤ m + p + m , d j is defined using d j − m − p of CK . Precisely speaking, weput d j ( u ; x , · · · , x n )= ( d j u ; x , . . . , x n ) ( j = 0)( d j i u ; x , . . . , x n ) ( m ≤ i − + p ≤ i + 1 ≤ j ≤ m ≤ i + p ≤ i , j = j − m ≤ i − − p ≤ i )( u ; x , . . . , d j x i , . . . , x n ) ( m ≤ i − + p ≤ i − + 1 ≤ j ≤ m ≤ i − + p ≤ i , j = j − m ≤ i − − p ≤ i − )( d m n +1 n u ; x , . . . , x n ) ( j = m ≤ n + p ≤ n + 1) ,s j ( u ; x , · · · , x n )= (cid:26) ( s j i u ; x , . . . , x n ) ( m ≤ i − + p ≤ i ≤ j ≤ m ≤ i + p ≤ i − , j = j − m ≤ i − − p ≤ i )( u ; x , . . . , s j x i , . . . , x n ) ( m ≤ i + p ≤ i ≤ j ≤ m ≤ i + p ≤ i +1 − , j = j − m ≤ i − p ≤ i )where p i = | x i | , u ∈ CK ( m , . . . , m n ), 0 ≤ i ≤ n and m ≤ t = m + · · · + m t and p ≤ t is similar,and m ≤− = p ≤− = p ≤ = 0. The following lemma easily follows from Lemma 4.4.7 andthe definition of the equivalence relation ∼ . Lemma 4.4.10. The morphisms d j and s j defined above are well defined and satisfy thecosimplicial identity.Now, we put CK ( X ) = G n ≥ CK n ( X, . . . , X ) . A unit id −→ CK and a product CK ◦ CK −→ CK are defined by using the unit and thecomposition of CK . By definition, these natural transformations are compatible with cofacesand codegeneracies and give a well-defined structure of a monad on the functor CK . Definition 4.4.11. For objects X , . . . , X n of C ∆ , A morphism U n,X ,...,X n : CK n ( X , . . . , X n ) →K ( n ) ˆ ⊗ X (cid:3) · · · (cid:3) X n is defined by U n ( u ; x , . . . , x n ) = ( U ( u ) , d m x , d m x , . . . , d m n − d m n x n ) . , see Definition 4.4.8 and d and d denote the first and last cofaces. It is easy to see this mor-phism actually commutes with the cosimplicial operators. The collection { U n,X ,...,X n } X ,...,X n defines a natural transformation U n : CK n ⇒ K ( n ) ˆ ⊗ ( − ) (cid:3) n . ohen-Jones isomorphism Lemma 4.4.12. The natural transformation U = ⊔ n ≥ U n : CK = ⇒ e K : C ∆ → C ∆ is a morphism of monads. (See sub-subsection 2.4.1 for e K .) Proof. The commutativity of the morphism U with the products follows from the fact that U : CK → CK is a morphism of operads and the following cosimplicial identities: d i d m x = d m +1 x, d | x | + i +1 d m x = d m +1 x ( m ≥ , ≤ i ≤ m ). A ∞ -operad CB In this sub-subsection, we define an A ∞ -operad CB which acts on g Tot of an CK -algebra. Wedefine a morphism of cosimplicial objects¯ U n : CK n ( X , . . . , X n ) −→ X (cid:3) · · · (cid:3) X n as the composition of U n and the obvious morphism K ( n ) ˆ ⊗ X (cid:3) · · · (cid:3) X n → X (cid:3) · · · (cid:3) X n . Lemma 4.4.13. Suppose C = CG . Let X , . . . , X n be Reedy cofibrant objects in C ∆ . Thenthe map ¯ U n : CK n ( X , . . . , X n ) −→ X (cid:3) · · · (cid:3) X n is a weak homotopy equivalence. Proof. We first define a poset ˜ (cid:3) ln . An object of ˜ (cid:3) ln is a sequence of non-negative integers( m , . . . , m n , p , . . . , p n ) such that m + · · · + m n + p + · · · + p n = l . The partial order ≤ isgenerated by( m , . . . , m n , p , . . . , p n ) < ( m , . . . , m i − , . . . m n , p . . . p i + 1 . . . , p n ) (1 ≤ i ≤ n )( m , . . . , m n , p , . . . , p n ) < ( m , . . . , m i − , . . . m n , p . . . p i +1 + 1 . . . , p n ) (0 ≤ i ≤ n − F : ˜ (cid:3) ln −→ CG . For each object ( m , . . . , m n , p , . . . , p n ) ∈ ˜ (cid:3) ln we put F ( m , . . . , m n , p , . . . , p n ) = X p ×· · ·× X p n n . We associate the map id × i − × d × id × n − i − to the inequality ( m , . . . , m n , p , . . . , p n ) < ( m , . . . , m i − , . . . , m n , p . . . p i + 1 . . . , p n ) (1 ≤ i ≤ n ) and the map id × i × d × id × n − i − tothe other generating inequality. It is easy to see there is a natural isomorphism colim ˜ (cid:3) ln F ∼ =( X (cid:3) · · · (cid:3) X n ) l .Fix a non-negative integer l and let l , l be two integers such that 0 ≤ l ≤ l ≤ l .˜ (cid:3) ( l , l ) denote the subposet of ˜ (cid:3) ln consisting of objects ( m , . . . , m n , p , . . . , p n ) such that l ≤ P ni =1 p i ≤ l . Let l ′ be an integer with l ′ ≤ l and put ˜ (cid:3) ( l ′ ) = ˜ (cid:3) ( l ′ , l ′ ) for simplicity.The following diagram ( P ) is a pushout diagram. F ( m ,...,p n ) ∈ ˜ (cid:3) ( l ′ ) (colim ˜ (cid:3) ( l ′ − ,l ′ − / ( m ,...,p n ) F ) / / (cid:15) (cid:15) F ( m ,...,p n ) ∈ ˜ (cid:3) ( l ′ ) F ( m , . . . , p n ) (cid:15) (cid:15) colim ˜ (cid:3) (0 ,l ′ − F / / colim ˜ (cid:3) (0 ,l ′ ) F , where ˜ (cid:3) ( l ′ − , l ′ − / ( m , . . . , p n ) denotes the subposet of elements smaller than ( m , . . . , m n , p , . . . , p n ),and the all arrows are induced by inclusions of posets. We shall define a similar pushout di-agram for CK n ( X , . . . , X n ). Put CK n (0 , l ′ ) = G ( m ,...,p n ) ∈ ˜ (cid:3) (0 ,l ′ ) CK ( m , . . . , m n ) × X p × · · · × X p n n / ∼ . ohen-Jones isomorphism ∼ is the equivalence relation used to define CK n ( X , . . . , X n ).) Then, there exists a pushoutdiagram ( P ) as follows: F ( m ,...,p n ) ∈ ˜ (cid:3) ( l ′ ) ( CK ′ × colim ˜ (cid:3) ( l ′ − ,l ′ − / ( m ,...,p n ) F ) / / (cid:15) (cid:15) F ( m ,...,p n ) ∈ ˜ (cid:3) ( l ′ ) CK ′ × F ( m , . . . , p n ) (cid:15) (cid:15) CK n (0 , l ′ − / / CK n (0 , l ′ ) , where CK ′ denotes CK ( m , . . . , m n ), and the left vertical map is induced by the inclusions CK ( m , . . . , m n ) × F ( m , . . . , m i + 1 , . . . , p i − , . . . , p n ) ∼ = ( CK ( m , . . . , m n ) ◦ i d ) × F ( m , . . . , m i + 1 , . . . , p i − , . . . , p n ) ⊂ CK ( m , . . . m i + 1 , . . . , m n ) × F ( m , . . . , m i + 1 , . . . , p i − , . . . , p n ) (1 ≤ i ≤ n ) , CK ( m , . . . , m n ) × F ( m , . . . , m i + 1 , . . . , p i +1 − , . . . , p n ) ∼ = ( CK ( m , . . . , m n ) ◦ i +1 d ) × F ( m , . . . , m i + 1 , . . . , p i +1 − , . . . , p n ) ⊂ CK ( m , . . . , m i + 1 , . . . , m n ) × F ( m , . . . , m i + 1 , . . . , p i +1 − . . . , p n ) (0 ≤ i ≤ n − , and the other arrows are defined similarly to the corresponding arrows of ( P ). We de-fine a map of diagram P → P similarly to ¯ U n . The top horizontal arrow of ( P ) :colim ˜ (cid:3) ( l ′ − ,l ′ − / ( m ,...,p n ) F → F ( m , . . . , p n ) is a cofibration because each X i is Reedy cofi-brant and colim ˜ (cid:3) ( l ′ − ,l ′ − / ( m ,...,p n ) F is the latching object modeled by ∪ i ∆ p × · · · × (∆ p i ∪ ∆ p i p i +1 ) × · · · × ∆ p n (∆ pk denotes the k -th face of ∆ p ). This and the fact that CK ′ is a cellcomplex imply ( P ) and ( P ) are homotopy pushout diagrams. As CK ′ is contractible byLemma 4.4.4, the map P → P induces a weak equivalence between the resulting pushouts.So by induction on l ′ , we have proved the assertion.Now we define a topological operad CB . Let ˜∆ • be a projective cofibrant replacement ofthe cosimplicial space ∆ • (see subsection 1.1). We put CB ( n ) = Map CG ∆ ( ˜∆ • , CK n ( ˜∆ • , . . . , ˜∆ • ))From the monad structure of CK , we define an operad structure on CB ( n ) by the way exactlyanalogous to the definition of e B or e B ′ given in subsection 2.4. The morphism U : CK → e K inLemma 4.4.12 induces a morphism of operads U : CB −→ e B ′ . Theorem 4.4.14. (1) U is weak equivalence of topological operads. In particular, CB is an A ∞ -operad.(2) g Tot induces a functor ALG CK ( C ∆ ) −→ ALG CB ( C ) which satisfy the following condi-tion: Let X • be a e K -algebra and Y • be a CK -algebra and let α : Y • → X • ∈ C ∆ be a mor-phism compatible with U : CK → e K . Then the induced morphism g Tot α : g Tot Y • → g Tot X • iscompatible with the morphism U : CB → e B ′ . Here, g Tot X • is considered as a e B ′ -algebra byProposition 2.4.8. Proof. (1) follows from Lemma.4.4.13 and the standard fact that mapping space betweena projective cofibrant cosimplicial space and a termwise fibrant cosimplicial space is weakhomotopy invariant. (Note that a similar proof of a similar statement where ˜∆ • is replacedwith ∆ • does not work as CK n (∆ • , . . . , ∆ • ) may not be Reedy fibrant). The proof of (2) issimilar to Proposition 2.4.7. ohen-Jones isomorphism CK on IM • In this subsection we prove the following theorem: Theorem 4.5.1. There exists an action of the monad CK on IM • such that the both mor-phisms p : IM • → L • and ¯ q : IM • → THC • ( A ′ , B ) defined in Definition 4.3.1 are compatiblewith the morphism U : CK → e K . Here, THC • ( A ′ , B ) are regared as e K -algebra as explainedbelow Lemma 2.5.2. Hence p and ¯ q induce the morphisms g Tot( L • ) ( p ) ∗ ←−−− g Tot( IM • ) (¯ q ) ∗ −−−→ g Tot(THC • ( A ′ , B ))which is compatible with the morphism of A ∞ -operads U : CB → e B ′ by Theorem 4.4.14.In subsection 4.7, we prove the two morphisms between g Tot in the above theorem arestable equivalences. According to the natural decomposition L n × T n ( T n ) [0 , = L n × T n ( T n ) [0 , × T n ( T n ) [1 , , wedecompose IM • into two parts, IM • and f IM • so that IM • ∼ = IM • × T f IM • . More explicitly IM n is a subspectrum of L n × T n ( T n ) [0 , consisting of elements which satisfythe the same conditions as IM • satisfies (see sub-subsection 4.3.2) and f IM • = ( T • ) [1 , . Wewill define three actions of monads as follows:1. an action of CK on IM • , Ω : CK ( IM • ) −→ IM • ,2. an action of e K on T • , b Υ : e K ( T • ) −→ T • ,3. an action of e K on f IM • , e Υ : e K ( f IM • ) −→ f IM • .We shall fix notations. T • regarded as a e K -algebra as explained below Lemma 2.5.2 is denotedby the same notation T • while we denote by b T • the e K -algebra ( T , b Υ).The above three actions satisfy the following conditions:1. p : IM • → L • and p : IM • → b T • are compatible with the morphism CK → e K ,2. q : f IM • → b T • and q : f IM • → T • are compatible with the actions of e K .Here, p t and q t denote the evaluations at t ∈ [0 , CK on IM • is given by CK ( IM • ) → CK ( IM • ) × CK ( b T • ) CK ( f IM • ) → CK ( IM • ) × e K ( b T • ) e K ( f IM • ) → IM • × b T • f IM • ∼ = IM • , where the first morphism is induced by the universal property of fiber products, the secondby the morphism U : CK → e K , and the third by the three actions Ω, b Υ and e Υ. As themorphism ϕ ∗ φ ∗ defined in Definition 4.3.1 preserves (cid:3) -monoid structures, by the above twoconditions, it is obvious that the action of CK on IM • satisfies the condition of Theorem4.5.1. Thus the proof of the theorem is reduced to the construction of Ω, b Υ, and e Υ.The construction of Ω is lengthy and occupies most of this subsection (sub-subsections4.5.2, 4.5.3, and 4.5.4) and the construction of b Υ and e Υ is very short (sub-subsection 4.5.5). ohen-Jones isomorphism ΩWe put e F k ,...,k n ( m , . . . , m n )(= e F ( m , . . . , m n )):= ( U × id) − D nk ,...,k n ⊂ CK ( m , . . . , m n ) × M [ k , . . . , k n ], where n ≥ m , . . . , m n ≥ 0, and U : CK ( m , . . . , m n ) → K ( n ) is the forgetful morphismand see sub-subsection 3.1.1 for D nk ,...,k n . Put I = [0 , CK on IM • , we will define the following three kind of data : • maps e ψ = ( e ψ i ) : D n × I −→ Map( M, M ) n ( i = 1 , . . . , n ), • maps ω = ( ω ji ) : e F ( m , . . . , m n ) × I −→ Map( M, M ) m + ··· + m n ( i = 0 , . . . , n , j =1 , . . . , m i ) • maps z = ( z ji ) : e F ( m , . . . , m n ) −→ M m + ··· + m n ( i = 0 , . . . , n , j = 1 , . . . , m i )Similarly to the construction of the A ∞ -action on LM − τ in subsection 3.1, we first givethe formula for the action Ω in this sub-subsection, then state conditions on these mapswhich ensure well-definedness of Ω in next sub-subsection, and finally construct them insub-subsection 4.5.4.Using e ψ , ω , and z , the structure map Ω : CK ( IM • ) −→ IM • is defined as follows. For u ∈ CK ( m , . . . , m n ), ( λ i , h i ) ∈ IM p i k i ,(˜ λ, ˜ h ) = Ω( u ; ( λ , h ) , . . . , ( λ n , h n )) ∈ IM m + ··· + m n + p + ··· + p n k + ··· + k n is defined as follows. We put ˜ λ = Ψ( U ( u ); λ , λ , . . . , λ n ) (see subsection 4.2 for Ψ). Let( f , . . . , f m + ··· + m n + p + ··· + p n ) be a seqence of elements of F ( M ). To write down ˜ h t ( f , . . . , f m + ··· + p n )we rename this sequence as( g , . . . , g m , f , . . . , f p , g , . . . , g m , . . . , f n , . . . , f p n n , g n , . . . , g m n n ) . If ( u ; λ , . . . , λ n ) can not be represented by an element of e F ( m , . . . , m n ), (˜ λ, ˜ h ) is the basepoint. Suppose ( u ; λ , . . . , λ n ) is represented by d ∈ e F ( m , . . . , m n ) (such d is unique by thecondition ( ǫ -4) in sub-subsection 3.1.2. We abbreviate e ψ i,s ( U × id( d )), ω ji,s ( d ), and z ji ( d ) as e ψ i,s , ω ji,s , and z ji respectively in the following formulae ( s ∈ I ). We put¯ g i,s = ( ω i,s ◦ H z i ,s ) ∗ g i · · · ( ω m i i,s ◦ H z mii ,s ) ∗ g m i i (0 ≤ i ≤ n, s ∈ I ) , ¯ h i,s = h i,s (( e ψ i,s ) ∗ f i , . . . , ( e ψ i,s ) ∗ f p i i ) (1 ≤ i ≤ n, s ∈ I ) . Here H : M × I → M is the homotopy fixed in sub-subsection 4.3.2. Then, we put˜ h s ( f , . . . , f m + ··· + p n ) = σ − · (¯ g ,s · ¯ h ,s · ¯ g ,s · · · ¯ h n,s · ¯ g n,s ) . (3)Here σ is the permutation corresponding to the transposition h , . . . , h n , g , f , . . . f n , g n g , h , f , g , h , f , . . . h n , f n , g n (for example, g i represents the sequence g i , . . . , g m i i . f i is similar abbreviation.) More ex-plicity, σ ( i ) = i + P l ≤ k − lev g l + lev f l if P l ≤ k − lev h l + 1 ≤ i ≤ P l ≤ k lev h l for some k ,and otherwise σ ( i ) = i − P k +1 ≤ l ≤ n lev h l . Here lev denotes the level.Note that the formula (3) is somewhat analogous to the unsuccessful definition of anaction of e K on IM • given in sub-subsection 4.4.1. ohen-Jones isomorphism e ψ , z , and ω satisfyConditions on e ψ . e ψ satisfies the following six conditions ( e ψ -1) - ( e ψ -6) for any n ≥ u ; v , . . . , v n ) ∈ D n and s ∈ I . (These conditions are much analogous to those on ψ .)( e ψ -1) e ψ s (1 , v ) = id M ( i is the unit of K ).( e ψ -2) For a number t ∈ R if ( u ; v , . . . , t × v i , . . . , v n ) ∈ D n , then e ψ s ( u ; . . . , v i , . . . ) = e ψ s ( u ; . . . , t × v i , . . . ). Here t × v i denotes the element (0 × φ i , ǫ i , ( t, v i )).( e ψ -3) For σ i ∈ Σ k i , e ψ s ( u ; σ · v , . . . , σ n · v n ) = e ψ s ( u ; v , . . . , v n ).( e ψ -4) e ψ i ( u ; v , . . . , v n ) = ψ i ( u ; v , . . . , v n ) and e ψ i ( u ; v , . . . , v n ) = id M for each i .( e ψ -5) The following diagram commutes for each n, m , and i : β − i D n + m − k ,...,k n + m − × I τ ◦ ( α i × γ i × ∆) / / β i (cid:15) (cid:15) ( D n...,k i + ··· + k i + m − ,..., × I ) × ( D mk i ,...,k i + m − × I ) e ψ × e ψ (cid:15) (cid:15) Map( M, M ) n × Map( M, M ) mComp i (cid:15) (cid:15) D n + m − k ,...,k n + m − × I e ψ / / M ap ( M, M ) n + m − Here, ∆ : I → I × I denotes the diagonal and τ denotes the transposition D ×D × I × ∼ = D × I × D × I .( e ψ -6) | e ψ s ( u ; v , . . . , v n )( y ) − y | ≤ n d (( v , . . . , v n ) , φ ( M )) where φ = φ × · · · × φ n . Conditions on z . z satisfies the following conditions for any n ≥ m , . . . , m n ≥ u ; v , . . . , v n ) ∈ D n and s ∈ I .( z -1) For t ∈ R if ( u ; . . . , t × v i , . . . ) ∈ e F ( m , . . . , m n ), then z ( u ; . . . , t × v i , . . . ) = z ( u ; v , . . . , v n ).( z -2) For ( u ; v , . . . , v n ) ∈ e F ( m , . . . , m n ) and permutations σ , . . . , σ n , z ( u ; σ v , . . . , σ n v n ) = z ( u ; v , . . . , v n ).( z -3) The following diagram is commutative for each n , n , i , m , . . . , m n , m ′ , . . . , m ′ n : β − i e F ( . . . , m i − + m ′ , . . . , m ′ n + m i , . . . ) α i × γ i / / β i (cid:15) (cid:15) e F ( m , . . . , m n ) × e F ( m ′ , . . . , m ′ n ) T ◦ ( z × z ) (cid:15) (cid:15) e F ( m , . . . , m i − + m ′ , . . . , m ′ n + m i , . . . ) z / / M m + ··· + m n + m ′ + ··· + m ′ n T is the transposition given by T ( x , . . . , x m , . . . , x m n n , y , . . . , y m ′ n n )= ( x , . . . , x m i i , y , . . . , y m ′ n n , x i +1 , . . . , x m n n ).( z -4) The following diagrams are commutative for each i , j . e F ( · · · m i · · · ) d ji / / z (cid:15) (cid:15) e F ( · · · m i + 1 · · · ) z (cid:15) (cid:15) M m + ··· + m n ¯ δ ji / / M m + ··· + m n +1 e F ( · · · m i · · · ) s ji / / z (cid:15) (cid:15) e F ( · · · m i − · · · ) z (cid:15) (cid:15) M m + ··· + m n ¯ σ ji / / M m + ··· + m n − ohen-Jones isomorphism δ ji is the map repeating P i − k =0 m k + j -th component if ( i, j ) = (0 , , ( n, m n + 1),and ¯ δ (resp. ¯ δ m n +1 n ) is the map putting π φ ×···× φ n ( v , . . . , v n ) at the first (resp. thelast) component, and ¯ σ ji is the map skipping the P i − k =0 m k + j + 1-th component.( z -5) | v − φ ( z ji ( u )) | ≤ n | φ | · d ( v, φ ( M )) for each i , j , where v = ( v , . . . , v n ) and φ = φ × · · · × φ n . Conditions on ω . ω satisfies the following conditions for any n ≥ m , . . . , m n ≥ u ; v , . . . , v n ) ∈ D n and s ∈ I .( ω -1) For t ∈ R , if ( u ; . . . , t × v i , . . . ; s ) ∈ e F ( m , . . . , m n ) × I , then ω s ( u ; . . . , t × v i , . . . ) = ω s ( u ; . . . , v i . . . ).( ω -2) For permutations σ , . . . , σ n , ω s ( u ; σ v , . . . , σ n v n ) = ω s ( u ; v , . . . , v n ).( ω -3) The following diagram is commutative for each n , n , i , m , . . . , m n , m ′ , . . . , m ′ n : β − i e F ( ˜ m k ) n + n − k =0 × I τ ◦ ( α i × γ i × ∆) / / β i (cid:15) (cid:15) ( e F ( m k ) n k =0 × I ) × ( e F ( m ′ k ) n k =0 × I ) ω × ( e ψ ( − ) ◦ ω ) (cid:15) (cid:15) Map( M, M ) m + ··· + m n × Map( M, M ) m ′ + ··· + m ′ n T (cid:15) (cid:15) e F ( ˜ m k ) n + n − k =0 ω / / M ap ( M, M ) m + ··· + m n + m ′ + ··· + m ′ n Here, ( ˜ m k ) n + n − k =0 = ( m , . . . , m i − + m ′ , . . . , m ′ n + m i , . . . m n ), and for x ∈ e F ( m j ) n j =0 × I and y ∈ e F ( m ′ j ) n j =0 × I , ω × ( e ψ ( − ) ◦ ω )( x, y ) = ω ( x ) × ( e ψ i ( x ) ◦ ω ( y ) , . . . , e ψ i ( x ) ◦ ω m ′ n n ( y )). τ and T are the transpositions defined by the same formula as in ( e ψ -5) and ( z -3) above.( ω -4) The following diagrams are commutative for each ( i, j ): e F ( m k ) nk =0 × I d ji / / ω (cid:15) (cid:15) e F ( · · · m i + 1 · · · ) × I ω (cid:15) (cid:15) Map( M, M ) × m ≤ n δ ji / / Map( M, M ) × m ≤ n +1 e F ( m k ) nk =0 × I s ji / / ω (cid:15) (cid:15) e F ( · · · m i + 1 · · · ) × I ω (cid:15) (cid:15) Map( M, M ) × m ≤ n σ ji / / Map( M, M ) × m ≤ n +1 Here, δ ji is the map repeating the P i − k =0 m k + j -th component if ( i, j ) = (0 , , ( n, m n +1), δ (resp. δ m n +1 n ) is the map putting id M on the first component (resp. the lastcomponent). σ ji is the map deleting the P i − k =0 m k + j + 1-th component.( ω -5) For each ( i, j ), the equalities ω ji, ( u ; v , . . . , v n )( z ji ( u ; v , . . . , v n )) = π φ ×···× φ n ( v , . . . , v n ),and ω ji, ( u ; v , . . . , v n ) = id M hold.Assuming the above conditions, we shall show Proposition 4.5.2. Assuming the conditions ( e ψ -1), ( e ψ -2), ( e ψ -3), ( e ψ -5), ( e ψ -4), ( z -1), ( z -2),( z -3), ( z -4), ( ω -1), ( ω -2), ( ω -3), ( ω -4), ( ω -5), the definition of the action of CK on IM • givenby the formula (3) in sub-subsection 4.5.2 is well-defined. Proof. We must show the mapΩ : CK ( m , . . . , m n ) ⊗ IM p ⊗ · · · ⊗ IM p n −→ IM m + ··· + m n + p + ··· + p n . ohen-Jones isomorphism u ∈ CK ( m , . . . , m n ) and ( λ i , h i ) ∈ IM p i k i where λ i ∈ L p i k i and h i ∈ ( T I ) p i k i . Put (˜ λ, ˜ h ) = Ω( u ; ( λ , h ) , . . . , ( λ n , h n )). We beginby proving (˜ λ, ˜ h ) belongs to the pullback L • × T • ( T • ) I . Suppose ( U ( u ) , λ , . . . , λ n ) is repre-sented by an element d of D n as otherwise (˜ λ, ˜ h ) is the base point. Let y ∈ M be a pointsatisfying | v − φ ( y ) | ≤ e ǫ . An easy calculation shows that | z ji ( d ) − y | ≤ L e / ω -5) imply ω ji, ( d ) ◦ H z ji ( d ) , ( y ) = π φ ( v ). So for a sequence( g , . . . , g m , f , . . . f p , . . . ) of elements of F ( M ), we have p (˜ h )( g , f , g , . . . , g n )( y ) = σ − ( g ( π φ ( v )) · h , ( ψ ∗ f ) · · · h n, ( ψ ∗ n f n ) · g n ( π φ ( v )))( y ) . Here, we use abbreviations in the formulae in sub-subsection 4.5.2, and g i ( π φ ( v )) means g i ( π φ ( v )) · · · g m i i ( π φ ( v )) ∈ S . For y with | v − φ ( y ) | > e ǫ , the above equality still holds as theboth sides are the base points by the definition of IM • . By straightforward calculation, theleft hand side is seen to be equal to e ρ (˜ λ )( y ) and we have verified the assertion. It is obviousthat (˜ λ, ˜ h ) actually belongs to the sub-spectrum IM • ⊂ L × T T I . The equivariance withspheres and symmetric groups follows from conditions ( e ψ -2), ( z -1), ( ω -1), ( e ψ -3), ( z -2), ( ω -2).We shall check Ω( u ; . . . d ( λ i , h i ) , . . . ) = Ω( u ◦ i d ; . . . , ( λ i , h i ) , . . . ). Unwinding the formulaof Ω, we see it is enough to show H ∗ π φi ( v i ) e ψ i,s ( u ) ∗ f = ( ω m i − +1 i − ( u ◦ i d ) ◦ H z mi − i − ( u ◦ i d ) ) ∗ f. (In view of the formula, f is presented as f i in the left hand side and as g m i − +1 i − in the righthand side.) By the conditions ( ω -3), ( ω -4), ( z -3), and ( z -4), we have ω m i − +1 i − ( u ◦ i d ) = e ψ i,s ( u ) ◦ id , z m i − +1 i − ( u ◦ i d ) = π φ i ( v i ) . So the assertion holds. Similarly we see the corresponding equality holds for d . Thus, wehave checked that Ω induces a morphism of symmetric spectra CK ( IM ) p −→ IM p . Thecommutativity between Ω and cofaces and codegeneracies follows from the conditions ( z -4)and ( ω -4) obviously.Finally, we shall show commutativity with the monad product CK ◦ CK −→ CK . it isenough to showΩ( u ◦ i u ; ( λ , h ) , . . . , ( λ i − , h i − ) , ( λ ′ , h ′ ) , . . . , ( λ ′ l , h ′ l ) , ( λ i , h i ) , . . . , ( λ n − , h n − ))= Ω( u ; ( λ , h ) , . . . , ( λ i − , h i − ) , Ω( u ; ( λ ′ , h ) , . . . , ( λ ′ l , h ′ l )) , ( λ i , h i ) , . . . , ( λ n − , h n − ))for numbers n, l ≥ u ∈ CK ( m , . . . , m n ), u ∈ CK ( m ′ , . . . , m ′ l ), ( λ , h ), . . . ,( λ n − , h n − ) , ( λ ′ , h ′ ) , . . . , ( λ ′ l , h ′ l ) ∈ IM . Let g , . . . , g m , f , . . . , f p , . . . , g m i − i − , ˜ g , . . . , ˜ g m ′ , ˜ f , . . . , ˜ f q , . . . ˜ g m ′ l l ,g i , . . . , g m i i , . . . , f n − , . . . , f p n − n − , g n , . . . , g m n n be m ≤ n + m ′≤ l + p ≤ n − + q ≤ l elements of F ( M ) ( p k and q k are cosimplicial degrees of elements( λ k , h k ) and ( λ ′ k , h ′ k ) respectively). Unwinding the formula of Ω, we haveΩ( u ; . . . , Ω( u ; ( λ ′ , h ′ ) , . . . , ( λ ′ l , h ′ l )) , . . . )( g , . . . , g m n n )= · · · h i − ( e ψ i − ( u ) ∗ f i − ) · ( ω i − ( u ) ◦ H z i − ( u ) ) ∗ g i − · [Ω( u ; ( λ ′ , h ′ ) , . . . , ( λ ′ l , h ′ l ))( e ψ i ( u ) ∗ ˜ g , . . . , e ψ i ( u ) ∗ ˜ g m ′ l l )] · ( ω i ( u ) ◦ H z i ( u ) ) ∗ g i · h i ( e ψ i ( u ) ∗ f i ) · · · = · · · h i − ( e ψ i − ( u ) ∗ f i − ) · ( ω i − ( u ) ◦ H z i − ( u ) ) ∗ g i − · [( ω ( u ) ◦ H z ( u ) ) ∗ e ψ i ( u ) ∗ ˜ g · h ′ ( e ψ ( u ) ∗ e ψ i ( u ) ∗ ˜ f ) · · · ( ω l ( u ) ◦ H z l ( u ) ) ∗ e ψ i ( u ) ∗ ˜ g l ] · ( ω i ( u ) ◦ H z i ( u ) ) ∗ g i · h i ( e ψ i ( u ) ∗ f i ) · · · ohen-Jones isomorphism h i ′ ( e ψ i ′ ( u ) ∗ f i ′ ) = h i ′ ( e ψ i ′ ( u ) ∗ f i ′ . . . e ψ i ′ ( u ) ∗ f m i ′ i ′ )( ω i ′ ( u ) ◦ H z i ′ ( u ) ) ∗ g i ′ = ( ω i ′ ( u ) ◦ H z i ′ ( u ) ) ∗ g i ′ · · · ( ω m i ′ i ′ ( u ) ◦ H z mi ′ i ′ ( u ) ) ∗ g m i ′ i ′ for notational simplicity. On the other hand, by definition, we haveΩ( u ◦ i u ; ( λ , h ) , . . . , ( λ n − , h n − ))( g , . . . , g m n n )= · · · h i − ( e ψ i − ( u ◦ i u ) ∗ f i − ) · ( ω i − ( u ◦ i u ) ◦ H z i − ( u ◦ i u ) ) ∗ g i − · [( ω i − ( u ◦ i u ) ◦ H z i − ( u ◦ i u ) ) ∗ ˜ g · h ′ ( e ψ i ( u ◦ i u ) ∗ ˜ f ) · · · ( ω i + l − ( u ◦ u ) ◦ H z i + l − ( u ◦ i u ) ) ∗ ˜ g l ] · ( ω i + l − ( u ◦ i u ) ◦ H z i + l − ( u ◦ i u ) ) ∗ g i · h i ( e ψ i + l ( u ◦ i u ) ∗ f i ) · · · In view of these expansions, using the conditions ( e ψ -5), ( ω -3) and ( z -3), we easily see thedesired equality holds (actually, these conditions are deduced from this equality). e ψ , z , and ω Construction of e ψ , z , and ω is much similar to that of ψ . Construction of e ψ . This is completely analogous to the construction of ψ given insub-subsection 3.1.3, so we only give a sketch here. When n = 1, e ψ s ( pt )( v ) = id M .When n = 2, we put e ψ i,s ( v , v )( y ) = π φ ((1 − s )( v + φ ( π φ i ( v i ))) + φ ( y )) . For general n , suppose we have constructed for ≤ n − 1. Then, we put e ψ i,s ( tu ; v , . . . , v n )( y ) = π φ ((1 − t ) { (1 − s )( v + φ ( π φ i ( v j )) + φ ( y ) } + t e ψ i,s ( u )( y ) . Construction of z . We give the set { ( n, m , . . . , m n ) | n ≥ , m , . . . , m n ≥ } the lexicographical order. Construction of z proceeds on by induction on this ordered set.For n = 1, we put z j = z j = π φ ( v )Suppose we have constructed z ji for ( l, m ′ , . . . , m ′ l ) such that ( l, m ′ , . . . , m ′ l ) < ( n, m , . . . , m n ).We define a diagram B e F : T c ( m , . . . , m n ) , −→ CG as follows. For an element T ∈ T c ( m , . . . , m n ) , we denote by β T the following composition: B c ( T ) × M [ k , . . . , k n ] Comp × id −−−−−−→ CK ( T ) × M [ k , . . . , k n ] U × id −−−→ K ( n ) × M [ k , . . . , k n ] . See sub-subsection 4.4.3 for T c ( m , . . . , m n ) , and B c . We put B e F ( T ) = ( β T ) − D . Thenwe easily see there is a natural identification:colim T c ( m ,...,m n ) , B e F ∼ = δ e F := { ( u ; v , . . . , v n ) | u ∈ ∂ CK ( m , . . . , m n ) } Using this identification, we define z ji on δ e F . More precisely, for T of codimension one, z : B e F ( T ) → M ap ( M, M ) m ≤ n by the composition in the diagram of the condition ( z -3),and by exactly analogous way to the case of ψ , we can prove these maps fit together to ohen-Jones isomorphism B e F ⇒ M ap ( M, M ) m ≤ n . Thus we get z on δ e F . For a point tu ∈ CK ( m , . . . , m n ) ( t ∈ I and u ∈ CK ( ∂T ( m , . . . , m n ))), we put z ji ( tu ; v , . . . , v n ) = π φ ( b z ji ( tu ; v , . . . v n )) b z ji ( tu ; v , . . . v n ) = (cid:26) (1 − t )( v , . . . , v n ) + 2 tz ji ( u ; v , . . . , v n ) (0 ≤ t ≤ / z ji ( u ; v , . . . , v n ) (1 / ≤ t ≤ φ = φ × · · · × φ n . (The condition ( z -5)is used to ensure that b z ji belongs to the domainof π φ , see below.) We will see this formula satisfies the conditions on z below. Construction of ω : Construction of ω is slightly more delicate. We actually constructa map b ω = ( b ω ji ) : e F k ,...,k n ( m , . . . , m n ) × I −→ Map( M, R k + ··· + k n ) m + ··· + m n by induction and put ω ji ( u ; v , . . . , v n ) = π φ ( b ω ji ( u ; v , . . . , v n )) . (4)This is because the value | ω ( u ; v , . . . , v n )( y ) − y | , which we want to estimate would divergeif we try to construct ω by induction. We first list the conditions which b ω satisfies. It is clearthat the following conditions on b ω implies the conditions ( ω -3) - ( ω -5) on ω defined by theformula (4). Conditions on b ω . b ω satisfies the following conditions for any n ≥ m , . . . , m n ≥ u ; v , . . . , v n ) ∈ D n and s ∈ I .( b ω -1) For t ∈ R , if ( u ; . . . , v i × t, . . . ; s ) ∈ e F ( m , . . . , m n ) × I , then b ω s ( u ; . . . , v i × t, . . . ) − b ω s ( u ; . . . , v i . . . ) ∈ R e k ≤ i +1 . Here e j is the unit vector whose unique non-zero compo-nents is j -th component.( b ω -2) For permutations σ , . . . , σ n , b ω s ( u ; σ v , . . . , σ n v n ) = σ ( k , . . . , k n ) · b ω s ( u ; v , . . . , v n ).( b ω -3) Let n , n , i , m , . . . , m n , m ′ , . . . , m ′ n be non-negative integers such that 1 ≤ i ≤ n .If n ≥ 2, the following diagram is commutative: β − e F ( ˜ m k ) n + n − k =0 × I τ ◦ ( α i × γ i × ∆) / / β i (cid:15) (cid:15) ( e F ( m k ) n k =0 × I ) × ( e F ( m ′ k ) n k =0 × I ) b ω × ( e ψ ′ ( − ) ◦ ω ) (cid:15) (cid:15) Map( M, R K ) m + ··· + m n × Map( M, R K ) m ′ + ··· + m ′ n T (cid:15) (cid:15) e F ( ˜ m k ) n + n − k =0 ω / / M ap ( M, R K ) m + ··· + m n + m ′ + ··· + m ′ n Here, • ( ˜ m k ) n + n − k =0 , τ and T are the transpositions defined by the same formula in ( z -3)above. • when we write e F ( ˜ m k ) n + n − k =0 = e F k ,...,k n n − ( ˜ m k ) n + n − k =0 , we define the num-ber K by K = k + · · · + k n + n − , • for x = ( u ; v ′ , . . . , v ′ n ; t ) ∈ e F ( m j ) n j =0 × I and y ∈ e F ( m ′ j ) n j =0 × I , we put ω × ( e ψ ′ ( − ) ◦ ω )( x, y ) = ω ( x ) × ( φ ◦ [ e ψ i ( x )] ◦ ω ( y ) , . . . , φ ◦ [ e ψ i ( x )] ◦ ω m ′ n n ( y )) , where φ = φ ′ × · · · × φ ′ n . ohen-Jones isomorphism n = 1, the following diagram is commutative: β − e F ( m + m ′ , . . . , m ′ n + m ) × I τ ◦ ( α i × γ i ) / / β i (cid:15) (cid:15) ( e F ( m , m ) × I ) × ( e F ( m ′ k ) n k =0 × I ) b ω × b ω (cid:15) (cid:15) Map( M, R K ) m + m × Map( M, R K ) m ′ + ··· + m ′ n T (cid:15) (cid:15) e F ( ˜ m k ) n + n − k =0 b ω / / M ap ( M, R K ) m + m + m ′ + ··· + m ′ n ( b ω -4) The following diagrams are commutative for each pair ( i, j ): e F ( m k ) nk =0 × I d ji / / b ω (cid:15) (cid:15) e F ( · · · m i + 1 · · · ) × I b ω (cid:15) (cid:15) Map( M, R K ) × m ≤ n δ ji / / Map( M, R K ) × m ≤ n +1 e F ( m k ) nk =0 × I s ji / / b ω (cid:15) (cid:15) e F ( · · · m i + 1 · · · ) × I b ω (cid:15) (cid:15) Map( M, R K ) × m ≤ n σ ji / / Map( M, R K ) × m ≤ n +1 Here, δ ji and σ ji are the maps defined by the same formula in ( ω -4).( b ω -5) For each i, j , b ω ji, ( u ; v , . . . , v n )( z ji ( u ; v , . . . , v n )) belongs to the fiber over π φ ( v , . . . , v n )of the projection π φ , and b ω ji, ( u ; v , . . . , v n ) = φ .( b ω -6) | b ω ( u )( y ) − φ ( y ) | ≤ n | φ | · d ( v, φ ( M )) for each y ∈ M . Construction of b ω . The construction proceeds on by induction on the set { ( n, m , . . . , m n ) } with the lexicographical order. For n = 1, we put b ω j ,s = b ω j ,s ≡ φ .Suppose we have constructed b ω ji for ( l, m ′ , . . . , m ′ l ) < ( n, m , . . . , m n ). We define b ω ji ( u ; v , . . . , v n )for ( u ; v , . . . , v n ) ∈ δ e F similarly to construction of z . We put b ω ji,s ( tu )( y ) = ( (1 − s )[ v + φ ( y − z ji ( tu ))] + s [(1 − t ) φ ( y ) + t b ω ji,s ( u )( y )] (0 ≤ t ≤ / − s )[(2 − t )( v + φ ( y − z ji ( u ))) + (2 t − b ω ji,s ( u )( y )] + s [(1 − t ) φ ( y ) + t b ω ji,s ( u )( y )] (1 / ≤ t ≤ u ∈ δ e F and t ∈ [0 , φ = φ ×· · ·× φ n , v = ( v , . . . , v n ) and b ω ( u ) and b ω ( tu ) denotes b ω ( u ; v , . . . , v n ) and b ω ( tu ; v , . . . , v n ). We will see this formula satisfies the conditions on b ω below.@ Verification on construction of z and b ω : (On z ) The condition ( z -3) is satisfied bydefinition. Verification of the conditions ( z -1), ( z -2) is trivial routine work.We shall verify the condition ( z -5) and the claim that b z ji belongs to the domain of π φ . Weshall show these condition and claim by induction on the same poset as we use in constructionof z . Let u ∈ ∂ CK ( m , . . . , m n ) be an element. By definition, there exist two elements u , u which are not unit, such that u = u ◦ i u for some i . We first show z ji ( u ) satisfies theinequality of the condition ( z -5). By construction, z ji ( u ) = z j i ( u ) or z ji ( u ) = z j i ( u ) forsome ( i , j ) or ( i , j ). In the former case, we observe | v − φ ( z ji ( u )) | = | v − φ ( z ji ( u )) | ≤ n | φ | · d ( v, φ ( M )) ≤ n | φ | · d ( v, φ ( M ))by inductive hypothesis. In the latter case, | v − φ ( z j i ( u )) | ≤ | v − φ ( π φ ( v )) | + | φ ( π φ ( v )) − φ ( π φ ( i ( v ( i ) )) | + | φ ( π φ ( i ( v ( i ) )) − φ ( z j i ( u )) | ohen-Jones isomorphism φ ( i ) = φ i × · · · × φ i + n − , and v ( i ) = ( v i , . . . , v i + n − ). We have | φ ( π φ ( i ( v ( i ) )) − φ ( z j i ( u )) | ≤ | φ || φ ( i ) | · ( | φ ( i ) ( π ( i ) ( v ( i ) )) − v ( i ) | + | v ( i ) − φ ( i ) ( z j i ( u )) | ) ≤ | φ || φ ( i ) | · [ d ( v, φ ( M )) + 4 n | φ ( i ) | · d ( v, φ ( M ))] , | φ ( π φ ( v )) − φ ( π φ ( i ( v ( i ) )) | ≤ | φ || φ ( i ) | · ( | π φ ( i ( v ( i ) ) − v ( i ) | + | v ( i ) − π φ ( v ) | ) ≤ | φ || φ ( i ) | · d ( v, φ ( M )) . So we have | v − φ ( z j i ( u )) | ≤ (cid:18) | φ || φ ( i ) | + 4 n | φ | (cid:19) · d ( v, φ ( M )) . As | φ ( i ) | ≥ 1, if n < n , we see | v − φ ( z j i ( u )) | ≤ n | φ | · d ( v, φ ( M )). If n = n , as v = v ( i ) and φ = φ ( i ) , clearly, | v − φ ( z j i ( u )) | ≤ n | φ | · d ( v, φ ( M )). Thus the inequality in thecondition ( z -5) is satisfied for u ∈ ∂ CK ( m , . . . , m n ).Now we turn to prove that b z ji belongs to the domain of π φ . It is enough to prove d ( b z ji , φ ( M )) < | φ | L e . By definition of b z ji , we have | v − b z ji ( tu ) | ≤ t | v − φ ( z ji ( u )) | for t ∈ [0 , 1] and u ∈ ∂ CK ( m , . . . , m n ). So d ( b z ji ( tu ) , φ ( M )) ≤ | b z ji ( tu ) − φ ( π φ ( v )) | ≤ | b z ji ( tu ) − v | + | v − φ ( π φ ( v )) |≤ t | v − φ ( z ji ( u )) | + d ( v, φ ( M )) ≤ (8 tn | φ | + 1) d ( v, φ ( M )) ≤ (8 tn | φ | + 1) e ǫ ≤ | φ | L e by the conditions ( ǫ -4) and ( z -5) for u ∈ ∂ CK ( m , . . . , m n ). We shall prove the condition( z -5) for general element tu . We have | b z ji ( tu ) − φ ( z ji ( tu )) | ≤ | b z ji ( tu ) − φ ( z ji ( u )) | ≤ (1 − t ) | v − φ ( z ji ( u )) | . So | v − φ ( z ji ( tu )) | ≤ | v − b z ji ( tu ) | + | b z ji ( tu ) − φ ( z ji ( tu )) |≤ | v − φ ( z ji ( u )) | ≤ n | φ | · d ( v, φ ( M )) . We shall prove the condition ( z -4) is satisfied. We consider coface maps { d jk } . We firstconsider the case ( k, j ) = (0 , , ( n, m n ). Let T = T ◦ i T be a cofacial n -tree of codimension1 with T , T indecomposable. d jk ( T ) is a composition of d j ′ k ′ ( T ) and T or T and d j ′′ k ′′ ( T )with appropriate ( k ′ , j ′ ) or ( k ′′ , j ′′ ). Consider the case of d jk ( T ) = d j ′ k ′ ( T ) ◦ i T . Underthe presentation: CK ( m , . . . , m n ) ∼ = Cone (colim B c ), d jk on B c ( T ) is equal to the followingcomposition. B c ( T ◦ i T ) = CK ( T ) × CK ( T ) d j ′ k ′ × id −−−−→ CK ( d j ′ k ′ ( T )) × CK ( T ) ⊂ CK ( T ′ ) × CK ( T ) , where T ′ is an indecomposable cofacial tree such that d jk ( T ) ≤ T ′ . Using this expressionand inductive hypothesis, we see z commutes with d jk on δ e F . For general point t · u ∈ Cone (colim B c ) , we have d jk ( t · u ) = td jk ( u ). By the construction, if z ji ( u ) = z j +1 i ( u ),then z ji ( tu ) = z j +1 i ( tu ) so the compatibility with d jk holds on the whole e F . The case of d jk ( T ) = T ◦ i d j ′′ k ′′ ( T ) is similarly verified. If ( k, j ) = (0 , 0) or ( n, m n + 1), d jk ( T ) = d ◦ T where d = d or d . So the compatibility with d jk follows from the condition ( z -3). ohen-Jones isomorphism b ω ) The condition ( b ω -3) is satisfied by definition. The verification of ( b ω -4) is com-pletely analogous to that of ( z -4). ( b ω -5) is easily verified using the formula of z given in theconstruction. Verification of ( b ω -1) and ( b ω -2) is trivial.We shall verify the condition ( b ω -6). We put k ωn = 10 n | φ | , k ψn = 6 n . Let u = u ◦ i u ∈ ∂ CK ( m , . . . , m n ) be an element and suppose ar( u ) ≥ 2. For 0 ≤ t ≤ / | b ω ji,s ( tu )( y ) − φ ( y ) | ≤ (1 − s ) | v − φ ( z ji ( tu ) | + st | b ω ji,s ( u )( y ) − φ ( y ) | . By the conditions ( b ω -3), ( e ψ -6) and inductive hypothesis, we have | b ω ji,s ( u )( y ) − φ ( y ) | = | φ ◦ e ψ i,s ( u ) ◦ ω ji,s ( u )( y ) − φ ( y ) |≤ | φ | · ( | e ψ i,s ( u )( ω ji,s ( u )( y )) − ω ji,s ( u )( y ) | + | ω ji,s ( u )( y ) − y | ) ≤ | φ | · (cid:18) k ψn d ( v, φ ( M )) + 2 k ωn d ( v ( i ) , φ ( i ) ( M )) | φ ( i ) | (cid:19)(cid:18) ∵ | ω ji,s ( u )( y ) − y | ≤ | φ ( i ) | | b ω ji,s ( u )( y ) − φ ( i ) ( y ) | (cid:19) ≤ | φ | · (cid:18) k ψn + 2 k ωn | φ ( i ) | (cid:19) d ( v, φ ( M )) . So by the condition ( z -5), | b ω ji,s ( tu )( y ) − φ ( y ) | ≤ (cid:20) (1 − s )4 n | φ | + st | φ | · (cid:18) k ψn + 2 k ωn | φ ( i ) | (cid:19)(cid:21) d ( v, φ ( M )) ≤ max (cid:26) n | φ | , | φ | · (cid:18) k ψn + 2 k ωn | φ ( i ) | (cid:19)(cid:27) d ( v, φ ( M ))So it is enough to prove | φ | · (cid:18) k ψn + 2 k ωn | φ ( i ) | (cid:19) ≤ k ωn ( n = n + n − , n ≥ . This inequality is equivalent to 10 n − · n − n ≥ 0. But this inequality is verified byeasy calculation.The evaluation in the case 1 / ≤ t ≤ | b ω ji,s ( tu )( y ) − φ ( y ) | ≤ (1 − s )(2 − t ) | v − φ ( z ji ( u )) | + ((1 − s )(2 t − 1) + st ) | b ω ji,s ( u )( y ) − φ ( y ) |≤ max (cid:26) n | φ | , | φ | · (cid:18) k ψn + 2 k ωn | φ ( i ) | (cid:19)(cid:27) d ( v, φ ( M )) ≤ k ωn d ( v, φ ( M )) . as above. The evaluation of the case ar( u ) = 1 is much easier (so we omit it). b Υ and e ΥIn this sub-subsection, we construct the actions b Υ and e Υ, see sub-subsection 4.5.1. This com-pletes the proof of Theorem 4.5.1. Put J = [1 , n : ( K ( n ) × J ) ˆ ⊗ Γ( M ) ⊗ n −→ Γ( M ) ohen-Jones isomorphism n ≥ M )) Let u ∈ K ( n ), s ∈ J , and h φ i , ǫ i , s i i ∈ Γ( M ) k i be elements ( i = 1 , . . . , n ). We putΥ n ( u, s ; h φ , ǫ , s i , . . . , h φ n , ǫ n , s n i ) = h φ × · · · × φ n , e ǫ s , s ∧ · · · ∧ s n i , e ǫ s = (2 − s ) e ǫ ( u ; ( φ , ǫ ) , . . . , ( φ n , ǫ n )) + ( s − 1) min { ǫ , . . . , ǫ n } (see sub-subsection 3.1.3 for the definition of e ǫ ). To define an action e K ( f IM • ) −→ f IM • , weshall define a morphism e Υ p ,...,p n : K ( n ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) J −→ Map( F ( M ) ⊗ p + ··· + p n , Γ( M )) J for each n ≥ , p , . . . , p n ≥ 0. By adjointness, this is equivalent to define a morphism( K ( n ) × J ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) J ⊗ F ( M ) ⊗ p + ··· + p n −→ Γ( M ) . We define this morphism as the following composition:( K ( n ) × J ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) J ⊗ F ( M ) ⊗ p + ··· + p n id × ∆ J × id −−−−−−−→ ( K ( n ) × J × n +1 ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) J ⊗ F ( M ) ⊗ p + ··· + p n T −→ ( K ( n ) × J ) ˆ ⊗ n O i =1 [ J ˆ ⊗ Map( F ( M ) ⊗ p i , Γ( M )) J ] ⊗ F ( M ) ⊗ p + ··· + p n id × ev × nJ −−−−−−→ ( K ( n ) × J ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) ⊗ F ( M ) ⊗ p + ··· + p n T −→ ( K ( n ) × J ) ˆ ⊗ O [Map( F ( M ) ⊗ p i , Γ( M )) ⊗ F ( M ) ⊗ p i ] id × ev × nF ( M ) −−−−−−−→ ( K ( n ) × J ) ˆ ⊗ Γ( M ) ⊗ n Υ n −−→ Γ( M )Here, ∆ J : J → J × n +1 is the diagonal, and T is the appropriate transposition, and ev − isthe evaluation. We also define a morphism b Υ p ,...,p n : K ( n ) ˆ ⊗ n O i =1 Map( F ( M ) ⊗ p i , Γ( M )) −→ Map( F ( M ) ⊗ p + ··· + p n , Γ( M ))similarly to e Υ using the restriction of Υ n to ( K ( n ) × { } ) ˆ ⊗ Γ( M ) ⊗ n . Proposition 4.5.3. The collection { e Υ p ,...,p n } n,p ,...,p n induce a well-defined action of e K on f IM • , which we denote by e Υ, and the collection { b Υ p ,...,p n } n,p ,...,p n induce a well-definedaction of e K on T • , which we denote by b Υ. These two actions e Υ, b Υ and the action Ωconstructed in sub-subsections 4.5.2, 4.5.4 satisfy the compatibility conditions 1,2 stated insub-subsection 4.5.1. Proof. These are clear from the definition of Ω given in sub-subsections 4.5.2, 4.5.4 and theconditions on e ǫ , and ( e ψ -4), ( ω -5).If we replace e ǫ s with min { ǫ , . . . , ǫ n } in the defintion of Υ n , the action of e K on f IM • becomes the one naturally induced by the associative product on T • .By Proposition 4.5.3, we have completed the proof of Theorem 4.5.1. ohen-Jones isomorphism A ∞ -structures on Tot and g Tot In this subsection, we prove Tot( X • ) and g Tot( X • ) are weak equivalent as nu- A ∞ -ring spectrafor a cs-spectrum X • under some assumptions, see Corollary 4.6.3. To do this we use thenotion of colored operads. We mainly consider the case of non-symmetric operad as it isenough for the A ∞ or little 1-cubes case, but to prove Theorem 1.0.2 for general n , we needto consider the symmetric case, which is completely analogous to the non-symmetric case.In the last part of this subsection, we indicate the necessary changes for the symmetric caseand prove Theorem 1.0.2. Definition 4.6.1. (1) A (non-symmetric topological) colored operad O consists of1. a set Ob( O ) whose elements we call objects (or colors) of O ,2. a family of spaces {O ( c , . . . , c n ; d ) ∈ CG | n ≥ , c , . . . , c n , d ∈ Ob( O ) } ,3. a family of morphisms in CG ,( − ◦ − ) : O ( c , . . . , c n ; d ) × n Y i =1 O ( e i , . . . , e ik i ; c i ) −→ O ( e , . . . , e k , . . . , e n , . . . , e nk n ; d ) , where c , . . . , c n , d, e i , . . . e ik i ∈ Ob( O ), which we call the composition of O and4. an element id c ∈ O ( c ; c ) which we call the identity, for each c ∈ Ob( O ),which satisfy the associativity and unity lows which are exactly analogous to those ofoperad. For a colored operad O and an object c ∈ Ob( O ), we define an operad O c by O c ( n ) = O ( c n ; c ) with the composition induced by the composition of O . Here c n denotesthe n -tuple ( c, c, . . . , c ).(2) Let O be a colored operad. An O - algebra X is a collection of objects of SP , { X c } c ∈ Ob( O ) equipped with a morphism O ( c , . . . , c n ; d ) ˆ ⊗ X c ⊗ · · · ⊗ X c n −→ X d for each integer n ≥ c , . . . , c n , d ) ∈ Ob( O ) n +1 which satisfies compatibility conditionsexactly analogous to those on algebras over an operad. For c ∈ Ob( O ), we always regard X c as an O c -algebra by the obvious way. Theorem 4.6.2. Let O be a colored operad, and a, b two objects of O , and α ∈ O ( a, b ) anelement. Let X be an O -algebra. Then there exist • a topological operad e O with two morphisms of operads ζ a : e O → O a , ζ b : e O → O b , and • an e O -algebra ˜ X with two morphisms of symmetric spectra η a : ˜ X → X a , η b : ˜ X → X b such that1. ζ c is compatible with η c for c = a, b ,2. In Ho( CG ) there exists an isomorphism e O ( n ) ∼ = O a ( n ) × h O ( a n ; b ) O b ( n ) such that thefollowing diagram commutes for c = a, b : e O ( n ) ∼ = / / ζ c ' ' PPPPPPPPPPPPP O a ( n ) × h O ( a n ; b ) O b ( n ) p c (cid:15) (cid:15) O c ( n ) . Here O a ( n ) × h O ( a n ; b ) O b ( n ) denotes the homotopy fiber product of the diagram: O a ( n ) α ◦− −−−→ O ( a n ; b ) −◦ α n ←−−−− O b ( n ), and p c the natural projection. ohen-Jones isomorphism η a is a level equivalence and the following diagram commutes in Ho( SP ):˜ X η a / / η b ❅❅❅❅❅❅❅❅ X aα ∗ (cid:15) (cid:15) X b Here, α ∗ is the evaluation of the structure morphism O ( a ; b ) ˆ ⊗ X a −→ X b at α . Corollary 4.6.3. Let X • be a cs-spectrum. Suppose X • is given a structure of (cid:3) -object or e K -algebra. If the morphism Tot( X • ) → g Tot( X • ) induced by a weak equivalence f : ˜∆ • → ∆ • is a stable equivalence, then Tot( X • ) and g Tot( X • ) are equivalent as nu- A ∞ -ring spectra. Proof of Corollary 4.6.3 assuming Theorem 4.6.2. We only prove the case of a (cid:3) -object. Thecase of e K -algebra is completely analogous.We define a colored operad O as follows: a = ∆ • , b = ˜∆ • , O ( c , . . . , c n ; d ) = Map CG ∆ ( d, c (cid:3) · · · (cid:3) c n ) , where the composition is naturally defined using the composition of ( CG ∆ ) op . Clearly O a = B and O b = B ′ . We put α = f . We define an O -algebra Y by Y c = Map CG ∆ ( c, X • ) withstructure morphisms induced by the product on X • . Clearly Y a = Tot( X • ) and Y b = g Tot( X • ). We apply Theorem 4.6.2 to ( O , Y, α ) and obtain an e O -algebra ˜ Y with morphisms η c : ˜ Y → Y c ( c = a, b ). e O is an A ∞ -operad by the second condition of the theorem, andthe first and third conditions and the assumption of the corollary imply η a and η b are weakequivalences of nu- A ∞ -ring spectra.The rest of this subsection is devoted to the proof of Theorem 4.6.2. The main taskis the construction of the operad e O . If the pullback O a ( n ) × O ( a n ; b ) O b ( n ) has ’correct’homotopy type, we may define the n -th space of e O as this pullback and the composition asthe component-wise composition. Our task is to construct a model of the homotopy pullback O a ( n ) × h O ( a n ; b ) O b ( n ) which carries a unital and associative composition. e O is defined in thepart (5) of the following definition. It is something like ’multiple version’ of Moore’s loopspace. The reader who quickly wants to get some intuition may jump to Example 4.6.6. Definition 4.6.4. Let O be a colored operad and a , b two objects of O , and α an ele-ment of O ( a ; b ). Let R ≥ denote the space of non-negative real numbers with the usualtopology.(1) We define an operad L as follows: Put L ( n ) = ( R ≥ ) × n . For l = ( l , . . . , l n ) ∈ L ( n )and l ′ = ( l ′ , . . . , l ′ m ) ∈ L ( m ), the composition l ◦ i l ′ ∈ L ( n + m − 1) is given by l ◦ i l ′ =( l , . . . , l i − , l i + l ′ , . . . , l i + l ′ m , l i +1 , . . . , l n ) . The unit is 0 ∈ R ≥ .(2) We give the set C := { a, b } an order b < a and give the n times product C n the productpartial order. We regard C n as a category in the usual manner. We define a functor F O : C n −→ CG as follows: We put F O ( c ) = O ( c ; b ) for an element (object) c ∈ C n , andfor two elements c = ( c , . . . , c n ) and c ′ = ( c ′ , . . . , c ′ n ) with c ≤ c ′ , we put F O ( c ≤ c ′ )( f ) = f ◦ ( β , . . . , β n ) , where β i = α if c i < c ′ i , and β i = id if c i = c ′ i .(3) Let l ∈ L ( n ) and c = ( c , . . . , c n ) ∈ C n . We put l = max { l i | i = 1 , . . . , n } , l ( c ) = (cid:26) c = ( a, a, . . . , a )max { l i | c i = b } otherwise . We define a functor G l : C n −→ CG by G l ( c ) = [ l ( c ) , l ] and G l ( c ≤ c ′ ) being the usualinclusion (Note that l ( c ′ ) ≤ l ( c ) if c ≤ c ′ .) ohen-Jones isomorphism l ∈ L ( n ) let ¯ O ( l ) denote the set of all natural transformations G l → F O .So an element of ¯ O ( l ) is a collection { p c } c of paths p c : [ l ( c ) , l ] → O ( c ; b ) which satisfycertain compatibility condition. We define a space O ( n ) ∈ T OP as follows: As a set, weput ¯ O ( n ) = a l ∈L ( n ) ¯ O ( l ) . The topology of O ( n ) is analogous to the compact-open topology of a mapping space.The open subbasis is defined as follows: Let U be an open subset of L ( n ), K a compactsubset of R ≥ and { V c } c ∈ C n a collection of open subsets V c of O ( c ; b ). We put B ( U, K, { V c } ) = { ( l, p c ) ∈ ¯ O ( n ) | l ∈ U, p c ( K ∩ [ l ( c ) , l ]) ⊂ V c ∀ c ∈ C n } The open sub-basis of ¯ O ( n ) is { B ( U, K, { V c } ) } U,K, { V c } , where U (resp. K , V c ) runsthrough all open sets (resp. compact sets, open sets). We define a map λ n : ¯ O ( n ) →O ( a n , b ) by λ n ( l, { p c } c ∈ C n ) = p a n (0). Clearly λ n is continuous.(5) We define an operad e O as follows: e O ( n ) is the k-ification of the fiber product (in T OP )of the following diagram: O a ( n ) α ◦− −−−→ O ( a n , b ) λ n ←−− ¯ O ( n ) , Let ( f ′ , l ′ , { p c ′ } c ′ ∈ C n ) ∈ e O ( n ), and ( f ′′ , l ′′ , { q c ′′ } c ′′ ∈ C m ) ∈ e O ( m ). The i -th composition( f, l, { r c } ) = ( f ′ , l ′ , { p c ′ } ) ◦ i ( f ′′ , l ′′ , { q c ′′ } )is defined as follows: we put f = f ′ ◦ i f ′′ and l = l ′ ◦ i l ′′ . For an element c =( c , . . . , c n + m − ) ∈ C n + m − , we define three elements ¯ c a , ¯ c b ∈ C n , and c ∈ C m by¯ c a = ( c , . . . , c i − , a, c i + m , . . . , c n + m − ), ¯ c b = ( c , . . . , c i − , b, c i + m , . . . , c n + m − ), and c =( c i , . . . , c i + m − ). If l ( c ) ≤ l ′ i , we put r c ( t ) = (cid:26) p ¯ c a ( t ) ◦ i f ′′ ( l ( c ) ≤ t ≤ l ′ i ) p ¯ c b ( t ) ◦ i q a m ( t − l ′ i ) ( l ′ i < t ≤ l ) . If l ( c ) > l ′ i , we put r c ( t ) = p ¯ c b ( t ) ◦ i q c ( t − l ′ i ) . Here, we regard p ¯ c a , p ¯ c b etc. as continuous maps from R by extending the domain usingconstant maps if necessary.(6) We define a morphism of operads ζ a : e O → O a as the projection to the first component,and another morphism ζ b : e O → O b by ( f, l, { p c } ) p b n ( l ). Remark 4.6.5. A more intuitive (but less convenient) description of O ( l ). Let L < · · · Let O be a colored operad, and a, b two objects of O , and α ∈ O ( a ; b ).Let X be an O -algebra.(1) A symmetric spectrum ˜ X is defined as follows: We first define a space ¯ X ′ k for each k ≥ 0. An element of ¯ X ′ k is a pair ( L, h ) of non-negative number L ∈ R ≥ and a path h : [0 , L ] → X b,k . ¯ X ′ k is topologized analogously to O ( n ) in Definition 4.6.4. We define¯ X k as the k-ification of the quotient ¯ X ′ k / { ( L, ∗ L ) | L ≥ } where ∗ L denotes the constantpath at the base point, and regard the point represented by { ( L, ∗ L ) | L ≥ } as thebase point of ¯ X k . The sequence ¯ X = { ¯ X k } k is regarded as a symmetric spectrum in theobvious manner. Then, we put ˜ X := X a × α,X b ,ev ¯ X, where ev : ¯ X → X b is the evaluation at 0 ∈ [0 , L ]. We shall define an action of e O on ˜ X .Let ( f, l, { p c } ) ∈ e O ( n ) and ( x i , h L i , h i i ) ∈ ˜ X k i ( i = 1 , . . . , n ). The value of the structuremorphism e O ( n ) ˆ ⊗ ˜ X ⊗ n → ˜ X ,( x, h L, h i ) = ( f, l, { p c } )(( x , h L , h i ) , . . . , ( x n , h L n , h n i ))is defined as follows: x = f ( x , . . . , x n ) ,L = max { l + L , . . . , l n + L n } ,h ( t ) = p c ( t )( h ′ ( t ) , . . . , h ′ n ( t )) . In the last formula, we write c = min { c ′ | l ( c ′ ) ≤ t } and h ′ i ( t ) = (cid:26) x i t ≤ l i h i ( t − l i ) otherwise,and extend the domains of p c and h ′ i using constant maps if necessary.(2) We define a morphism of symmetric spectra η a : ˜ X → X a as the projection to the firstcomponent, and another morphism η b : ˜ → X b by ( x, h L, h i ) h ( L ).We shall show the definitions given above are well-defined. ohen-Jones isomorphism Lemma 4.6.8. For each n ≥ 0, the space e O ( n ) is compactly generated. Proof. It is enough to see the k-ification of ¯ O ( n ) is weak Hausdorff. We shall define acontinous monomorphism ¯ O ( n ) → L ( n ) ˆ × ˆ Q c Map( R ≥ , O ( c , b )) by ( l, { p c } ) ( l, { ˜ p c } ).Here, ˜ p c is the extension of p c by a constant map and ˆ × and ˆ Q denote the products in T OP . Clearly, the k-ification of L ( n ) ˆ × ˆ Q c Map( R ≥ , O ( c , b )) is weak Hausdorff and so is thek-ification of ¯ O ( n ). Lemma 4.6.9. The topological operad e O given in Definition 4.6.4, (5) is well-defined. Proof. We use the notations in Definiton 4.6.4, (5). By naturality, p ¯ c a ( l ′ i ) ◦ i f ′′ = ( p ¯ c b ( l ′ i ) ◦ i α ) ◦ i f ′′ = p ¯ c b ◦ i q a m (0) , which implies r c : [ l ( c ) , l ] → O ( c ; b ) is continuous. It is a trivial routine work to verify { r c } forms a natural transformation and the composition of e O is associative. The unit is(id a , l = 0 , α, id b ).We shall show the composition is continuous. Let e O ′ ( n ) denote the fiber product of thediagram O a ( n ) α ◦− −−−→ O ( a n , b ) λ n ←−− ¯ O ( n ) , in T OP . As the k-ification preserves products (see [11]), it is enough to show the map( − ◦ i − ) : e O ′ ( n ) × e O ′ ( n ) −→ e O ′ ( n + m − e O is continuous. Take elements ( f ′ , l ′ , { p c ′ } ) ∈ e O ′ ( n ) and ( f ′′ , l ′′ , { q c ′′ } ) ∈ e O ′ ( m ). Let R × B ( U, K, { V c } ) be an open neighborhood of( f, l, { r c } ) := ( f ′ , l ′ , { p c ′ } ) ◦ i ( f ′′ , l ′′ , { q c ′′ } ). Let L < · · · < L N be all the different values of l , . . . , l n + m − . We put A j = { c ∈ C n + m − | l ( c ) = L j } . Let j be the number such that L j ≤ l ′ i < L j +1 . Let j be a number with 1 ≤ j ≤ j . Byan elementary argument, we see there exist1. a number ǫ j > M j ≥ K j,k ⊂ R ≥ for each k = 1 , . . . , M j ,4. an open subset W j,k (¯ c b ) ⊂ O (¯ c b ; b ) for each k = 1 , . . . , M j and c ∈ A j ,5. an open subset X j,k ( c ) ⊂ O ( c ; b ) for each k = 1 , . . . , M j and c ∈ A j ,6. an open subset Y j (¯ c a ) ⊂ O (¯ c a ; b ) for each c ∈ A j ,7. an open subset Z j ⊂ O a ( n )which satisfy the following conditions:1. K j, ∪ · · · ∪ K j,M j = K ∩ [ l ′ i , l ],2. p ¯ c b ( K j,k ± ǫ j ) ⊂ W j,k (¯ c b ), q c ( K j,k ± ǫ j − l ′ i ) ⊂ X j,k ( c ),3. p ¯ c a (([ L j , l ′ i ] ∩ K ) ± ǫ j ) ⊂ Y j (¯ c a ) and f ′′ ∈ Z j ,4. ( − ◦ i − )( W j,k (¯ c b ) × X j,k ( c )) ⊂ V c and ( − ◦ i − )( Y j (¯ c a ) × Z j ) ⊂ V c . ohen-Jones isomorphism S ⊂ R ≥ and a number δ > 0, we use the following notations: S ± δ = { t ∈ R ≥ | [ t − δ, t + δ ] ∩ S = ∅} , S − δ = { t | t + δ ∈ S } . Similarly, for a number j with j < j < N , we can take1. a number ǫ j > M j ≥ K j,k ⊂ R ≥ for each k = 1 , . . . , M j ,4. an open subset W j,k (¯ c b ) ⊂ O (¯ c b ; b ) for each k = 1 , . . . , M j and c ∈ A j ,5. an open subset X j,k ( c ) ⊂ O ( c ; b ) for each k = 1 , . . . , M j and c ∈ A j ,satisfying conditions similar to the above first and second conditions (but in the first condi-tion, l ′ i is replaced by L j ).We put ǫ = min {| L j − L j ′ | | j = j ′ } and put ǫ = min { ǫ , ǫ , . . . , ǫ N } . Let U × U ⊂L ( n ) × L ( m ) be an open neighborhood of ( l ′ , l ′′ ) such that ( − ◦ i − )( U × U ) ⊂ U , and foreach (¯ l ′ , ¯ l ′′ ) ∈ U × U , max {| ¯ l ′ i − l ′ i | , | ¯ l ′′ i − l ′′ i |} < ǫ . Let R × R ⊂ O a ( n ) × O a ( m ) be an openneighborhood of ( f ′ , f ′′ ) such that ( − ◦ i − )( R × R ) ⊂ R . We put D j,k = K j,k ± ǫ j , E j = ([ L j , l ′ i ] ∩ K ) ± ǫ j ,S = R × (cid:16) N \ j =1 M j \ k =1 B (cid:0) U , D j,k , { W j,k (¯ c b ) | c ∈ A j } (cid:1) ∩ \ j ≤ j B (cid:0) U , E j , { Y j (¯ c a ) | c ∈ A j } (cid:1)(cid:17) ⊂ e O ′ ( n ) S = (cid:0) R ∩ \ j ≤ j Z j (cid:1) × N \ j =1 M j \ k =1 B (cid:0) U , D j,k , { X j,k ( c ) | c ∈ A j } (cid:1) ⊂ e O ′ ( m )Here precisely speaking, { W j,k (¯ c b ) | c ∈ A j } denotes the collection { W j,k c ′ } c ′ ∈ C n defined by¯ W j,k c ′ = (cid:26) W j,k (¯ c b ) if c ′ = ¯ c b for some c ∈ A j O ( c ′ ; b ) otherwise , and { Y j (¯ c a ) | c ∈ A j } and { X j,k ( c ) | c ∈ A j } are similarly understood.By definition, ( f ′ , l ′ , { p c ′ } ) ∈ S and ( f ′′ , l ′′ , { q c ′′ } ) ∈ S . We shall show ( − ◦ i − )( S × S ) ⊂ R × B ( U, K, { V c } ). Take elements ( ¯ f ′ , ¯ l ′ , { ¯ p ′ c ′ } ) ∈ S and ( ¯ f ′′ , ¯ l ′′ , { ¯ q c ′′ } ) ∈ S andput ( ¯ f , ¯ l, { ¯ r c } ) = ( ¯ f ′ , ¯ l ′ , { ¯ p c ′ } ) ◦ i ( ¯ f ′′ , ¯ l ′′ , { ¯ q c ′′ } ). Take c ∈ C n + m − and let l ( c ) = L j . Then¯ l ( c ) ∈ ( L j − ǫ, L j + 2 ǫ ). Take t ∈ [¯ l ( c ) , ¯ l ] ∩ K . If t ≤ ¯ l ′ i , as ¯ l ′ i < l ′ i + ǫ , we have t ∈ E j ,which implies ¯ r c ( t ) = ¯ p ¯ c a ( t ) ◦ i ¯ f ′′ ∈ V c . On the other hands, if t > ¯ l ′ i , as ¯ l ′ i ≥ l ′ i − ǫ , we have t ∈ D j,k and t − l ′ i − ǫ < t − ¯ l ′ i < t − l ′ i + ǫ , which imply ¯ p ¯ c b ( t ) ◦ i ¯ q c ( t − ¯ l ′ i ) ∈ V c . Thus, wehave ( ¯ f , ¯ l, { ¯ r c } ) ∈ R × B ( U, K, { V c } ).Proof of the following lemma is similar to that of Lemma 4.6.9 and much easier so omitted. Lemma 4.6.10. The e O -algebra ˜ X given in Definition 4.6.7, (1) is well-defined.To prove Theorem 4.6.2, we need some lemmas. We shall define two continuous maps F : O ( n ) −→ O b ( n ), F ′ : O b ( n ) −→ O ( n ) by F ( l, { p c } ) = p b n , and F ′ ( x ) = (1 n , { x c } ), where1 n denotes (1 , . . . , ∈ L ( n ) and x c is defined by x c ( t ) = F O ( b n ≤ c )( x ). Lemma 4.6.11. F and F ′ are homotopy equivalences which are homotopy inverse to eachother. Proof. Clearly, F ◦ F ′ is the identity, so it is enough to give a homotopy H : O ( n ) × [0 , −→O ( n ) such that H = id and H = F ′ ◦ F . ( l s , { p s c } ) = H ( l, { p c } , s ) is defined as follows.When 0 ≤ s ≤ 1, we put l s ( i ) = (1 − s ) l ( i )+ sl (0), and p s c ( t ) = p c ( t ). (Note that l s ( i ) ≥ l ( i ) sothis is possible.) When 1 ≤ s ≤ 2, we put l s ( i ) = (2 − s ) l (0) + s − p s c ( t ) = p c ( l (0)). ohen-Jones isomorphism Lemma 4.6.12. The map λ n : O ( n ) −→ O a ( n ) is a Serre fibration. Proof. Consider a commutative diagram D k − × { } f / / i (cid:15) (cid:15) O ( n ) λ n (cid:15) (cid:15) D k − × [0 , g / / O ( a n , b )We define a lift h : D k − × [0 , → O ( n ), D k − × [0 , ∋ ( u, s ) ( l u,s , p u,s c ) by( l u, , { p u, c } ) = f ( u ) , l u,s ( i ) = l u, ( i ) + 1 − s,p u,s c ( t ) = (cid:26) g ( u, t ) (0 ≤ t ≤ − s ) p c ( t + s − 1) (1 − s ≤ t ≤ l u,s ( i )) . Lemma 4.6.13. η a defined in Definition 4.6.7,(2) is a level equivalence. Proof. In fact, for each k ≥ η a,k : ˜ X k → X a,k has a homotopy inverse: η ′ k : X a,k → ˜ X k , x (0 , α ( x )). A homotopy between η ′ k ◦ η a,k and id is given by (( x, h L, h i ) , s ) ( x, h sL, h | [0 ,sL ] i ). Proof of Theorem 4.6.2. The topological operad e O defined in Definition 4.6.4,(5) is well-defined by Lemmas 4.6.8 and 4.6.9. Verification of the first condition of the theorem (com-patibility of ζ c with η c ) is trivial. The second condition easily follows from Lemmas 4.6.11,4.6.12 and the equality λ n ◦ F ′ = α b n ,a n . The third condition also easily follows from Lemma4.6.13. Remark 4.6.14. It is straightforward to construct certain homotopy invariant version ofcolored operad of morphisms by extending e O . Here, the colored operad of morphisms ofa colored operad O is defined by saying that objects are 1-array morphisms of O and n -array morphisms are pairs of n -array morphisms of O compatible with 1-array morphismsat the sources and targets. In fact, e O will fit in it as the endomorphism operad of α . Thisconstruction should be equivalent to a special case of internal hom-objects of dendoroidalsets introduced by Moerdijk and Weiss [21]. Our construction is more intuitive and efficientfor our purpose i.e., we can easily determine the homotopy type of e O ( n ) and construct theaccompanying algebra ˜ X .We shall prove Theorem 1.0.2. We first state it more precisely. In [17] two operads D n and e D n both of which are weak equivalent to the little n -cubes operads are introduced (seesections 9 and 15 of [17]). In our notation, D = B and e D = e B . Let X • be a cosimplicialspace or symmetric spectrum. It is proved that a Ξ n -algebra structure on X • (see Definition4.5 and 8.4 of [17]) induces an action of D n (resp. e D n ) on Tot( X • ) (resp. g Tot( X • )) in afunctorial way. Theorem 4.6.15 (precise statement of Theorem 1.0.2) . Let X • be a Ξ n -algebra. Supposethe morphism Tot( X • ) → g Tot( X • ) induced from a weak equivalence ˜∆ • → ∆ • is a weakequivalence. Then, there exist a symmetric operad e O , e O -algebra ˜ Y , weak equivalences ζ : e O → D n , ζ : e O → e D n of symmetric operads, and weak equivalences η : ˜ Y → Tot( X • ), η : ˜ Y → g Tot( X • ) such that η i is compatible with ζ i for i = 1 , 2. Here a weak equivalence ofspaces or symmetric spectra means any of a weak homotopy equivalence, level equivalence,or stable equivalence. ohen-Jones isomorphism Proof. A symmetric colored operad is a non-symmetric colored operad equipped with anaction of the k -th symmetric group Σ k on each morphism space O ( c , . . . , c k ; d ) for each k ≥ O is a symmetric colored operad, the space e O ( k ) has a natural actionof Σ k induced from that of O and permutations of components of L ( k ) = R k ≥ . Theorem4.6.2 is still valid if we replace all morphisms of operads in the statement by morphisms ofsymmetric operads. Now the proof of Theorem 4.6.15 is completely analougous to Corollary4.6.3. The only necessary change is to replace c (cid:3) . . . (cid:3) c k by Ξ nk ( c , . . . , c k ) (see Definition8.4 of [17]). t ¯ p ( t ) p ( t ) ◦ f q ◦ p ′ ( t − l )0 l l + l ′ Figure 5: Composition of two elements ( f, l, p, q ), ( f ′ , l ′ , p ′ , q ′ ) ∈ e O (1) p ( t ) ∈ O ( a ; b ) q ( t ) ∈ O ( b, a ; b )0 l l ( l ≤ l ) p ( t ) ∈ O ( a ; b ) q ( t ) ∈ O ( a, b ; b )0 l l ( l > l )Figure 6: An element of e O (2) ( f ∈ O a (2) and r ∈ O b (2) are omitted) The following lemma easily follows from compatibility of totalization and homotopy pushoutin [5] Lemma 4.7.1. The morphism Tot( L • ) → g Tot( L • ) induced by a weak equivalence ˜∆ • → ∆ • is a level equivalence.Let F : L • → X • be a term-wise stable fibrant replacement. We must prove the inducedmorphism g Tot( L • ) → g Tot( X • ) is a stable equivalence. Definition 4.7.2. (1) We say a symmetric spectrum X is a strongly semi-stable object ( sss-object for short) if there exists a number α > k , thecanonical map π i ( X k ) → π i +1 ( X k +1 ) is an isomorphism for 0 ≤ i ≤ αk .(2) H Z ∈ SP denotes a fixed cofibrant model of the Eilenberg-MacLane spectrum of Z and − ⊗ L H Z denotes the derived tensor product.The first part of the following lemma trivially follows from the fact that stable fibrantobjects are Ω-spectra (in the sense of [12]) and the second part is proved in section 5.6 of[12]. Lemma 4.7.3 ([12]) . (1) Any stable fibrant object is an sss-object.(2) Any stable equivalence between sss-objects is a π ∗ -isomorphism. Lemma 4.7.4. (1) A finite homotopy limit of sss-objects in the level model structure is alsoa sss-object and is (weak equivalent to) the homotopy limit in the stable model structure. ohen-Jones isomorphism X is a sss-object, X ⊗ L H Z is also a sss-object. (Note that X ⊗ L H Z is well-definedup to level equivalences, so this statement makes sense.)(3) ( − ⊗ L H Z ) preserves finite homotopy limits in the stable model structure.(4) For each p ≥ L p is a sss-object. Proof. (1) follows from Lemma 4.7.3. (2) and (4) are trivial. (3) follows from the factthat homotopy pullback squares and homotopy pushout squares coincide in the stable modelstructure of SP . Proposition 4.7.5. The morphism g Tot( L • ) → g Tot( X • ) induced by F is a stable equivalence. Proof. Let G : X • ⊗ L H Z → Y • be a termwise stable fibrant replacement. (Note that wemust replace X • by a (stable) cofibrant object when we apply the derived tensor, but thisdoes not matter as a cofibrant replacement can be taken as a level equivalent object.)We first prove the composition g Tot( X • ) ⊗ L H Z −→ g Tot( X • ⊗ L H Z ) G ∗ −−→ g Tot( Y • )is a stable equivalence.By Thom isomorphism, we have ˜ H t ( L sk ) ∼ = H t − k + d ( M × n × ( M × e V k )) ( d = dimM ). So N s ˜ H t ( L • k ) = 0 if t − k + d ≤ s . So N s ˜ H St ( L • ) = 0 if t ≤ s − d . Here N s is the part ofdegree s of the usual normalization and ˜ H St ( X ) = colim k H t + k ( X k ). So we have H St ( N s ( X • )) ∼ = π St (( N s X • ) ⊗ L H Z ) ∼ = π St ( N s ( X • ⊗ L H Z )) ∼ = N s π St ( X • ⊗ L H Z ) ∼ = N s ˜ H St ( L • ) = 0if t ≤ s − d . (See [26] for N s of a cosimplicial space and we apply it in the levelwisemanner.) Here, the first isomorphism is trivial, and the second one follows from Lemmas4.7.3 and 4.7.4, (1),(2),(3) as N s is a finite homotopy limit in the level model structure, andthe third is proved in [26], and the forth follows from Lemmas 4.7.3 and 4.7.4,(4). (For thedefinition of N s for cosimplicical spaces, see [26] and we apply it in a levelwise manner.) Bythe Hurewicz theorem, this imply π St ( N s ( X • )) = 0 for t ≤ s − d . This and the homotopyfiber sequence Ω s N s X • → g Tot s ( X • ) → g Tot s − ( X • )(see [26]) implies g Tot s ( X • ) → g Tot s − ( X • ) is s − d -connected. Here g Tot l is the homotopylimit of the restriction to the full subcategory of simplices of dimension ≤ l . Consider thefollowing diagram: g Tot( L • ) ⊗ L H Z F ∗ / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ g Tot( X • ) ⊗ L H Z / / G ∗ (cid:15) (cid:15) g Tot s ( X • ) ⊗ L H Z G ∗ (cid:15) (cid:15) g Tot( Y • ) / / g Tot s ( Y • ) . By the preceding argument and Hurewicz theorem, the top right arrow is s − d -connected.Similarly, the right bottom arrow is also s − d -connected. As g Tot s is a finite homotopy limit,The right vertical arrow is a stable equivalence by Lemma 4.7.4 (3). Thus, the middle verticalarrow is s − d -connected for all s ≥ g Tot( L • )and g Tot( X • ) are connected, i.e., π Sk = 0 for all sufficiently small k ∈ Z , and sss-objects byLemma 4.7.1, we see the morphism F ∗ : g Tot( L • ) → g Tot( X • ) is a stable equivalence by theHurewicz theorem. ohen-Jones isomorphism A ∞ -ringspectra. LM − τ (A) ∼ = Tot( L • ) (B) ≃ g Tot( L • ) (C) ≃ g Tot( IM • ) (D) ≃ g Tot(THC • ( A ′ , B )) (E) ≃ Tot(THC • ( A ′ , B )) = THC( A ′ , B ) (F) ≃ THC( Q, Q )The isomorphism (A) is given in subsection 4.2, and the equivalences (B) and (E) follow fromCorollary 4.6.3, Lemma 4.7.1 and the fact that THC • ( A, B ′ ) is a Reedy fibrant object ,whichimplies the morphism Tot(THC • ( A, B ′ )) → g Tot(THC • ( A, B ′ )) is a (level) equivalence.By Lemma 4.3.2, the morphism p : IM • → L • is a termwise level equivalence so inducesa level equivalence ( p ) ∗ : g Tot( IM • ) → g Tot( L • ). 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