The May-Milgram filtration and E k -cells
aa r X i v : . [ m a t h . A T ] M a y THE MAY-MILGRAM FILTRATION AND E k -CELLS INBAR KLANG, ALEXANDER KUPERS, AND JEREMY MILLER
Abstract.
We describe an E k -cell structure on the free E k +1 -algebra on apoint, and more generally describe how the May-Milgram filtration of Ω m Σ m S k lifts to a filtration of the free E k + m -algebra on a point by iterated pushouts offree E k -algebras. Introduction
Cell structures on topological spaces have many uses, and operadic cell structuresplay similar roles in the study of algebras over an operad; they appear when definingmodel category structures on categories of algebras over operads, and in the case ofthe little k -cubes operad E k they have found applications to homological stability[KM18, GKRW18, GKRW19].Despite the usefulness of such E k -cell structures, few have been described ex-plicitly. In this paper we give an E k -cell decomposition of an E k -algebra weaklyequivalent to the free E k +1 -algebra on a point. We generalize this to a filtrationof the free E k + m -algebra on a point by E k -algebras, and explain how this can bethought of as a lift of the May-Milgram filtration of the iterated based loop spaceΩ m Σ m S k .We shall state our results without giving definitions (which appear in Section 2),with the exception of that of a cell attachment in the category Alg E k ( Top ). Thisdefinition uses the free E k -algebra functor E k = F E k , which is left adjoint to theforgetful functor U E k : Alg E k ( Top ) → Top sending an E k -algebra to its underlyingspace. Let A be an E k -algebra in Top , and let e : ∂D d → U E k ( A ) be a map oftopological spaces. To attach a d -dimensional E k -cell to A , we take the adjoint map E k ( ∂D d ) → A of e and take the pushout of the following diagram in Alg E k ( Top ): E k ( ∂D d ) AE k ( D d ) A ∪ E k e D d . To make this homotopy invariant we need to require that A is cofibrant, or derivethe cell-attachment construction.An E k -algebra is called cellular if it is built by iterated E k -cell attachments (suchalgebras are always cofibrant). Giving an E k -cell structure on A is giving a weakequivalence between A and a cellular E k -algebra. Note that the following colimitis also a homotopy colimit: Date : May 8, 2020.Alexander Kupers is supported by NSF grant DMS-1803766.Jeremy Miller was supported by NSF grant DMS-1709726.
Theorem 1.1. E k +1 ( ∗ ) is weakly equivalent as an E k -algebra to a cellular E k -algebra with exactly one cell in dimensions divisible by k and no other cells. Thatis, it is weakly equivalent to the colimit colim r ∈ N A r of algebras A r obtained bysetting A − = ∅ and taking iterated pushouts in Alg E k ( Top ) E k ( ∂D rk ) A r E k ( D rk ) A r +1 . An E k -cell structure on A induces an ordinary cell structure on its k -fold deloop-ing B k A (see Remark 2.6). The one induced on B k E k +1 ( ∗ ) ∼ = ΩΣ S k by the E k -cellstructure of Theorem 1.1 is that coming from the James construction [Jam55].The filtration coming from the James construction was generalized by May andMilgram [May72, Mil66] to a filtration on Ω m Σ m S k , and we will construct a fil-tration of E k + m ( ∗ ) by E k -algebras which deloops to the May-Milgram filtration on B k E k + m ( ∗ ) ≃ Ω m Σ m S k .To state this result precisely, we need to introduce some notation. Let I denotethe open interval (0 , F r ( I m ) the space of ordered configurations of r points in I m , and C r ( I m ) := F r ( I m ) / S r the space of unordered configurations of r points in I m . Furthermore, let φ m,r denote the vector bundle F r ( I m ) × S r R r − → C r ( I m )with R r − the representation of the symmetric group S r given by the orthogonalcomplement to the trivial representation in the permutation representation withits usual metric. This vector bundle inherits a Riemannian metric. For E → B avector bundle with Riemannian metric, let D ( E ) denote its unit disk bundle, S ( E )denote its unit sphere bundle and kE denote its k -fold Whitney sum. Theorem 1.2. E k + m ( ∗ ) is weakly equivalent as an E k -algebra to the colimit colim r ∈ N A r of algebras A r obtained by setting A − = ∅ and taking iterated pushouts in Alg E k ( Top ) E k ( S ( kφ m,r +1 )) A r E k ( D ( kφ m,r +1 )) A r +1 . This implies that the homotopy cofiber of A r → A r +1 in Alg E k ( Top ) is the free E k -algebra on the Thom space of kφ m,r +1 viewed as a based space. These Thomspaces and their corresponding Thom spectra are well-studied, e.g. being relatedto Brown-Gitler spectra when k = 2 [CMM78, CCKN83]. When m = 1, C r ( I m )is contractible and the vector bundle kφ m,r +1 has dimension kr . Thus the spherebundle is homotopy equivalent to ∂D rk and hence Theorem 1.1 is a consequence ofTheorem 1.2.In Corollary 4.14 we give a configuration space model F [ r ] ( I k × I m )( ∗ ) for the E k -algebras A r . The space F [ r ] ( I k × I m )( ∗ ) is given by the spaces of configurationspaces of points in I m × I k such that each subset { x } × I k contains at most r points.We use this in Theorem 5.2 to prove that the k -fold delooping of A r is homotopyequivalent to the r th stage of the May-Milgram filtration of Ω m Σ m S k . Remark . Our results bear a resemblance to the Dunn–Lurie additivity theorem[Dun88] [Lur17, Theorem 5.1.2.2]. This result says that E k + m ≃ E k ⊗ E m fora suitable tensor product of operads, and our result says that E k + m ( ∗ ) can be HE MAY-MILGRAM FILTRATION AND E k -CELLS 3 obtained as E k -algebra from the cardinality filtration on E m ( ∗ ), twisted by thevector bundles kφ m,r +1 . It would be interesting to know whether it is possible todeduce Theorem 1.2 from the additivity theorem. Acknowledgments.
The authors would like to thank Ralph Cohen, Søren Galatius,David Gepner, Mike Mandell, and Oscar Randal-Williams for helpful conversations.We would also like to thank the anonymous referee for helpful comments and sug-gestions.
Contents
1. Introduction 12. Recollections of homotopy theory for algebras over an operad 33. Rank completion 64. An E k -algebraic analogue of the May-Milgram filtration 115. Relation to the May-Milgram filtration 20References 242. Recollections of homotopy theory for algebras over an operad
We work in the setting of [GKRW18] and use similar notation when possible.
Assumption 2.1. S is a simplicially enriched complete and cocomplete categorywith closed symmetric monoidal structure such that the tensor product ⊗ commuteswith sifted colimits. Assumption 2.2. S comes equipped with a cofibrantly generated model struc-ture which is both simplicial and monoidal, and such that the monoidal unit iscofibrant.The first version of [GKRW18] required that homotopy equivalences are weakequivalences but this is in fact always the case by Proposition 9.5.16 of [Hir03].When G is a symmetric monoidal category, then we may endow the category S G of functors G → S with the Day convolution tensor product; this will also besymmetric monoidal. Similarly, the category S ∗ of pointed objects in S with smashproduct inherits these properties. Example . The examples of S most relevant to this paper are: (i) the category sSet of simplicial sets with the Quillen model structure and cartesian product, and(ii) the category Top of compactly generated weakly Hausdorff spaces with theQuillen model structure and cartesian product (see [Str09] for more details aboutthe point-set topology).Let FB ∞ denote the category of (possibly empty) finite sets and bijections, thenthe objects of the category ( S G ) FB ∞ of functors FB ∞ → S G are called symmetricsequences . In addition to the Day convolution tensor product, ( S G ) FB ∞ admitsa composition product ◦ (which is rarely symmetric); for X , Y ∈ ( S G ) FB ∞ theevaluation of the composition product X ◦ Y on the set { , , . . . , r } is given by X ◦ Y ( r ) = G d ≥ X ( d ) ⊗ S d G k + ... + k d = r S r × S k + ... + S kd Y ( k ) ⊗ . . . ⊗ Y ( k d ) ! . INBAR KLANG, ALEXANDER KUPERS, AND JEREMY MILLER
A (symmetric) operad is a unital monoid with respect to this composition prod-uct. An O -algebra A is an object A ∈ S with a left O -module structure on A considered as a symmetric sequence concentrated in cardinality 0. Equivalently wecan use the associated monad on S , for which we also use the notation O , O ( X ) := G r ≥ O ( r ) ⊗ S r X ⊗ r , and define an O -algebra to be an algebra over this monad. The category Alg O ( S G )of O -algebras is both complete and cocomplete.A free O -algebra is one of the form O ( X ) with O -algebra structure maps inducedby the monad multiplication and unit. We use the notation F O : S G → Alg O ( S G )for the free O -algebra functor, which is the left adjoint to the forgetful functor U O : Alg O ( S G ) → S G sending an algebra to its the underlying object. Note O = U O F O .Any O -algebra admits a canonical presentation as a reflexive coequalizer of free O -algebras: F O ( O ( U O ( A ))) F O ( U O ( A )) A , the top map coming from the O -algebra structure map O ( U O ( A )) → U O ( A ), andthe bottom map coming from the natural transformation F O O → F O inducedby the monad multiplication. Thus free O -algebras generate the category of O -algebras under sifted colimits, and the category of right O -module functors C → D preserving sifted colimits is equivalent to the category of functors Alg O ( C ) → D preserving sifted colimits; one constructs the latter from the former using thecanonical presentation, and one constructs the former from the latter by evaluatingon free O -algebras.For X ∈ S G and g ∈ G , the evaluation X X ( g ) ∈ S has a left adjoint;given Y ∈ S we denote its image under this left adjoint by Y g . Given a map ∂D d → U O ( A )( g ) (where ∂D d stands for ∂D d ⊗ , the copowering of ∂D d with themonoidal unit), we obtain by adjunction first a map ∂D g,d → U O ( A ) and then amap F O ( ∂D g,d ) → A . An O -cell attachment is defined to be the following pushoutin Alg O ( S G )(1) F O ( ∂D g,d ) AF O ( D g,d ) A ∪ O D g,d . Explicitly this pushout may be constructed as the following reflexive coequalizer F O ( O ( U O ( A )) ∪ D g,d ) F O ( U O ( A ) ∪ D g,d ) A ∪ O D g,d . The left vertical map in (1) is a cofibration, so cell attachments are homotopy-invariant when
Alg O ( S G ) is left proper. In general we need to derive the construc-tion; we will momentarily explain when this can be done using a monadic barresolution.Using the copowering of S G over sSet , any operad in simplicial sets gives rise toan operad in S G , and using the strong monoidal functor Sing : Top → sSet so doesany operad in compactly-generated weakly Hausdorff topological spaces. We shallrestrict our attention to operads O in sSet which are Σ-cofibrant, i.e. for all r ≥ HE MAY-MILGRAM FILTRATION AND E k -CELLS 5 the S r -action on O ( r ) is free. We may attempt to define a model structure on Alg O ( S G ) by declaring the (trivial) fibrations and weak equivalences to be those ofunderlying objects. If it exists, this is called the projective model structure. Assumption 2.4.
The projective model structure exists on
Alg O ( S G ).When this assumption is satisfied, the projective model structure will be a cofi-brantly generated model structure with generating (trivial) cofibrations obtainedby applying O to the generating (trivial) cofibrations of the model structure on S G . Since S G is a simplicial and monoidal model category, it is automatic that theforgetful functor U O : Alg O ( S G ) → S G preserves (trivial) cofibrations, cf. Lemma9.5 of [GKRW18]. When O is a Σ-cofibrant operad in simplicial sets, the projectivemodel structure exists in the settings of Example 2.3, cf. Section 9.2 of [GKRW18].When U O ( A ) ∈ S G is cofibrant, we may use the monadic bar resolution to findan explicit cofibrant replacement of A and thus compute derived functors. Definition 2.5.
The monadic bar resolution is the augmented simplicial object B • ( F O , O , A ) with p -simplices given by F O ( O p ( U O ( A ))) for p ≥ A for p = −
1. The face maps and augmentation are induced by the monad multiplicationand the O -algebra structure on A , and the degeneracies by the unit of the monad.This is a special case of the two-sided monadic bar construction , which is usedthroughout the paper. It takes as input a monad T , a right T -functor F and a T -algebra A with underlying object A , and has p -simplices given by B • ( F, T, A ) = F ( T p ( A )). The face maps and degeneracy maps are similar to above, for detailssee e.g. Section 9 of [May72].Let | − | denote the (thin) geometric realization, and introduce the notation B ( F O , O , A ) := | B • ( F O , O , A ) | . Note that here we take geometric realization in thecategories of O -algebras, but U O commutes with geometric realization by Section8.3.3 of [GKRW18]. The augmentation induces a map B ( F O , O , A ) → A , which isalways a weak equivalence using an extra degeneracy argument. It is a free simplicialresolution in the sense of Definition 8.16 of [GKRW18] when the bar construction isReedy cofibrant. Because O is Σ-cofibrant, this is the case when U O ( A ) is cofibrant,using the Reedy cofibrancy criterion of Lemma 9.14 of [GKRW18]. Remark . In [KM18], O -algebra cell attachments were defined using partialalgebras. The formula in Definition 3.1 of [KM18], written in our notation, is | [ p ] F O ( O p ( A ) ∪ D d ) | . This may be obtained by inserting B ( F O , O , A ) into theunderived formula for cell attachment. We explained above that this gives derivedcell attachment when U O ( A ) is cofibrant, but in Top and sSet this assumption isunnecessary. In sSet , every object is cofibrant. In
Top , we use that geometric real-ization sends levelwise weak equivalences between proper simplicial spaces to weakequivalences, even if the simplicial spaces are not levelwise cofibrant. This allowsus to cofibrantly replace A in the category of O -algebras (which will be cofibrantin topological spaces because O is Σ-cofibrant). Thus results of [KM18] apply: inparticular, Proposition 6.12 of [KM18] implies that an E k -cell structure on an E k -algebra in topological spaces deloops to an ordinary cell structure on the k -folddelooping. The reason for this is that delooping preserves homotopy pushouts and E k ( ∂D d ) → E k ( D d ) deloops to ∂D d ֒ → D d . INBAR KLANG, ALEXANDER KUPERS, AND JEREMY MILLER Rank completion
We shall define a rank completion filtration in the case that we are working ina category of functors S G where G has a notion of rank, and the operad O and O -algebra A satisfy mild conditions. Later in this paper, O will be E k and G willbe N ; the rank function will be used to keep track of the number of points in aconfiguration, and though we shall not use this, G can be used to record groupactions on configurations. Assumption 3.1. G is a symmetric monoidal groupoid equipped with strongmonoidal functor κ : G → N , which we call a rank functor .Let G ≤ r denote the full subcategory on G on those objects g such that κ ( g ) ≤ r ,and G r denote the full subcategory on objects g such that κ ( g ) = r . Precompositiongives restriction functors ( ≤ r ) ∗ and ( r ) ∗ participating in adjunctions S G ≤ r S G , ( ≤ r ) ∗ ( ≤ r ) ∗ S G r S G . ( r ) ∗ ( r ) ∗ There are further relative restriction and extension functors between S G r , S G ≤ r fordifferent r , participating in analogous adjunctions. The functors ( ≤ r ) ∗ and ( r ) ∗ are themselves left adjoints; though we will not use their right adjoints, we will usethat ( ≤ r ) ∗ and ( r ) ∗ commute with colimits.It follows from the formula for Day convolution that S G ≤ r inherits a symmetricmonoidal tensor product, an alternative expression for which is given by X ⊗ Y = ( ≤ r ) ∗ (( ≤ r ) ∗ ( X ) ⊗ ( ≤ r ) ∗ ( Y )). This makes visible that ( ≤ r ) ∗ is strong monoidal andsimplicial. In particular, the functor ( ≤ r ) ∗ takes O -algebras in S G to O -algebrasin S G ≤ r . Its left adjoint ( ≤ r ) ∗ in general does not. However, we may use thecanonical presentation of O -algebras explained in the previous section to constructa left adjoint ( ≤ r ) alg ∗ : Alg O ( S G ≤ r ) → Alg O ( S G ) to ( ≤ r ) ∗ : Alg O ( S G ) → Alg O ( S G ≤ r ).Explicitly it is the following reflexive coequalizer F O (( ≤ r ) ∗ O ( U O ( A ))) F O (( ≤ r ) ∗ U O ( A )) ( ≤ r ) alg ∗ ( A ) . It is defined uniquely up to isomorphism by demanding that ( ≤ r ) alg ∗ F O ( X ) = F O (( ≤ r ) ∗ ( X )) and that it preserves sifted colimits. Definition 3.2.
We define the r th rank completion functor T r : Alg O ( S G ) → Alg O ( S G )to be ( ≤ r ) alg ∗ ( ≤ r ) ∗ .This functor underlies the monad associated to the adjunction ( ≤ r ) alg ∗ ⊣ ( ≤ r ) ∗ and has a right adjoint. The counit gives a natural transformation T r ⇒ id, andthe commutative diagram of groupoids GG ≤ G ≤ G ≤ · · · , gives rise to a tower of natural transformations of functors Alg O ( S G ) → Alg O ( S G )id T T T · · · . HE MAY-MILGRAM FILTRATION AND E k -CELLS 7 Since colimits are computed objectwise and the map ( g ) ∗ T r ( A ) → ( g ) ∗ A is theidentity as soon as r ≥ κ ( g ), the natural transformation colim r ∈ N T r ⇒ id is anatural isomorphism.The functor ( ≤ r ) ∗ obviously preserves fibrations and weak equivalences, so( ≤ r ) alg ∗ is a left Quillen functor. However, ( ≤ r ) ∗ also preserves cofibrations,as these are retracts of iterated pushouts along free O -algebra maps, which arepreserved by ( ≤ r ) ∗ . Hence T r = ( ≤ r ) alg ∗ ( ≤ r ) ∗ preserves trivial cofibrationsbetween cofibrant objects, and thus admits a left derived functor by precompositionwith a functorial cofibrant replacement. Moreover, as explained above, when U O ( A )is cofibrant we may use a monadic bar resolution to cofibrantly replace A . As acomposition of two left adjoints, T r commutes with geometric realization. Thus weobtain the following formula for T L r ( A ):(2) T L r ( A ) = B ( F O ( ≤ r ) ∗ , O , ( ≤ r ) ∗ U O ( A )) = B ( F O ( ≤ r ) ∗ ( ≤ r ) ∗ , O , U O ( A )) , the latter equality following from the fact that ( ≤ r ) ∗ commutes with O .We next restrict our attention to a setting where the underlying object of F O ( X )agrees with X in rank ≤ r up to homotopy, for those X which are concentrated inrank r . To see when this occurs, note that for any operad O and X concentratedin rank r , O ( X ) is isomorphic to O (0) in rank 0 and O (1) ⊗ X in rank r . Hencethe following assumption suffices: Assumption 3.3. O is a non-unitary operad in simplicial sets, i.e. O (0) = ∅ , and O (1) ≃ ∗ . Definition 3.4.
We say X ∈ S G is reduced if it is concentrated in rank >
0, thatis, X ( g ) is initial when κ ( g ) = 0.The horizontal maps in the following proposition are obtained from the identitymorphisms of ( r + 1) ∗ U O T L r ( A ) and ( r + 1) ∗ U O T L r +1 ( A ) respectively, the verticalmaps from the natural transformation T r ⇒ T r +1 . Proposition 3.5.
For reduced A there is a homotopy cocartesian square in Alg O ( S G ) F O (( r + 1) ∗ ( r + 1) ∗ U O T L r ( A )) T L r ( A ) F O (( r + 1) ∗ ( r + 1) ∗ U O T L r +1 ( A ))) T L r +1 ( A ) , where we remark that ( r + 1) ∗ U O T L r +1 ( A ) ∼ = ( r + 1) ∗ U O ( A ) .Proof. This diagram is obtained by applying ( ≤ r +1) alg ∗ to a diagram in Alg O ( S G ≤ r +1 ),a functor which preserves homotopy cocartesian squares as it is a left Quillenfunctor. Hence it suffices to prove that the following is homotopy cocartesian in Alg O ( S G ≤ r +1 ): F O (( r + 1) ∗ U O T L r ( A )) ( ≤ r ) alg ∗ ( ≤ r ) ∗ AF O (( r + 1) ∗ U O A ) ( ≤ r + 1) ∗ A , where F O now denotes the free algebra functor S G ≤ r +1 → Alg O ( S G ≤ r +1 ), ( ≤ r ) ∗ and( r + 1) ∗ denote the left adjoints to ( ≤ r ) ∗ : S G ≤ r +1 → S G ≤ r and ( ≤ r + 1) ∗ S G ≤ r +1 → INBAR KLANG, ALEXANDER KUPERS, AND JEREMY MILLER S G r +1 respectively, and ( ≤ r ) alg ∗ denotes the left adjoint to ( ≤ r ) ∗ : Alg O ( S G ≤ r +1 ) → Alg O ( S G ≤ r ).The result now follows from the next lemma: substitute in its statement X ( r + 1) ∗ U O T L r ( A ) Y ( r + 1) ∗ U O AA ( ≤ r ) alg ∗ ( ≤ r ) ∗ AB A . Verifying condition (ii) uses that O (1) ≃ ∗ . (cid:3) Lemma 3.6.
Suppose A , B ∈ Alg O ( S G ≤ r +1 ) are cofibrant in S G ≤ r +1 and reduced,and X, Y ∈ S G ≤ r +1 are cofibrant and concentrated in rank r +1 . Then a commutativesquare F O ( X ) AF O ( Y ) B , is homotopy cartesian in Alg O ( S G ≤ r +1 ) if the following two conditions hold:(i) the map ( ≤ r ) ∗ A → ( ≤ r ) ∗ B is a weak equivalence,(ii) the commutative square ( r + 1) ∗ X ( r + 1) ∗ U O A ( r + 1) ∗ Y ( r + 1) ∗ U O B is homotopy cocartesian.Proof. We may assume without loss of generality that A and B are cofibrant in S G ≤ r +1 and X → Y is a cofibration between cofibrant objects. We can factor thecommutative square as F O ( X ) B ( F O , O , U O ( A )) AF O ( Y ) B ( F O , O , U O ( B )) B , ≃≃ where the horizontal maps are weak equivalences because A and B are cofibrant in S G ≤ r +1 .The left square is the geometric realization of the following square of simplicialobjects (cid:0) [ p ] F O ( X ) (cid:1) (cid:0) [ p ] F O ( O p ( U O ( A ))) (cid:1)(cid:0) [ p ] F O ( Y ) (cid:1) (cid:0) [ p ] F O ( O p ( U O ( B ))) (cid:1) . All these simplicial objects are Reedy cofibrant; this is evident for the left en-tries, and for the right entries follows from another application of Lemma 9.14 of
HE MAY-MILGRAM FILTRATION AND E k -CELLS 9 [GKRW18]. Geometric realization of Reedy cofibrant simplicial objects is a homo-topy colimit, and thus commutes with homotopy pushouts. In particular, a levelwisehomotopy cocartesian diagram of Reedy cofibrant simplicial objects geometricallyrealizes to a homotopy cocartesian diagram. It thus suffices to prove that each ofthe levels F O ( X ) F O ( O p ( U O ( A ))) F O ( Y ) F O ( O p ( U O ( B )))is homotopy cocartesian.Since X → Y is a cofibration between cofibrant objects, the map from thehomotopy pushout to the bottom-right corner is given by F O ( O p ( U O ( A )) ∪ X Y ) −→ F O ( O p ( U O ( B ))) . This is a weak equivalence in
Alg O ( S G ≤ r +1 ) if and only if the map on underlyingobjects is. Since F O preserves weak equivalences between cofibrant objects, itsuffices to prove that the map O p ( U O ( A )) ∪ X Y −→ O p ( U O ( B ))is a weak equivalence. Indeed, both objects are cofibrant since X
7→ O ( X ) preservescofibrant objects, as do pushouts along a cofibration.We do this by induction over p . For p = 0, we observe that since X and Y areconcentrated in rank r + 1, we have a commutative diagram( ≤ r ) ∗ U O ( A ) ( ≤ r ) ∗ U O ( B )( ≤ r ) ∗ U O ( A ) ∪ X Y ( ≤ r ) ∗ U O ( B ) . ∼ = Thus for ranks ≤ r the result follows from assumption (i). In rank r + 1, the case p = 0 follows from assumption (ii). This completes the proof of the initial case.To prove the induction step, it suffices to prove the following statement: if Z, W are reduced and
X, Y are concentrated in degree r + 1, then if (i) ( ≤ r ) ∗ Z → ( ≤ r ) ∗ W is a weak equivalence and (ii) the commutative square( r + 1) ∗ X ( r + 1) ∗ Z ( r + 1) ∗ Y ( r + 1) ∗ W is homotopy cocartesian, then (i’) ( ≤ r ) ∗ O ( Z ) → ( ≤ r ) ∗ O ( W ) is a weak equivalenceand (ii’) the commutative square(3) ( r + 1) ∗ X ( r + 1) ∗ O ( Z )( r + 1) ∗ Y ( r + 1) ∗ O ( W )is homotopy cocartesian. Deducing (i) from (i’) is done by noting that ( ≤ r ) ∗ commutes with O and O preserves weak equivalences between cofibrant objects. To deduce (ii’) from (i) and(ii), we use the formula( r + 1) ∗ O ( Z ) = G n ≥ O ( n ) ⊗ S n G ≤ r ,...,r n ≤ r +1 P r i = r +1 ( r ) ∗ ( r ) ∗ Z ⊗ · · · ⊗ ( r n ) ∗ ( r n ) ∗ Z and a similar one for W . To restrict the r i to positive integers, we used that O isnon-unitary, and that Z and W are reduced. From this expression, we see that (3)is a coproduct of two commutative diagrams. The first is i ( r + 1) ∗ O (( ≤ r ) ∗ ( ≤ r ) ∗ Z ) i ( r + 1) ∗ O (( ≤ r ) ∗ ( ≤ r ) ∗ W ) , where i is the initial object, which is homotopy cocartesian because the right mapis a weak equivalence as a consequence of (i). The second is( r + 1) ∗ X O (1) ⊗ ( r + 1) ∗ Z ( r + 1) ∗ Y O (1) ⊗ ( r + 1) ∗ W, which homotopy cocartesian by (ii) since O (1) ≃ ∗ . (cid:3) We thus get a sequence of maps T L ( A ) −→ T L ( A ) −→ T L ( A ) −→ · · · whose homotopy colimit is naturally weakly equivalent to A and whose homotopycofibers we understand. This is the rank completion filtration .When we can make sense of homology, e.g. in one of the settings mentioned inSection 10.1 of [GKRW18], we get a corresponding spectral sequence convergingconditionally to the homology of U O ( A ). The E -page will be rather unwieldy,and we believe the following spectral sequence may be more useful: Remark . Let ( − ) + denote the monad whose underlying functor takes the co-product with the terminal object (so that algebras over it are pointed objects). As O is a non-unitary operad in simplicial sets, cf. Assumption 3.3, there is a canonicalmap of monads from O to ( − ) + which can be viewed as an augmentation of O . Thisaugmentation is given on X ∈ S G by the map O ( X ) = F n ≥ O ( n ) ⊗ S n X ⊗ n → X + which on the summand O (1) ⊗ X is the map O (1) ⊗ X → ∗ ⊗ X = X and on thesummands O ( n ) ⊗ S n X ⊗ n for n ≥ O -indecomposables functor Q O : Alg O ( S G ) → S G ∗ determined uniquely up to isomor-phism by demanding that Q O F O ∼ = ( − ) + and that Q O commutes with sifted colim-its. Applying its left-derived functor Q O L to the diagram in the previous proposition, HE MAY-MILGRAM FILTRATION AND E k -CELLS 11 we get that if A is reduced there is a homotopy cocartesian square in S G ∗ :( r + 1) ∗ ( r + 1) ∗ U O T L r ( A ) + Q O L ( T L r ( A ))( r + 1) ∗ ( r + 1) ∗ U O T L r +1 ( A ) + Q O L ( T L r +1 ( A )) . When we can make sense of homology, we can define O -homology by H O g,d ( A ) :=˜ H d (( g ) ∗ Q O L ( A )). The result of the previous discussion is a conditionally convergentspectral sequence (suppressing the filtration degree, so in particular the p in E p,q refers to rank) E p,q = H p + q (( p ) ∗ T L p ( A ) , ( p ) ∗ T L p − ( A )) = ⇒ H O p,p + q ( A ) , where it may be helpful to recall that ( p ) ∗ T L p ( A ) ∼ = ( p ) ∗ A .4. An E k -algebraic analogue of the May-Milgram filtration To deduce our results, we specialize the results of the previous section to O = E k ,the non-unital little k -cubes operad. Recall that I denotes the open interval (0 , rect ( F n I k , I k ) denote the space of ordered n -tuples of rectilinear em-beddings I k → I k with disjoint image (that is, they are a composition of translationand dilation by positive real numbers in each of the k directions). Definition 4.1.
The non-unital little k -cubes operad E k has topological space E k ( n )of n -ary operations given by E k ( n ) := ( ∅ if n = 0,Emb rect ( F n I k , I k ) if n > S n permuting the n -tuples. The unit in E k (1) is the identitymap I k → I k , and composition is induced by composition of embeddings.This satisfies Assumption 3.3 and hence gives rise to an operad in S G , all ofwhose objects are concentrated on the monoidal unit of G . E k -algebras in S G arealgebras over this operad, and we shall adopt the shorter notation E k for the free E k -algebra functor F E k . (If S = Top and G = ∗ , as a consequence of our conventionsthese are algebras over the operad | Sing( E k ) | in topological spaces.)We shall take G = N , with κ : N → N the identity functor. Let U E k + m E k : Alg E k + m ( S N ) −→ Alg E k ( S N )denote the forgetful functor induced by the map of operads E k → E k + m given bysending a cube e : I k → I k to e × id I m : I k × I m → I k × I m . For the sake of brevitywe will often write U for U E k + m E k .We are interested in free algebras on a point, which we will consider concentratedin rank 1. In this section we will more generally study E k + m ( X ) for X ∈ S N satisfying a similar condition: Assumption 4.2. X ∈ S N is concentrated in rank 1, i.e. X ( g ) is initial unless g = 1 (so in particular reduced), and X is cofibrant.We will give an elementary geometric model for the E k -algebra T L r ( U E k + m ( X )),and use Proposition 3.5 to describe U E k + m ( X ) up to weak equivalence as a colimit •• •••• •• I k = II m = I Figure 1.
An element of F ( I k × I m ) for k = 1, m = 1 (suppressingthe labels on the point for the sake of clarity) which is in F [ r ]8 ( I k × I m )when r ≥
4, but not when r < of iterated pushouts along maps of free E k -algebras. The following definition wasmentioned in the introduction: Definition 4.3.
For a manifold M and n ≥
1, the topological space F n ( M ) of or-dered configurations of n points in M is given by { ( m , . . . , m n ) | m i = m j if i = j } ⊂ M n . For n = 0 we define F n ( M ) = ∅ .We choose to define F ( M ) to be empty since we work with non-unital E k -algebras. For n >
0, the topological space F n ( M ) is homeomorphic to the space ofembeddings of the set { , . . . , n } into M , and precomposition by permutations of { , . . . , n } defines a S n -action on the space F n ( M ). Taking the singular simplicialset, these assemble to a symmetric sequence F ( M ) in sSet . For M = I k × I m , thisis a left E k + m -module, where the action is given by composition of embeddings.Just like we used the enrichment of copowering of S N over sSet to make the operad E k in sSet into an operad in S N , we use it to make the left E k + m -module F ( I k × I m ) in sSet into a left E k + m -module in S N . Analogously to the free E k -algebra construction,we can take the composition product of F ( I k × I m ) ∈ ( S N ) G with an object X ∈ S N considered as a symmetric sequence concentrated in cardinality 0. We refer to thisas “applying” F ( I k × I m ) to X . The resulting object F ( I k × I m )( X ) ∈ S N comesendowed with an E k + m -algebra structure. This construction is natural in X , andthus we obtain a functor F ( I k × I m ) : S N → Alg E k + m ( S N ). Definition 4.4.
We let F [ r ] n ( I k × I m ) denote the subspace of F n ( I k × I m ) consistingof ordered configurations η = ( m , . . . , m n ) such that for all x ∈ I k the intersection η ∩ ( { x } × I m ) has cardinality at most r .Since the condition defining F [ r ] n ( I k × I m ) is invariant under the S n -action, thesetopological spaces may be assembled into a symmetric sequence F [ r ] ( I k × I m ) ⊂ F ( I k × I m ) in sSet and by the copowering also in S N . The left E k + m -module structureon F ( I k × I m ) does not restrict. However, using the map of operads E k → E k + m induced by the inclusion I k → I k × I m on the first k coordinates, we get a left E k -module structure on F ( I k × I m ) which does restrict and application of this symmetricsequence gives a functor F [ r ] ( I k × I m ) : S N → Alg E k ( S N ). HE MAY-MILGRAM FILTRATION AND E k -CELLS 13 As we assumed that X is cofibrant, we can use a monadic bar resolution to givean explicit formula for T L r ( U E k + m ( X )) ∈ Alg E k ( S N ): T L r ( U E k + m ( X )) = B ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ U E k + m ( X )) . We take this specific model for the domain of the map in the following proposition:
Proposition 4.5.
There are weak equivalences α r : T L r ( U E k + m ( X )) −→ F [ r ] ( I k × I m )( X ) , of E k -algebras, which fit into commutative diagrams for r ≥ T L r ( U E k + m ( X )) T L r +1 ( U E k + m ( X )) F [ r ] ( I k × I m )( X ) F [ r +1] ( I k × I m )( X ) . α r α r +1 Let us start by defining the maps:
Lemma 4.6.
There are maps α r : T L r ( U E k + m ( X )) −→ F [ r ] ( I k × I m )( X ) of E k -algebras making Diagram (4) commute.Proof. The map E k + m → F ( I k × I m ) which sends a cube to its center is a homotopyequivalence of left E k + m -modules in symmetric sequences, so we have an inducedweak equivalence E k + m ( X ) → F ( I k × I m )( X ) of E k + m -algebras. To define α r , wefirst insert this weak equivalence into the right entry of the bar construction | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ U E k + m ( X )) || B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ U F ( I k × I m )( X )) | . ≃ The assumption that X is concentrated in rank 1 gives us an isomorphism( ≤ r ) ∗ U F ( I k × I m )( X )) ∼ = U (( ≤ r ) ∗ F ( I k × I m ))(( ≤ r ) ∗ X ) . Here ( ≤ r ) ∗ F ( I k × I m ) is an object in the truncated symmetric sequence category(functors from the category of possibly empty finite sets of cardinality ≤ r into sSet with tensor product the restriction of the composition product) and ( ≤ r ) ∗ F ( I k × I m )is the functor given by tensoring with ( ≤ r ) ∗ F ( I k × I m ).Because ⊗ commutes with colimits in each variable and geometric realization,the target is obtained by applying the symmetric sequence | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m ) | in Top to X (as always, via Sing and the simplicial copowering). We define a mapof left E k -modules in symmetric sequences in Top a r : | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m )) | −→ F [ r ] ( I k × I m )by describing an augmentation from B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m )) to F [ r ] ( I k × I m ). The 0-simplices of the former are given by E k ( ≤ r ) ∗ ( ≤ r ) ∗ F ( I k + m ) = G n ≥ E k ( n ) × S n G ≤ k ,...,k n ≤ r F k i ( I k × I m )where the rank of each component is k + ... + k n . Given a collection of embeddings e i : I k → I k and configurations ξ i ∈ F k i ( I k × I m ), we may take the union of the images ( e i × id I m )( ξ i ) in I k × I m and obtain an ordered configuration of k + . . . + k m points such that no subset { x } × I m contains more than r points. This map iseasily seen to be compatible with the left E k -module structures. That the diagramcommutes is clear from the definition. (cid:3) We next prove that each a r is a weak homotopy equivalence, using a microfibra-tion argument. Definition 4.7.
A map π : E → B of topological spaces is a microfibration if foreach i ≥ D i × { } ED i × [0 , B, h πH there exists an ǫ > H : D i × [0 , ǫ ] → E , i.e. π ◦ ˜ H = H | D i × [0 ,ǫ ] and ˜ H | D i ×{ } = h . Lemma 4.8 (Lemma 2.2 of [Wei05]) . If π : E → B is a microfibration with weaklycontractible fibers, then π is a weak homotopy equivalence. Our strategy is to prove that a r is a microfibration with weakly contractiblefibers. To do this, we use the following lemma in point-set topology. Lemma 4.9.
Let X • be a levelwise Hausdorff simplicial space. Let X , • ⊂ X , • ⊂ · · · ⊂ X • be an N > -indexed sequence of simplicial subspaces such that: (i) X s,p ⊂ X p iscompact for all s, p , and (ii) each point x ∈ X p has an open neighborhood containedin some X s,p . If C is compact, then any continuous map C → | X • | factors as C → | X s, • | → | X • | for some s .Proof. The strategy is to first identify | X • | with the sequential colimit colim s | X s, • | and then show that this particular sequential colimit commutes with maps out ofthe compact space C .The inclusions X s,p → X p induce a continuous bijection colim s X s,p → X p . Toshow it is a homeomorphism we need to prove it is open: V ⊂ colim s X s,p being openmeans that all V ∩ X s,p are open, and by the hypothesis for all x ∈ V , V containsan open neighborhood of x in X p , which means it is open in X p . Since colimitsof simplicial spaces are computed levelwise, colim s X s, • → X • is an isomorphismof simplicial spaces. Since geometric realization commutes with filtered colimits(it has a right adjoint when working with compactly generated weakly Hausdorffspaces), the canonical map colim s | X s, • | → | X • | is a homeomorphism.In CGWH spaces, maps out of a compact space commute with sequential colimitsof closed inclusions by Lemma 3.6 of [Str09]. Thus we shall verify that each map | X s, • | → | X s +1 , • | is a closed inclusion, using its description as a colimit of the mapsof skeleta: sk | X s, • | sk | X s, • | · · · sk | X s +1 , • | sk | X s +1 , • | · · · . HE MAY-MILGRAM FILTRATION AND E k -CELLS 15 We claim all maps in this diagram are closed inclusions. All maps are clearlycontinuous injections and a continuous injection between compact Hausdorff spacesis always a closed inclusion, so it suffices to prove that each space is compactHausdorff. They are compact because each sk p | X s, • | is a quotient of the compactspace F k ≤ p ∆ k × X s,k . They are Hausdorff because we may freely add degeneraciesto write sk p | X s, • | as the geometric realization of a levelwise Hausdorff simplicialspace and apply Theorem 1.1 of [dSP13]. Furthermore, from the construction it isclear each square is a pullback square. The result then follows from the followingresult about CGWH spaces, Lemma 3.9 of [Str09]: given a commutative diagram A A · · · B B · · · f f with all maps closed inclusions and all squares pullbacks, the induced map colim s A s → colim s B s is also a closed inclusion. (cid:3) Lemma 4.10.
The map a r : | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m ) | → F [ r ] ( I k × I m ) isa microfibration.Proof. Fixing a cardinality n , we need to prove that the component a r ( n ) is amicrofibration. Suppose we are given a commutative diagram D i × { } | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m )) | ( n ) D i × [0 , F [ r ] n ( I k × I m ) . h a r ( n ) H Since D i × [0 ,
1] is compact, there exists a δ > H factors over thecompact subspace of configurations ξ where(a) the points in ξ have distance ≥ δ from each other,(b) for all closed cubes C ⊂ R k with equal sides of length < δ , the set C × I m contains at most r points of ξ .Let us abbreviate B p ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m ))( n ) by X p , and by X δp thesubspace of X p of elements whose image under α r ( n ) satisfies (a) and (b).Let ρ p : X δp → (0 , ∞ ) be the minimum of the distances from the points in theimage ξ to the boundaries of the images of the cubes. Then X δ • is a simplicial spacewith a sequence of continuous functions ρ p : X δp → (0 , ∞ ) such that ρ p +1 ◦ s i = ρ p and ρ p − ◦ d i ≥ ρ p . For each integer s ≥
1, the subspaces X δs,p := ρ − p ([1 /s, ∞ )) ⊂ X δp assemble to a simplicial space X δs, • .This satisfies the hypotheses of Lemma 4.9 (condition (i) of that lemma is thereason we use X δ • instead of X • , and uses that only the interiors of cubes need tobe disjoint, not their closures). Hence the map h factors over some stage | X δs, • | with δ ≥ s > d ∈ D i , the configuration H ( d,
0) is given by a ξ which satisfies the properties(a) the points in ξ have distance ≥ s from each other,(b) for all closed cubes C ⊂ R k with equal sides of length < s , the set C × I m contains at most r points of ξ , (c) the points in ξ have distance ≥ s to the boundaries of the images of thecubes in h ( d ).By continuity of the map H , there is an ǫ > d ∈ D i and t ∈ [0 , ǫ ], the configuration H ( d, t ) is within distance s of H ( d, d, t ) ∈ D i × [0 , ǫ ] the element of | X • | represented by configuration H ( d, t ) inside the cubes coming from the unique non-degenerate representative of h ( d ). To see this is well-defined, note that (c) impliesa point in H ( d, t ) remains within the same cubes of h ( d ) as the corresponding pointin H ( d, d, t ) of (cid:0) [ − s , s ] k + m (cid:1) n (each ∆( d,
0) equals 0). That is, ∆( d, t ) is defined by H ( d, t ) = H ( d,
0) + ∆( d, t ).There is a simplicial map (cid:18) [ − s , s ] k + m (cid:19) n × X /ss, • −→ X • obtained by applying the displacement to the configuration. This is continuous, andwell-defined because whenever we move points in the configuration of an elementof X /ss,p at most s in any of the directions, they do not (a) collide with each other,(b) have more than r points in a subset { x } × I m , and (c) cross boundaries of cubes.That the lift is continuous then follows by observing that it can be realized as acomposition of continuous maps D i × [0 , ǫ ] ∆ × h −→ (cid:18) [ − s , s ] k + m (cid:19) n × | X /ss, • | ∼ = −→ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) [ − s , s ] k + m (cid:19) n × X /ss, • (cid:12)(cid:12)(cid:12)(cid:12) −→ | X • | . (cid:3) We now prove that the fibers of a r ( n ) are weakly contractible, so Lemma 4.8implies that a r ( n ) is a weak homotopy equivalence. Lemma 4.11.
The fibers of the map a r : | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m ) | → F [ r ] ( I k × I m ) are weakly contractible.Proof. Fix a configuration ξ ∈ F [ r ] n ( I k × I m ). As above, the fiber ǫ − ( ξ ) is given bythe geometric realization of the subsimplicial space of X • with underlying configu-ration ξ . Call this simplicial space X • ( ξ ). Another application of Lemma 9.14 of[GKRW18] tells us this is Reedy cofibrant.Let ξ ′ ∈ Sym n ( I k ) := ( I k ) n / S n be the configuration with multiplicities obtainedby projecting ξ onto I k . The p -simplices of X • ( ξ ) are given by p + 1 levels of nested k -dimensional cubes such that all points of ξ ′ are contained in an innermost cubeand all cubes except the outermost ones contain at most r points of ξ ′ countedwith multiplicity. Let ( e , e , . . . e l ) ∈ E k ( l ) be a collection of cubes such thatevery cube e i contains exactly one point of ξ ′ (counted without multiplicity) andevery point of ξ ′ (counted without multiplicity) is in one of the cubes e i . Let X • ( ξ, e ) denote the subsimplicial space of X • ( ξ ) where we require that if a cubecontains a point of ξ ′ , then it contains the corresponding e i . This is also Reedycofibrant. Thus, since the inclusion X • ( ξ, e ) ֒ → X • ( ξ ) induces a levelwise homotopy HE MAY-MILGRAM FILTRATION AND E k -CELLS 17 •••• •• I k = II m = I Figure 2.
An element of X in the case k = m = 1, r = 4, and n = 6.There are two innermost cubes and one outermost cube. equivalence, it induces a weak equivalence on geometric realizations. View X • ( ξ, e )as an augmented simplicial space by adding a point in degree −
1. There is an extradegeneracy X p ( ξ, e ) → X p +1 ( ξ, e ) given by inserting e i in the innermost cubes, andhence | X • ( ξ, e ) | is contractible. (cid:3) Proof of Proposition 4.5.
By combining Lemmas 4.8, 4.10, and 4.11, we see that themap a r : | B • ( E k ( ≤ r ) ∗ , E k , ( ≤ r ) ∗ F ( I k + m ) | → F [ r ] ( I k × I m ) is a weak equivalence.Since a r is a map of symmetric sequences and all of the symmetric group actionsare free, it is weak equivalence of symmetric sequences. The result follows becauseapplying a weak equivalence between Σ-cofibrant symmetric sequences in sSet to acofibrant object is a weak equivalence by Lemma 9.1 of [GKRW18], and geometricrealization preserves weak equivalences between Reedy cofibrant simplicial objects. (cid:3) The inclusion I k × F r +1 ( I m ) ֒ → F r +1 ( I k × I m ) given by ( x, ξ ) x × ξ , i.e. sendingeach point m i ∈ ξ to x × m i , has image given by the complement of F [ r ] r +1 ( I k × I m ) in F r +1 ( I k × I m ). Let ϕ m,r denote the trivial vector bundle over I k × F r ( I m )given by I k × F r ( I m ) × R r − → I k × F r ( I m ), with S r acting diagonally and with R r − the orthogonal complement to the trivial representation in the permutationrepresentation with its usual metric. The vector bundle ϕ m,r can be thought of asthe S r -equivariant analogue of φ m,r from the introduction. Lemma 4.12.
The normal bundle of I k × F r +1 ( I m ) in F r +1 ( I k × I m ) is S r +1 -equivariantly isomorphic to kϕ m,r +1 .Proof. The normal bundle to I k × F r +1 ( I m ) is the orthogonal complement in T ( F r +1 ( I k × I m )) to the tangent bundle T ( I k × F r +1 ( I m )). The former is the restriction of T ( I k × I m ) ⊕ r +1 ∼ = ( T I k ) ⊕ r +1 ⊕ ( T I m ) ⊕ r +1 , and the latter is the restriction of T I k ⊕ ( T I m ) ⊕ r +1 . The inclusion is the diagonal on the first term and the identityon the second, and equivariant for the S r +1 -action. Thus the normal bundle is S r +1 -equivariantly isomorphic to the restriction of the orthogonal complement ofthe diagonal T I k ⊂ ( T I k ) ⊕ r +1 . This is isomorphic to a k -fold Whitney sum of thetrivial R r -bundle, with S r +1 -action given by the standard representation. (cid:3) The vector bundle ϕ m,r +1 inherits a Riemannian metric, and we let S ( kϕ m,r +1 )be the sphere bundle with fiber over ( x, ξ ) ∈ I k × F r +1 ( I m ) those vectors of length d ( y, ∂I k ). This bounds a disk bundle D ( kϕ m,r +1 ), and both are clearly isomorphicto the unit sphere and disk bundles. Using the exponential map, we obtain thehorizontal maps in the following commutative diagram of S r +1 -spaces: S ( kϕ m,r +1 ) F [ r ] r +1 ( I k × I m ) D ( kϕ m,r +1 ) F [ r +1] r +1 ( I k × I m ) = F r +1 ( I k × I m ) . Proposition 4.13.
For all r ≥ , there is a zigzag of homotopy cocartesian squares E k (( r + 1) ∗ S ( kϕ m,r +1 ) ⊗ S r +1 X ⊗ r +1 ) · · · · · · T L r ( U E k + m ( X )) E k (( r + 1) ∗ D ( kϕ m,r +1 ) ⊗ S r +1 X ⊗ r +1 ) · · · · · · T L r +1 ( U E k + m ( X )) . ≃≃ Proof.
The above result implies that for cofibrant X , we have a homotopy cocarte-sian square S ( kϕ m,r +1 ) ⊗ S r +1 X ⊗ r +1 ( r + 1) ∗ U E k F [ r ] ( I k × I m )( X ) D ( kϕ m,r +1 ) ⊗ S r +1 X ⊗ r +1 ( r + 1) ∗ U E k F [ r +1] ( I k × I m )( X ) . By Proposition 4.5, we have a commutative diagram with horizontal maps weakequivalences( r + 1) ∗ U E k F [ r ] ( I k × I m )( X ) ( r + 1) ∗ U E k T L r ( U E k + m ( X ))( r + 1) ∗ U E k F [ r +1] ( I k × I m )( X ) ( r + 1) ∗ U E k T L r +1 ( U E k + m ( X )) . α r ≃ α r +1 ≃ Since applying E k and ( r + 1) ∗ preserves homotopy cocartesian squares, doingso gives us the left and middle squares. For the right square, specialize Proposition3.5 to O = E k and A = U E k + m ( X ) to obtain a homotopy cocartesian square E k (( r + 1) ∗ ( r + 1) ∗ U E k T L r ( U E k + m ( X ))) T L r ( U E k + m ( X )) E k (( r + 1) ∗ ( r + 1) ∗ U E k T L r +1 ( U E k + m ( X ))) T L r +1 ( U E k + m ( X )) . (cid:3) We now deduce Theorem 1.2 from this by taking S = Top and X = (1) ∗ ( ∗ ); weneed to resolve the issue that Proposition 4.13 only provides zigzags. HE MAY-MILGRAM FILTRATION AND E k -CELLS 19 Proof of Theorem 1.2.
We start with an elementary homotopy-theoretic observa-tion. Given a commutative diagram of topological spaces S ( kφ m,r +1 ) X X ′ D ( kφ m,r +1 ) Y Y ′≃≃ with decorated arrows weak equivalences, we can find maps S ( kφ m,r +1 ) → X ′ and D ( kφ m,r +1 ) → Y ′ such that in the following diagram S ( kφ m,r +1 ) X X ′ D ( kφ m,r +1 ) Y Y ′≃≃ the outer square commutes and the triangles commute up to homotopy. To provethis, first homotope S ( kφ m,r +1 ) → X until a lift exists (which is possible sincethe domain has the homotopy type of a CW-complex). Because S ( kφ m,r +1 ) ֒ → D ( kφ m,r +1 ) admits the structure of a NDR-pair, we may extend this to a homotopyof commutative diagrams. At this point it suffices to find a lift in the commutativediagram S ( kφ m,r +1 ) Y ′ D ( kφ m,r +1 ) Y, which exists as ( D ( kφ m,r +1 ) , S ( kφ m,r +1 )) is homotopy equivalent to a CW pair.Given this observation, we prove by induction over r that we may construct A r ≃ T L r +1 ( U E k + m ( ∗ )) by iterated pushouts along free algebras, obtaining in theprocess maps between the A r satisfying colim r A r = hocolim r A r ≃ F E k + m ( ∗ ).The initial case is A − = ∅ . For the induction step, let us assume we haveproduced A r as in the statement of Theorem 1.2 with a weak equivalence β r : A r → T L r ( U E k + m ( ∗ )). Using the observation in the diagram of Proposition 4.13, we mayassume we have a homotopy cocartesian commutative diagram S ( kφ m,r +1 ) ( r + 1) ∗ U E k T L r ( U E k + m ( ∗ )) D ( kφ m,r +1 ) ( r + 1) ∗ U E k T L r +1 ( U E k + m ( ∗ )) . Applying the observation again to lift along α r , we may assume we have a ho-motopy cocartesian commutative diagram S ( kφ m,r +1 ) ( r + 1) ∗ U E k A r ( r + 1) ∗ U E k T L r ( U E k + m ( ∗ )) D ( kφ m,r +1 ) ( r + 1) ∗ U E k T L r +1 ( U E k + m ( ∗ )) . ( r +1) ∗ U E k β r We take adjoints and define A r +1 as the pushout fitting in a commutative diagram E k (( r + 1) ∗ S ( kφ m,r +1 )) A r T L r ( U E k + m ( ∗ )) E k (( r + 1) ∗ D ( kφ m,r +1 )) A r +1 T L r +1 ( U E k + m ( ∗ )) . β r β r +1 The outer and left squares are homotopy cocartesian, the former by constructionand the latter as a homotopy pushout. Thus the right square is also homotopycocartesian, and hence the map β r +1 : A r +1 → T L r +1 ( U F E k + m ( ∗ )) is a weak equiv-alence. (cid:3) Combining the last sentence of this proof with Proposition 4.5, we get the de-scription of A r announced in the introduction. Corollary 4.14.
There are weak equivalences of E k -algebras A r β r −→ T L r ( U E k + m ( ∗ )) α r −→ F [ r ] ( I k × I m )( ∗ ) . Remark . It is plausible that Theorem 1.2 may be deduced from results anal-ogous to those in [GKRW18]. One would need an CW approximation theoremfor E k -algebras in Top N , and verify that one may desuspend the identification ofΣ k Q E k L ( F E k + m ( ∗ )) with the k -fold bar construction with respect to the canonicalaugmentation. 5. Relation to the May-Milgram filtration
We now explain the relationship between the results in the previous section andthe May-Milgram filtration on Ω m Σ m S k . Definition 5.1.
Given a based topological space (
X, x ), let C ( M ; X ) be the quo-tient of F n ≥ F n ( M ) × S n X n by the relation that ( m , . . . m n ; x , . . . x n ) is equiva-lent to ( m , . . . m n − ; x , . . . x n − ) if x n = x . We call this the configuration spaceof unordered points in M with labels in X .When X = S , we recover ordinary unordered configuration spaces and drop X from the notation. Work of Milgram and May implies that Ω m Σ m X has the weakhomotopy type of C ( I m ; X ) when X is connected [Mil66, May72]. The r th stage M r ( C ( I m ; X )) of the May-Milgram filtration of C ( I m ; X ) ≃ Ω m Σ m X is defined tobe the image of F r ( I m ) × S i X i in C ( I m ; X ).In this paper, we use only the case X = S k . In that case, Ω m Σ m S k is weaklyequivalent to the k -fold delooping of C ( I k × I m ) = F ( I k × I m )( ∗ ) ≃ E k + m ( ∗ ). Letus denote the E k -algebra F [ r ] ( I k × I m )( ∗ ) by C [ r ] ( I k × I m ). Theorem 5.2.
The k -fold delooping of C [ r ] ( I k × I m ) is homotopy equivalent to the r th stage in the May-Milgram filtration of Ω m Σ m S k . To prove this, we need to consider a generalization of C [ r ] ( I k × I m ) where pointscan vanish if they enter certain regions. Definition 5.3.
Let M be a manifold and N ⊂ M a subspace. Let C [ r ] ( M × I m )denote the subspace of C ( M × I m ) of configurations ξ where ξ ∩ ( { x } × I m ) hascardinality ≤ r for all x ∈ M . Let C [ r ] (( M, N ) × I m ) be the quotient of C [ r ] ( M × I m )by the equivalence relation that ξ ∼ ξ ′ if ξ ∩ (( M \ N ) × I m ) = ξ ′ ∩ ((( M \ N ) × I m ). HE MAY-MILGRAM FILTRATION AND E k -CELLS 21 We drop the superscript for r = ∞ and drop the − × I m for m = 0. There aretwo configuration space models for Ω m Σ m S k = Ω m S k + m . The first is a special caseof May’s approximation theorem from [May72], building on the work of Milgramin [Mil66], and the second is a specialization of Proposition 2 of [B¨od87]. Theorem 5.4 (May) . For k > , C ( I m ; S k ) is weakly homotopy equivalent to Ω m Σ m S k . Theorem 5.5 (B¨odigheimer) . For k > , C (( R k , R k \ I k ) × I m ) is weakly homotopyequivalent to Ω m Σ m S k . We will relate these two models of Ω m Σ m S k , and compare filtrations of thesespaces. The topological space C (( R k , R k \ I k ) × I m ) is filtered by the C [ r ] (( R k , R k \ I k ) × I m ). From now on, we view S k as R k / ( R k \ I k ) with base point given by theimage of R k \ I k . Define a map ρ by ρ : C ( I m ; S k ) −→ C (( R k , R k \ I k ) × I m )(( m ; x ) , . . . , ( m r , x r )) ( x × m , . . . , x r × m r ) , where m i ∈ I m and x i ∈ R k / ( R k \ I k ) = S k . This inclusion has image consistingof those configurations with at most one point in each fiber of R k × I m → I m . Wedenote its restrictions by ρ r : M r ( C ( I m ; S k )) → C [ r ] (( R k , R k \ I k ) × I m ). Lemma 5.6.
The maps ρ and ρ r are homotopy equivalences.Proof. The strategy is to scale the configurations so that in each fiber of R k × I m → I m all but at most one point is pushed into R k \ I k . To do so, we pick acontinuous function η : C (( R k , R k \ I k ) × I m ) → (0 , ∞ ) with the property that forall ξ ∈ C (( R k , R k \ I k ) × I m ) and x ∈ I m , there is at most one point in ξ that iswithin distance η ( ξ ) of 0 × x ∈ R k × I m . Let φ Rt : R k → R k be a continuous familyof maps, depending on t ∈ [0 ,
1] and
R >
0, such that: · φ R = id, · φ Rt | ( φ Rt ) − ( I k ) a homeomorphism onto its image, · φ Rt ( R k \ I k ) ⊂ R k \ I k and φ R ( y ) ∈ R k \ I k if || y || > R .Then we define H : [0 , × C (( R k , R k \ I k ) × I m ) −→ C (( R k , R k \ I k ) × I m )( t, ξ ) (id × φ η ( ξ ) t ) ∗ ( ξ ) , where the subscript ∗ means induced map on configuration spaces. For t = 1, allbut at most one point in each fiber are pushed into R k \ I k (where these pointsvanish). In particular, we can regard it as a continuous map h : C (( R k , R k \ I k ) × I m ) −→ C ( I m ; S k ) . The homotopy H then provides a homotopy from ρ ◦ h to the identity on C (( R k , R k \ I k ), and since it preserves the subspace C ( I m ; S k ) also a homotopy from h ◦ ρ tothe identity on C ( I m ; S k ). Thus ρ is a homotopy equivalence.Since ρ , h and H preserve the filtration, this also proves the ρ r are homotopyequivalences. (cid:3) Thus C [ r ] (( R k , R k \ I k ) × I m ) is homotopy equivalent to the r th stage of theMay-Milgram filtration. We claim that the k -fold delooping of C [ r ] ( I k × I m ) is C [ r ] (( R k , R k \ I k ) × I m ). The k -fold bar construction of an augmented E k -algebra is defined in full general-ity in Section 13.1 of [GKRW18]. We will specialize it to the E k -algebra C [ r ] ( I k × I m )in Top , with its canonical augmentation to ∗ , and make a minor modification to the“grids” for the sake of computational convenience, replacing [0 ,
1] with [1 / , /
4] inthe following definition:
Definition 5.7.
We write P k ( p , . . . , p k ) ⊂ Q j R p j +1 for the subspace of k -tuples { / < t j < · · · < t jp j < / } ≤ j ≤ k . We make [ p , . . . , p k ]
7→ P k ( p , . . . , p k ) into a k -fold semi-simplicial space by definingthe i th face map in the j th direction by forgetting t ji . Definition 5.8. B E k • , ··· , • ( C [ r ] ( I k × I m )) is the k -fold semi-simplicial space with( p , . . . , p k )-simplices given by the subspace of( { t ji } , ξ ) ∈ P k ( p , . . . , p k ) × C [ r ] ( I k × I m )such that ξ is contained in Q j [ t j , t jp j ] × I m and ξ is disjoint from [1 / , / j − ×{ t ji } × [1 / , / k − j × I m for 1 ≤ j ≤ k and 0 ≤ i ≤ p j .For 0 < i < p j , the i th face map in the j th direction is given by the correspondingface map on P k and the identity on C [ r ] ( I k × I m ). The 0th face map in the j thdirection is given by the corresponding face map on P k and by deleting all particlesin ξ which have j th coordinate < t j . Similarly, the p j th face map in the j thdirection is given by the corresponding face map on P k and by deleting all particlesin ξ which have j th coordinate > t jp j − . Definition 5.9.
The k -fold delooping of C r ( I k × I m ) is the pointed topologicalspace given by B k C [ r ] ( I k × I m ) := || B E k • , ··· , • ( C [ r ] ( I k × I m )) || . Combined with Lemma 5.6, the following proposition completes the proof ofTheorem 5.2.
Proposition 5.10.
There is a zig-zag of weak equivalences of E m -algebras B k C [ r ] ( I k × I m ) || f • || ←−−− || X • ,..., • || ǫ −→ C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) . Proof.
We start by defining the augmented k -fold semi-simplicial topological space X • , ··· , • : its topological space of ( p , . . . , p k )-simplices X p ,...,p k ⊂ P k ( p , . . . , p k ) × C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) is the subspace of ( { t ji } , ξ ) such that ξ is disjoint from [1 / , / j − × { t ji } × [1 / , / k − j × I m for each 1 ≤ j ≤ k and 0 ≤ i ≤ p j . This is augmented over C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) . The i th face map in the j th direction is given by forget-ting t ji and the augmentation forgets all t ji ’s.We denote the map || X • , ··· , • || −→ C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) by ǫ . To show this is a weak equivalence, we prove it is a microfibration withweakly contractible fibers and invoke Lemma 4.8. For ξ ∈ C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) ,let S jξ ⊆ R be the subspace of t ∈ (1 / , /
4) such that ξ ∩ (cid:0) [1 / , / j − × { t } × [1 / , / k − j × I m (cid:1) = ∅ . HE MAY-MILGRAM FILTRATION AND E k -CELLS 23 R k = R I k = I I m = I [1 / , / t t t ··•• ··• ··•••• Figure 3.
An element of X , for r = 4, k = 2, and m = 1. Points inthe configuration disappear when they leave the cube I × I , cannot hitthe walls { t i } × [1 / , / × I for i = 0 , / , / × { t } × I , andevery vertical line segment can contain at most 4 points. The fiber ǫ − ( ξ ) is the thick geometric realization of a k -fold semi-simplicial spacewith space of ( p , · · · , p k )-simplices homotopy equivalent to the product of sets oforder preserving-maps from { , . . . , p j } to π ( S jξ ) for 1 ≤ j ≤ k , which is productof simplices. Since levelwise weak equivalences induce weak equivalences on thickgeometric realizations (see e.g. Theorem 2.2 of [ERW19]), the fibers of ǫ are weaklycontractible.The proof that ǫ is a microfibration is similar to that of Lemma 4.10. The keyfact is that if ξ ∩ (cid:0) [1 / , / j − × { t j } × [1 / , / k − j × I m (cid:1) = ∅ , the same will be true for nearby configurations (this is why we use [1 / , /
4] insteadof I , otherwise new points could appear and hit the forbidden regions immediately).We will next construct a k -fold semi-simplicial map f • : X • ,..., • −→ B E k • , ··· , • ( C [ r ] ( I k × I m ))and prove its thick geometric realization is a weak equivalence. The map f p ,...,p k is defined on a ( p , . . . , p k )-simplex ( { t ji } , ξ ) by deleting from a configuration ξ ∈ C [ r ] (cid:0) ( R k , R k \ I k ) × I m (cid:1) those points outside Q kj =1 [ t j , t jp j ], and interpreting theremaining configuration as an element of C [ r ] (cid:0) I k × I m (cid:1) . Since we are taking thickgeometric realizations, to prove || f • || is a weak equivalence, it suffices to prove each f p , ··· ,p k is a weak homotopy equivalence. To prove this, we first observe that the inclusion of the subspace X ′ p ,...,p k of X p ,...,p k of those ( { t ji } , ξ ) such that for all 1 ≤ j ≤ k we have ξ ∩ (cid:16) [1 / , / j − × ([1 / , t j ] ∪ [ t jp j , / × [1 / , / k − j × I m (cid:17) = ∅ , is a homotopy equivalence. In other words, in X ′ p , ··· ,p k all points in ξ lie either in Q kj =1 [ t j , t jp j ] or have one of their first k coordinates < / > / g p , ··· ,p k : B E k p , ··· ,p k ( C [ r ] ( I k × I m )) −→ X ′ p ,...,p k , which regards a configuration in C [ r ] ( I k × I m ) as one in C [ r ] (( R k , R k \ I k ) × I m ), isa homotopy equivalence. Then the composition f p ,...,p k ◦ g p , ··· ,p k is the identityon B E k p , ··· ,p k ( C [ r ] ( I k × I m )). A homotopy from g p ,...,p k ◦ f p , ··· ,p k to the identity on X ′ p ,...,p k is given as follows: it is the identity on the points in ξ in Q kj =1 [ t j , t jp j ] andpushes the remaining points linearly outwards from (1 / , · · · , /
2) until all are inthe regions R k \ I k where they vanish. (cid:3) Remark . Snaith showed that the May-Milgram filtration stably splits [Sna74].However, its lift to a filtration of E k + m ( ∗ ) of E k -algebras does not split after takingsuspension spectra. Such a splitting would imply that C ( I × I ) ≃ R P stablysplits off a copy of F [1]2 ( I × I ) / S ≃ R P .However, this filtration does split after stabilizing in a different manner. Recall Q E k L denotes the derived indecomposables functor of Remark 3.7. Basterra-Mandellshowed that derived indecomposables can be considered as stabilization of an alge-bra over an operad [BM05], and derived indecomposables of an E k -algebra may becomputed using its k -fold bar construction, see [BM11] or Chapter 13 of [GKRW18].Thus the induced filtration of the stabilization Q E k L (Σ ∞ E k + m ( ∗ ) + ) agrees with thesuspension spectra of the May-Milgram filtration and hence splits by the work ofSnaith [Sna74]. References [BM05] M. Basterra and M. A. Mandell,
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