Symmetric Homology of Algebras

Symmetric Homology of Algebras S. Ault and Z. Fiedoro wicz August 11, 2007 Abstract In this note, w e outline the general deve lopment of a theory of symmet- ric homology of algebras, an an alog of cyclic homology w here the cyclic groups are replaced by symmetric groups . This theory is dev elop ed using the framework of crossed simplicial group s and the homological algebra of m o dule-v alued functors. The symmetric homology of group algebras is related to stable h omotop y theory . Two s p ectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discuss ed and some conjectures and questions to- w ards further work are discussed. 2000 MSC: 16E40, 55P45, 55S12 Symmetric ho mology is the analog of cyclic homology , w here the cyclic groups are replaced b y symmetric groups. The second author and Lo da y [6] dev el- op ed the no tion of cr osse d simplicia l gr oup as a fra mew ork for making this idea precise. Definition 1 A cr os s e d simplicial gr oup is a cat egory ∆ G whose ob jects are the sets [ n ] = { 0 , 1 , 2 , . . . , n } for n ≥ 0, whic h con tains the simplicial category ∆, a nd suc h tha t any morphism [ m ] → [ n ] factors uniquely as [ m ] ∼ = − → [ m ] γ − → [ n ] , where γ is a morphism in ∆. The collection of groups { G n = Aut ∆ G ([ n ]) op } n ≥ 0 are called the underlying gr oups of ∆ G . The comm utation relations implicit in ∆ G endow { G n } n ≥ 0 with the structure of a simplicial set (but not neces- sarily the structure of a simplicial g roup). 1 The standard w ell-known example of a crossed simplicial group is ∆ C , whose underlying groups are the cyclic g roups { Z n +1 } n ≥ 0 . Less w ell-know n is ∆ S , whose underlying gro ups a re the symmetric groups { Σ n +1 } n ≥ 0 . The descrip- tion of ∆ S giv en in [6] is difficult to w or k with. A m uch nicer construction of this category is due to Pirash vili [11 ]. Definition 2 The category ∆ S has ob jects [ n ] = { 0 , 1 , 2 , . . . , n } . A mor- phism f : [ m ] → [ n ] is a set function together with a specification of a total order on the p oint preimages { f − 1 ( i ) } 0 ≤ i ≤ n . Comp osition of mor- phisms [ m ] f − → [ n ] g − → [ p ] is giv en by sp ecifying the o rder on ( g f ) − 1 ( i ) = ∐ j ∈ g − 1 ( i ) f − 1 ( j ), a s the blo c k ordering specified by the ordering on g − 1 ( i ) and then within eac h blo ck b y the ordering sp ecified on eac h f − 1 ( j ). An y mor- phism f : [ m ] → [ n ] decomposes uniquely as the p erm utation on [ m ] sp ecified b y ∐ 0 ≤ i ≤ n f − 1 ( i ) follow ed b y an or der preserving function [ m ] → [ n ], whic h is th us in ∆. The cyclic category ∆ C is the eviden t sub category of ∆ S . Remark Pirash vili’s construction is a special case of a mor e general con- struction due to May and Thomason [10]. T his construction asso ciates t o an y top olo gical op erad {C ( k ) } n ≥ 0 a top ological category b C together with a functor from this catego ry to finite sets suc h that the inv erse image of an y function f : [ m ] → [ n ] is the space Q n i =0 C ( | f − 1 ( i ) | ). Composition in b C is defined using the comp osition o f the op erad. Ma y a nd Thomason refer to b C as the c ate gory of op er ators asso ciated to C . They w ere in terested in the case of an E ∞ op erad, but their construction eviden tly works for any operad. The category of op erators asso ciated to the discrete A ∞ op erad A ss , whic h parametrizes monoid structures, is precisely Pirash vili’s construction of ∆ S . No w g iv en any small catego ry C and a n y comm uta tiv e ring k , one can define homological algebra of cov arian t and contra v ar ian t functors F : C − → k -mo dules. The simplest w ay to describ e this is to consider the ring k [ C ], whic h is the free k -mo dule generated b y all the morphisms in C . Multiplication is defined on this basis by comp osition if the morphisms are comp osable and 0 other- wise. A co v aria n t functor F : C − → k - mo dules is the n exactly the same thing as a left k [ C ]-mo dule structure on L C ∈ Ob j ( C ) F ( C ). Similarly contra v aria n t functors corresp ond to right k [ C ]- mo dules (equiv a len tly , left k [ C op ]-mo dules). 2 One then defines f or con trav arian t F and co v ar ian t G , T or C ∗ ( F , G ) = T or k [ C ] ∗   M C ∈ Ob j ( C ) F ( C ) , M C ∈ Ob j ( C ) G ( C )   (There are some small technic alit ies that need to b e c hec ke d, as the ring k [ C ] do es not ha v e a multiplicativ e unit if C has infinitely man y ob jects. But it do es hav e lo cal units whic h are sufficien t to carry this out.) If A is a k -algebra, t hen the cyclic bar construction define s a functor B cy c A : ∆ C op − → k -mo dules, and cyclic ho mology can b e defined as H C ∗ ( A ) = T or ∆ C ∗ ( B cy c A, k ) , where k : ∆ C − → k - mo dules denotes the trivial functor whic h tak es ev ery ob ject to k and ev ery morphism to the identit y . Ho w ev er the results of [6] were discouraging as to the prosp ect of an ana logous definition of symmetric homology . First of a ll, the cyclic bar construction do es not extend to a functor ∆ S op − → k - mo dules. Secondly it w as sho wn that for an y functor F : ∆ S op − → k -mo dules, T or ∆ S ( F , k ) is just the ho- mology of the underlying simplicial mo dule of F , giv en b y restricting F to ∆ op . Subsequen tly , the second a uthor [5] noticed that the cyclic bar construction extends not to a con trav arian t functor on ∆ S but to a cov a rian t functor. Definition 3 The symmetric b ar c onstruction is the functor B sy m A : ∆ S − → k -mo dules whic h takes the ob ject [ n ] t o the ( n + 1)- fold tensor pro duct A ⊗ n +1 of A with itself ov er k . If f : [ m ] → [ n ] is a morphism in ∆ S , then B sy m ( f ) tak es a 0 ⊗ a 1 ⊗ a 2 ⊗ . . . ⊗ a m to b 0 ⊗ b 1 ⊗ b 2 ⊗ . . . ⊗ b n , where b i = Q j ∈ f − 1 ( i ) a j , where the pro duct is ta k en in the order sp ecified on f − 1 ( i ). The cyclic bar construction can b e iden tified with the comp osite ∆ C op D ∼ = ∆ C ⊂ ∆ S B sy m − → k -mo dules , where D is a suitable duality isomorphism. This now allo ws us to define symmetric homology as Definition 4 H S ∗ ( A ) = T or ∆ S ∗ ( k , B sy m A ) , where k : ∆ S op − → k -mo dules denotes the trivial functor whic h take s ev ery ob ject to k a nd ev ery morphism to the iden tity . 3 One can use the standard bar resolution of k to compute H S ∗ ( A ) as the homology of the simplicial ab elian group L ∗ ( A ), where L p ( A ) = M k h [ m 0 ] f 1 − → [ m 1 ] f 2 − → [ m 2 ] f 3 − → . . . f p − → [ m p ] i ⊗ A ⊗ m 0 +1 . Here the direct sum ranges ov er all comp osable chains o f morphisms in ∆ S of length p . The 0-th fa ce consists of deleting f 1 and acting on A ⊗ m 0 +1 via B sy m ( f 1 ). The p -th face consists of dropping f p . The o ther faces are giv en b y comp o sing f i +1 with f i . The degeneracies consist of inserting iden tities. If A = k [ M ] is a monoid ring, then H S ∗ ( A ) has a simpler des cription: it is the homology with k -co efficien ts of the nerv e o f the category whose set of ob j ects is the disjoin t union ∐ n ≥ 0 M n +1 . A morphism from ( m 0 , m 1 , m 2 , . . . , m p ) ∈ M p +1 to ( m ′ 0 , m ′ 1 , m ′ 2 , . . . , m ′ q ) ∈ M q +1 is a morphism f : [ p ] → [ q ] in ∆ S , suc h that m ′ i = Q j ∈ f − 1 ( i ) m j . In the sp ecial case when M = J ( X + ) is the free monoid on a generating set X ( for whic h w e ha ve A = k [ M ] = T ( X ), the tensor algebra o n X ) w e ha v e the following result. Theorem 1 H S ∗ ( T ( X )) = H S ∗ ( k [ J ( X + )]) = H ∗ ( C ∞ ( X + ); k ) , wher e C ∞ denotes the monad asso ciate d to the little ∞ -cub es op er ad ([8], [4 ] ) . W e ma y replace C ∞ ab ov e by the monad asso ciated to an y E ∞ op erad. In particular it is preferable to use the monad asso ciated to the op erad D (see Theorem 3 b elo w). If the monoid is a group G , we ha v e the fo llo wing result. Theorem 2 H S ∗ ( k [ G ]) = H ∗ (ΩΩ ∞ S ∞ ( B G ); k ) The sp ecial case when G is free ab elian o f rank n is of particular interest. In this case the group ring is the ring of Lauren t p o lynomials in n indetermi- nates. On the o ther hand B G is the n -torus whic h stably splits into a w edge of spheres. Thus w e o btain Corollary 1 H S ∗  k [ t ± 1 , t ± 2 , . . . , t ± n ]  = H ∗ (ΩΩ ∞ S ∞ ( W n i =1 W n ! / ( i !( n − i )!) j =1 S i ); k ) = H ∗ ( Q n i =1 Q n ! / ( i !( n − i )!) j =1 Ω ∞ S ∞ ( S i − 1 ); k ) Since the symmetric homology of t he group completion of a comm utativ e monoid is the gr oup completion of the symmetric homology o f the monoid, a natural conjecture w ould b e 4 Conjecture 1 H S ∗ ( k [ t 1 , t 2 , . . . , t n ]) = H ∗ ( Q n i =1 C ∞ ( S 0 ) × Q n i =2 Q n ! / ( i !( n − i )!) j =1 Ω ∞ S ∞ ( S i − 1 ); k ) In the case n = 1 , this conjecture is a sp ecial case of Theorem 1. The E ∞ structure visible in the ab ov e examples is a general phenomenon presen t in H S ∗ ( A ) for any algebra. In order to mak e this precise, we need to enlarge the categor y ∆ S by adding an initial ob ject [ − 1]. Call the resulting enlarged category ∆ S + , and let L + ∗ ( A ) b e the resulting enlarged bar com- plex. Then ∆ S + is a strict symmetric monoidal cat egory ( with the monoida l structure give n b y the copro duct) and w e ha ve Theorem 3 (a) H S ∗ ( A ) = H ∗ ( L + ∗ ( A )) (b) L + ∗ ( A ) is an E ∞ chain c omplex w i th r esp e ct to the action of the E ∞ op e r ad D . (c) If k = Z p , p a prime, then H S ∗ ( A ) is e quipp e d with Dyer-L ashof homolo gy op e r ation s . The E ∞ c hain op erad whic h acts on L + ∗ ( A ) is the c hain op erad asso ciated to the op erad D = { E Σ n } n ≥ 0 whic h acts on strict symmetric monoidal (a.k.a. p erm utative) categories [9]. This op erad, in its simplicial form, is kno wn as the Barratt- Eccles op erad. The follo wing related result is join t w ork with T omas Ba rros. Theorem 4 (a) The chain c omple x L + ∗ ( A ) is e quipp e d with a Smith filtr ation ([3], [14]). The n -stage of this filtr ation is a n E n chain c omplex. (b) If A = k [ G ] is a gr oup ring, then the homolo gy of the n -stage of the Sm i th filtr ation on L + ∗ ( A ) is isom orphic to H ∗ (Ω n S n − 1 ( B G ); k ) . While the chain complex L ∗ ( A ) fortuitously lends itself to computations of H S ∗ ( A ) in the sp ecial cases of the monoid rings of free monoids and g roup rings, it is m uc h to o un wieldy f or computations in general, as it is infinite dimensional in eac h degree. As a first step in obtaining a more tractable c hain complex, w e ha v e Prop osition 1 If A is e q uipp e d with an a ugm entation A → k a n d I de- notes the augmen tation ide al, then the inclusion L epi ∗ ( A ) ⊂ L ∗ ( A ) is a chain homotopy e q uiva l e nc e, wh er e L epi p ( A ) = M k  [ m 0 ] f 1 ։ [ m 1 ] f 2 ։ [ m 2 ] f 3 ։ . . . f p ։ [ m p ]  ⊗ B sy m m 0 I , 5 for p > 0 , wher e the f i ar e r e quir e d to b e ep i m orphisms. Her e B sy m m I =  A if m = 0 I ⊗ m +1 if m > 0 Thus H S ∗ ( A ) = H ∗ ( L epi ∗ ( A )) . The c hain complex L epi ∗ ( A ) in turn can b e filtered in a couple of wa ys, giving rise to sp ectral sequences for computing H S ∗ ( A ). The simplest such sp ectral sequence arises by filtering L epi ∗ ( A ) by the n umber of jumps: the n - th filtra- tion o f L epi ∗ ( A ) consists of c hains where at most n of the [ m i − 1 ] f i ։ [ m i ] are strict (i.e. m i − 1 > m i ). W e obtain Theorem 5 I f A is e quipp e d with an augmentation with augmentation ide al I , then ther e is a first quadr ant sp e ctr al se quenc e c o nver ging to H S ∗ ( A ) with E 1 p,q = M m 0 >m 1 >m 2 >...>m p ≥ 0 H q Σ op m p +1 ; B sy m m 0 I ⊗ k " p Y i =1 Epi ∆ ([ m i − 1 ] , [ m i ]) #! Here Epi ∆ ([ m ] , [ n ]) denotes the set of epimorphisms in ∆ b etw een [ m ] and [ n ]. The group homolo gy is defined with r esp ect to the group righ t action of Σ m p +1 giv en by the isomorphism B sy m m 0 I ⊗ k " p Y i =1 Epi ∆ ([ m i − 1 ] , [ m i ]) # ∼ = B sy m m 0 I ⊗ k [Σ m 0 +1 ] k [Epi ∆ S ([ m 0 ] , [ m 1 ])] ⊗ k [Σ m 1 +1 ] k [Epi ∆ S ([ m 1 ] , [ m 2 ])] ⊗ k [Σ m 2 +1 ] . . . ⊗ k [Σ m p − 1 +1 ] k [Epi ∆ S ([ m p − 1 ] , [ m p ])] . Here, the B sy m m 0 I comp onen t comes b efore the c hain of morphisms b ecause w e are viewing it as a right k Σ m 0 +1 -mo dule rather than a left k Σ op m 0 +1 -mo dule. The differen tial E 1 p,q → E 1 p − 1 ,q is an a lternating sum of faces. The 0-th face tak es a 0 ⊗ a 1 ⊗ . . . a m 0 ⊗  [ m 0 ] f 1 ։ [ m 1 ] f 2 ։ [ m 2 ] f 3 ։ . . . f p ։ [ m p ]  to B sy m ( f 1 )( a 0 ⊗ a 1 ⊗ . . . a m 0 ) ⊗  [ m 1 ] f 2 ։ [ m 2 ] f 3 ։ . . . f p ։ [ m p ]  . 6 The middle faces comp ose consecutiv e arrows . The last face is induced b y f ∗ p : Σ m p +1 → Σ m p − 1 +1 , whic h is part of the simplicial structure on the underlying groups { Σ n +1 } n ≥ 0 of ∆ S . No w, since the differen tial of L epi ∗ ( A ) reduces t he filtra tion degree by at most one, it can b e sho wn that the differen tials E r p,q → E r p − r,q + r − 1 m ust b e trivial for r ≥ 2. Hence, the sp ectral sequence collapses at the E 2 term. This spectral sequence is still not very computationally useful as the E 1 -term is infinitely generated in each degree. A b etter sp ectral sequenc e is obt ained b y filtering L epi ∗ ( A ) as follows : F m L epi p ( A ) = M m 0 ≤ m B sy m m 0 I ⊗ k  [ m 0 ] f 1 ։ [ m 1 ] f 2 ։ [ m 2 ] f 3 ։ . . . f p ։ [ m p ]  , W e obta in the following result. Theorem 6 I f A is e quipp e d with an augmentation whose augm entation ide al I is a fr e e k -m o dule with b asis X , then ther e is a sp e ctr al se quenc e c on ver ging str ongly to H S ∗ ( A ) with E 1 p,q = M u ∈ X p +1 / Σ p +1 e H p + q ( E G u ⋉ G u N S p / N S ′ p ; k ) Here G u is the isotropy subgroup of the orbit u ∈ X p +1 / Σ p +1 . N S p is the nerv e of the category S p , whic h is defined as follows. Let { z 0 , z 1 , z 2 , . . . } b e a coun table set of indeterminates. First w e define a larger category ˜ S p . The ob jects of ˜ S p are formal tensor pro ducts Z 0 ⊗ Z 1 ⊗ . . . ⊗ Z r where eac h Z i is a for mal (nonempty ) pro duct of t he indeterminates { z 0 , z 1 , . . . , z p } so that Z 0 Z 1 . . . Z r = z σ (0) z σ (1) . . . z σ ( p ) for some σ ∈ Σ p +1 . In other words eac h z i , i = 0 , 1 , 2 , . . . , p o ccurs once and only once a s a factor in exactly one of the pro ducts Z j , j = 0 , 1 , 2 , . . . , r . The re is precisely one morphism in ˜ S p Z 0 ⊗ Z 1 ⊗ . . . ⊗ Z r − → Y 0 ⊗ Y 1 ⊗ . . . ⊗ Y s iff eac h Y i is a pro duct of some of the monomials Z j ’s. W e then ta k e S p to b e a ske letal sub category of ˜ S p . S p is a p oset. The nerv e N S p is contractible, since S p con tains the initial ob ject z 0 ⊗ z 1 ⊗ . . . ⊗ z p . W e then take S ′ p to b e the subp oset obtained fro m S p b y deleting the initia l ob j ect. Thus the quotien t N S p / N S ′ p has the same homotop y t yp e as the susp ension of N S ′ p . The symmetric g roup Σ p +1 acts on S p b y p erm uting the generators { z 0 , z 1 , z 2 , . . . , z p } . This induces an action on N S p / N S ′ p . The differen tial E 1 p,q − → E 1 p − 1 ,q is induced b y the 0-th face map in N S p . 7 Th us a fundamen tal problem in computing symmetric homology is to deter- mine the homotopy t yp e of the spaces N S p / N S ′ p and to analyze the actions of t he symmetric gr oups on these spaces. If k is a field o f characterisitic 0, just knowing the ra tional homolo gy of these spaces and the action o f the symmetric g roups on the ho mology would suffice to determine the E 1 -term of the sp ectral sequence of Theorem 6. Ho w ev er the chain complex of the simplicial nerv e of N S p / N S ′ p is to o bulky to permit computations except for v ery small v alues of p . One can apply a similar techniq ue, as is used to derive Theorem 5 , to the nerv e of the nonsk eletal category ˜ S p to obtain a muc h smaller chain complex S y m ( p ) ∗ , whic h computes the homology of N S p / N S ′ p . The group of i -c hains S y m ( p ) i is the free ab elian group on the ob j ects of ˜ S p ha ving the form Z 0 ⊗ Z 1 ⊗ Z 2 ⊗ . . . ⊗ Z p − i , mo dded out by the equiv alence relation generated by Z 0 ⊗ Z 1 ⊗ . . . ⊗ Z j − 1 ⊗ Z j ⊗ . . . ⊗ Z p − i = ( − 1) ( | Z j − 1 | +1)( | Z j | +1) Z 0 ⊗ Z 1 ⊗ . . . ⊗ Z j ⊗ Z j − 1 ⊗ . . . ⊗ Z p − i where | Z | denotes t he length of the pro duct. The b oundary map in S y m ( p ) ∗ is an alternating sum of fa ces, where eac h face consists of splitting a pro duct Z j in to a tensor pro duct Z ′ j ⊗ Z ′′ j (so that Z j = Z ′ j Z ′′ j and the faces are ordered according to the p osition of the new ⊗ . F or example ∂ ( z 2 z 0 z 3 ⊗ z 1 z 4 ) = z 2 ⊗ z 0 z 3 ⊗ z 1 z 4 − z 2 z 0 ⊗ z 3 ⊗ z 1 z 4 + z 2 z 0 z 3 ⊗ z 1 ⊗ z 4 The action of Σ p +1 on S y m ( p ) ∗ is induced b y p erm utation of the generators { z 0 , z 1 , z 2 , . . . , z p } . The dir ect sum L p ≥ 0 S y m ( p ) ∗ forms a bigraded differen tial algebra, where S y m ( p ) i is assigned bigra ding ( p + 1 , i ). The pro duct ⊠ : S y m ( p ) i ⊗ S y m ( q ) j − → S y m ( p + q +1) i + j is giv en by Y ⊠ Z = Y ⊗ Z ′ , where Z ′ is obained fro m Z b y replacing each generator z r b y z r + p +1 for r = 0 , 1 , 2 , . . . , q . The pro duct is related to the b oundary map b y the the relation ∂ ( Y ⊠ Z ) = ∂ ( Y ) ⊠ Z + ( − 1) i Y ⊠ ∂ ( Z ) , when Y has bigrade ( p + 1 , i ). Th us there is an induced map in homology: ⊠ : H i ( S y m ( p ) ∗ ) ⊗ H j ( S y m ( q ) ∗ ) − → H i + j ( S y m ( p + q +1) ∗ ) 8 The product ⊠ , b oth o n the c hain lev el and the homolo gy lev el, is not strictly sk ew comm uta tiv e, but r ather sk ew comm utativ e in a t wisted sense: Y ⊠ Z = ( − 1) ij σ Z ⊠ Y where σ is the p erm utation whic h sends 0 , 1 , 2 , . . . q to p + 1 , p + 2 , . . . , p + q + 1 and q + 1 , q + 2 , . . . , p + q + 1 to 0 , 1 , 2 , . . . , p in an order preserving w a y . It is easy to compute the to p degree homology groups. Let b p = z 0 z 1 z 2 . . . z p + ( − 1) p z 1 z 2 . . . z p z 0 + ( − 1) 2 p z 2 z 3 . . . z p z 0 z 1 + . . . + ( − 1) p 2 z p z 0 z 1 z 2 . . . z p − 1 . Then b p is a cycle and th us a homology class. As a Z [Σ p +1 ]-mo dule, H p ( S y m ( p ) ∗ ) is g enerated b y b p and as a represen tation H p ( S y m ( p ) ∗ ) is either the sign rep- resen tation on Z p +1 (if p is o dd) or the trivial represen tation on Z p +1 (if p is ev en), induced up to Σ p +1 . Th us H p ( S y m ( p ) ∗ ) is free ab elian o f rank p !. W e summarize our calculations so far b elow. Theorem 7 F or p = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 H ∗ ( S y m ( p ) ∗ ) ar e fr e e ab elian and have the fol low ing Poinc ar´ e p olynomials: p 0 ( t ) = 1 , p 1 ( t ) = t, p 2 ( t ) = t + 2 t 2 , p 3 ( t ) = 7 t 2 + 6 t 3 , p 4 ( t ) = 43 t 3 + 24 t 4 , p 5 ( t ) = t 3 + 272 t 4 + 120 t 5 , p 6 ( t ) = 36 t 4 + 184 7 t 5 + 720 t 6 , p 7 ( t ) = 829 t 5 + 137 10 t 6 + 504 0 t 7 Ideally we would lik e to describ e generators and relations for L p ≥ 0 H ∗ ( S y m ( p ) ∗ ) with r esp ect to the mo dule structures o v er the group rings of the symme tr ic groups and the ⊠ pro duct. The calculatio ns summarized ab ov e sho w that b esides the generators b i constructed ab ov e, there ar e additio nal generators in H 3 ( S y m (4) ∗ ), H 4 ( S y m (5) ∗ ), H 5 ( S y m (6) ∗ ), and H 6 ( S y m (7) ∗ ). F or now w e only ha v e v ery limited understanding of these additional generators or of the r e- lations b et w een t he generators. F or instance we ha ve the fo llo wing relation in H 2 ( S y m (3) ∗ ) b 1 ⊠ b 1 = (1 + [0312] + [1230]) b 2 ⊠ b 0 , where [ abcd ] stands for the p erm utation 0 7→ a, 1 7→ b, 2 7→ c, 3 7→ d . The calculations a lso establish that N S p / N S ′ p has the homotopy type of a we dge of spheres for p ≤ 6. 9 In recen t work [15] V re ´ cica and ˇ Ziv alj evi ´ c ha v e connected S y m ( p ) ∗ to a certain w ell-studied class of geometric complexes, know n as c hessb oard complexes [16]. Using this they ha ve sho wn tha t S y m ( p ) ∗ is ⌊ 2 3 ( p − 1) ⌋ -connected. This result implies that the connectivit y of the spaces N S p / N S ′ p is an increasing function o f p , hence the sp ectral sequence of Theorem 6 conv erges in the strong sense. Indeed, for m > 3 2 ( i + 1), there is an isomorphism H i ( F m L epi p ( A )) ∼ = − → H S i ( A ) If we denote by ∆ ( m ) S the full sub category of ∆ S consisting o f the ob jects [0] , [1] , . . . , [ m ], then it follo ws that for m > 3 2 ( i + 1), H S i ( A ) = T or ∆ ( m ) S i ( k , B sy m A ) . Observ e that k  ∆ ( m ) S  is a finite-dimensional unital ring, hence if A is finitely generated ov er a No etherian ground ring k , then the increasing con- nectivit y of the spaces N S p / N S ′ p implies that H S ∗ ( A ) is finite dimensional o v er k in eac h degree. In the case when A = k [ G ] is the group ring of a fi- nite gr oup, this also follows from Theorem 2, the Atiy a h-Hirzebruc h sp ectral sequence for stable homot op y theory and Serre C - theory . Some questions a re suggested b y our part ial computations o f H ∗ ( S y m ( p ) ∗ ): Is it true that the homology is alw ays torsion-fr ee? Or mig h t it ev en b e true that the spaces N S p / N S ′ p are alw a ys w edges of sp heres? Can the V re ´ cica and ˇ Ziv alj evi ´ c connectivit y result b e improv ed to H i ( S y m ( p ) ∗ ) = 0 for i ≤ p − r , where r =  √ 8 p + 9 − 1 2  ≈ p 2 p ? If this w ere true, this would b e the b est p ossible connectivit y result, since the sign r epresen tation of Σ p +1 has nontrivial multiplicit y in all H i ( S y m ( p ) ∗ ) for p − r < i ≤ p . The computed multiplicities of the trivial represen tations are also consisten t with this h yp othesis. W e also ha v e the following results on symmetric homology in degrees 0 and 1. Prop osition 2 (a) H S i ( A ) for i = 0 , 1 is the homolo gy of the fol lowing p artial cha i n c omplex 0 ← − A ∂ 1 ← − A ⊗ A ⊗ A ∂ 2 ← − ( A ⊗ A ⊗ A ⊗ A ) ⊕ A 10 wher e ∂ 1 ( a ⊗ b ⊗ c ) = abc − cba ∂ 2 ( a ⊗ b ⊗ c ⊗ d ) = ab ⊗ c ⊗ d + d ⊗ ca ⊗ b + bca ⊗ 1 ⊗ d + d ⊗ bc ⊗ a, ∂ 2 ( a ) = 1 ⊗ a ⊗ 1 (b) H S 0 ( A ) = A/ [ A, A ] is the symmetrization of A (as an algebr a). W e also hav e an elab o ration o f Theorem 1, whic h describ es symmetric ho- mology as the ho mology of the E ∞ symmetrization of an algebra. The idea is to simplicially resolv e the algebra b y tensor alg ebras, then in each simplicial degree replace the tensor algebra by a free E ∞ c hain algebra o n the same generators, and finally to t ak e the double complex of the resulting simplicial c hain alg ebra. A more precise formu la tion is Theorem 8 H S ∗ ( A ) = H ∗ ( B ( D , T , A )) , whe r e B ( D , T , A ) is the 2-side d b ar c on struction, T is the functor whic h takes a k -mo dule to the tensor algebr a on that mo dule, D is the m o nad which takes a k -mo dule to the fr e e D chain algebr a over that mo dule (wher e D is the same op er ad as in The or em 3) , and B ( D , T , A ) is c onverte d fr om a simplicia l chai n c omplex to a double c omplex. Finally w e briefly discus s the relation b etw een symmetric homology and cyclic homolo gy . The relation b et we en t he cyclic bar construction and the symmetric bar construction, discussed ab o ve, leads to a na tural map H C ∗ ( A ) − → H S ∗ ( A ) . The same ana lysis a s in Theorems 5 and 6 can b e carried out fo r cyclic h o mol- ogy . The cyclic analog of N S ′ p can be iden tified as a simplicial complex with the barycen tric subdivision of the b oundary of a p -simplex. The cyclic group acts on this p − 1 sphere b y cyclicly p ermuting the ve rtices of the simplex. The cyclic analog of N S p / N S ′ p is homotop y equiv alen t t o the susp ension of this and is thus a p -sphere. One can then combine the cyclic analog of The- orem 6 with the Serre sp ectral sequenc e fo r computing the homology of the resulting half-smash pro ducts to o btain the standard sp ectral sequence f or cyclic homology . W e can use the partia l chain complex of Prop o sition 3 and an analogous one fo r cyclic homology (c.f. [7 ], page 59) to describ e the map H C i ( A ) − → H S i ( A ) for i = 0 , 1. These maps are induced by the fo llo wing par tial c hain map: 11 0 A A ⊗ A A ⊗ 3 ⊕ A 0 A A ⊗ 3 A ⊗ 4 ⊕ A o o   id o o ab − ba   a ⊗ b ⊗ 1 o o ∂ C 2   f o o o o abc − cba o o ∂ S 2 The map ∂ C 2 tak es a ⊗ b ⊗ c ∈ A ⊗ 3 to ab ⊗ c − a ⊗ bc + ca ⊗ b , and tak es a ∈ A to 1 ⊗ a − a ⊗ 1. The map ∂ S 2 is the map ∂ 2 from Prop osition 3. f is a map that is defined o n the first summand by a ⊗ b ⊗ c 7→ a ⊗ b ⊗ c ⊗ 1 − 1 ⊗ a ⊗ bc ⊗ 1 + 1 ⊗ ca ⊗ b ⊗ 1 +1 ⊗ 1 ⊗ abc ⊗ 1 − b ⊗ ca ⊗ 1 ⊗ 1 − 2 abc − cab and on the second summand by a 7→ 4 a − 1 ⊗ 1 ⊗ a ⊗ 1 The map H C 0 ( A ) − → H S 0 ( A ) is the quotient map whic h tak es the quotien t of A b y the k -mo dule generated b y all comm utators on to t he quotien t of A b y the ideal generated b y all comm utators. In a similar vein, Pirash vili and Ric hter (c.f. [12] and [13]) hav e show n that H C ∗ ( A ) = T or ∆ S ∗ ( b , B sy m A ) , where b is the contra v ariant functor on ∆ S whic h is the cok ernel of d 0 − d 1 : P 1 − → P 0 , where P i ( − ) = k [ hom ∆ S ( − , [ i ])] a nd d 0 − d 1 induces the comm u- tator map a ⊗ b 7→ ab − ba on B S y m A . Th us the natural map H C ∗ ( A ) − → H S ∗ ( A ) is induced by the unique natural transformation b → k . 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