FI-modules and the cohomology of modular representations of symmetric groups

Transcription

1 FI-modules and the cohomology of modula epesentations of symmetic goups axiv: v1 [math.rt] 16 May 2015 Rohit Nagpal Abstact An FI-module V ove a commutative ing K encodes a sequence (V n ) n0 of epesentations of the symmetic goups (S n ) n0 ove K. In this pape, we show that fo a "finitely geneated" FI-module V ove a field of chaacteistic p, the cohomology goups H t (S n,v n ) ae eventually peiodic in n. We descibe a ecusive way to calculate the peiod and the peiodicity ange and show that the peiod is always a powe of p. As an application, we show that if M is a compact, connected, oiented manifold of dimension 2 and conf n (M) is the configuation space of unodeed n-tuples of distinct points in M then the mod-p cohomology goups H t (conf n (M),K) ae eventually peiodic in n with peiod a powe of p. Contents 1 Intoduction An oveview of Chuch-Ellenbeg-Fab s theoy of FI-modules Intoduction to -filteed FI-modules Statements of the main theoems fo -filteed FI-modules Statements of the main theoem and its genealizations Statements of the esults on unodeed configuation spaces Theoy of -filteed FI-modules ove Noetheian ings 16 3 Peiodicity of invaiants Peliminaies The nice lift constuction Constuction of a nice lift of a twisted cycle Combinatoial lemmas and the definition of peiodicity Peiodicity of the nice lift constuction

2 3.5 Well-definedness and commutativity esults Poof of the main theoem fo -filteed FI-modules Poof of the main theoem fo finitely geneated FI-modules Genealizations: complexes of finitely geneated FI-modules mod-p cohomology of unodeed configuation spaces 55 5 Futhe questions and comments 57 1 Intoduction Nakaoka showed in his 1960 pape [Nak] that the cohomology goups H t (S n,v) stablizes in n when V is a finitely geneated abelian goup with a tivial action of S n. Moe pecisely he showed that H t (S n,v) = H t (S n 1,V), if n > 2t. (1) Immediately aftewads, A. Dold published his pape [Do] significantly simplifying Nakaoka s aguments. Ove the next two decades the cohomology goups H t (S n,z/pz) wee of geat eseach inteest. Nakaoka and othes studied theses goups extensively detemining the stuctue of the goups and the cohomology opeations on them; see [Nak1], [Nak2], [Mi], [May], [Man]. It is natual to ask what happens if we allow V to vay with n o let S n act nontivially on V. But, to make sense out of this we need coheent sequences of S n -epesentations. The coheent sequences of S n -epesentations wee fist obseved as a ubiquitously occuing phenomena by Chuch and Fab in [CF]. Soon afte, the theoy of FI-modules was intoduced and developed by Chuch, Ellenbeg and Fab in [CEF] to study such sequences. The notion of coheence of such a sequence was made pecise by the natual notion of finite geneation of an FI-module. In [CEF], it was shown that the finitely geneated FI-modules ove a chaacteistic 0 field fom an abelian categoy and seveal new theoems in topology, algeba, combinatoics and algebaic geomety wee poved as an application. Almost at the same time, Sam and Snowden studied a categoy equivalent to the categoy of finitely geneated FI-modules in [SS] and povided a detailed analysis of the algebaic stuctue of the categoy in chaacteistic 0. In [CEFN], Chuch-Ellenbeg-Fab and the cuent autho studied the categoy of finitely geneated FI-modules ove a geneal Noetheian ing showing that the categoy is abelian and that thee is an inductive desciption of a finitely geneated FI-module V = (V n ) n0 : thee exists some N 0 such that if n N then V n can be descibed in tems of V j, j N. Such an inductive desciption had appeaed befoe in chaacteistic 0 (see Putman s pape [Pu]) 2

3 and has seveal applications. In this pape, the techniques intoduced in [CEFN] ae used extensively. Recently, FI-modules and the elated theoies have been vey fuitful. Chuch-Ellenbeg- Fab has elated it to the point counts on vaieties defined ove finite fields via Gothendieck- Lefschetz fixed point theoem ([CEF2]). Related theoy of FI-goups has been developed by Chuch and Putman in [CP] to obtain esults about the Johnson filtation on the mapping class goups. The theoy has also been genealized to coheent sequences of epesentations of classical Weyl goups by Wilson which yields anothe set of applications ([Wi]). In this pape, we genealize Nakaoka s stability theoem and show that the cohomology goups H t (S n,v n ), whee the sequence of S n -epesentations (V n ) n0 is given by a finitely geneated FI-module V, ae eventually peiodic in n. A map f : U V of finitely geneated FI-modules detemine S n -equivaiant maps f n : U n V n fo each n 0. We show that kenel and image of f,n : H t (S n,u n ) H t (S n,v n ) ae eventually peiodic in n. Similaly, if 0 U V W 0 is an exact sequence of finitely geneated FI-modules then the kenel and the image of the connecting homomophisms in the cohomology long exact sequence ae eventually peiodic in n. The simplest example whee peiodicity can be obseved is constucted as follows. Woking ove F 2, let (V n ) n0 be the sequence consisting of pemutation epesentations (V n = (F 2 ) n ) and (U n ) n0 be the sequence consisting of tivial epesentations (U n = F 2 ). These sequences fom finitely geneated FI-modules V and U espectively and the map φ : V U given by n φ n (a 1,a 2,...,a n ) = n is an FI-module homomophism. Now W := keφ = (keφ n ) n0 foms an FI-module with H 0 (S n,w n ) = W S n n which is nontivial only when n 2 and 2 n. We emak that ou esults do not povide anothe poof of Nakaoka s theoem. Rathe we build on Nakaoka s theoem addessing a seemingly othogonal set of difficulties. Unlike Nakaoka s stability theoem ou peiodicity esult is nontivial even in the case when t = 0 (see example 3.35). As a diect application to ou esults, we show that the mod-p cohomology of the unodeed configuation space of a compact, connected and oiented manifold of dimension 2 is peiodic. The pecise statement and a shot histoy of elated woks on configuation spaces is povided in 1.5. Related wok on othe categoies. Paallel theoies fo the twisted cohomology (o homology) of seveal diffeent sequences of goups exist o ae being woked out. Some examples i=1 a i 3

4 include: cohomology of fee goups via polynomial functos [DPV]; ational cohomology of classical goups via polynomial functos [To], [Ku]; homology goups H t (GL n (R),T(Ad n (R)) whee T: Z-Mod Z-Mod is a functo of finite degee [Dw]; homology goups H t (S n,v n ) whee V is a finitely geneated FI-module [Wa] (note hee that the univesal coefficient theoem cannot be used to deduce the cohomology vesion fom the homology vesion and vice-vesa because the coefficients ae nontivial epesentations of S n ); stability of twisted homology goups H t (SL n (R),M n ) and H t (Sp 2n (R),M n ) whee R is a finite ing [PuSa]. A geneal twisted homological stability esult fo sequences of epesentations of weath poduct goups G S n, whee G is a polycyclic-by-finite goup, is poved in [SS4]. Andew Snowden and the cuent autho ae developing a cohomological vesion fo sequences of epesentations of the goups G S n [NS]. In paticula, when G is the tivial goup, [NS] povides an altenate poof of peiodicity of the cohomology goups H t (S n,v n ) with bette bounds on the peiodicity ange but without optimal bounds on the peiod. Outline of the pape. We stat with an oveview of FI-modules in 1.1 and the est of the 1 pesents pecise statements of ou esults and an outline of the poofs. In 2, we pove esults about the stuctue of finitely geneated FI-modules ove a Noetheian ing. Ou main theoem is poved in 3 and the application to configuation spaces is given in 4. In 5, we povide some elated open questions. Acknowledgements. Fist, I thank my adviso Jodan Ellenbeg fo intoducing me to the poblem. Fo his pecisely fomulated questions and innumeous suggestions that encouaged me thoughout and helped me pove the most geneal esults. Fo shaing his vision that made me investigate the spectal sequence fo the coveing map fom the configuation spaces of odeed points to the configuation spaces of unodeed points which lead to the main application. Next, I thank Thomas Chuch and Benson Fab fo inviting me to pesent this wok at the Geomety and Topology semina at Univesity of Chicago in Fall 2013 when the wok was only in its udimentay fom. I thank Benson Fab fo inviting me again to pesent this wok at Univesity of Akansas in Sping 2015 and fo his helpful convesations on configuation spaces. I am vey gateful to Thomas Chuch fo showing me a diect shot poof of Lemma 3.2 and fo seveal hous of cucial discussions we had in coffee shops of Chicago that geatly influenced the pape as it is now. I thank Eic Ramos and Jenny Wilson fo going though pats of the pape and fo thei comments. This wok is a pat of my thesis at Univesity of Wisconsin Madison and I thank the institution fo the suppot thoughout. 1.1 An oveview of Chuch-Ellenbeg-Fab s theoy of FI-modules Let FI be the categoy whose objects ae finite sets and a mophism between finite sets A and B is an injection f: A B. The categoy FI is equivalent to its full subcategoy whose 4

5 objects ae the sets {1, 2,...,n}, n 0. Fo simplicity we denote {1, 2,...,n} by [n] and { 1, 2,..., n} by [ n]. The empty set is denoted by [0]. An FI-module ove a commutative ing K is a covaiant functo fom the categoy FI to the categoy of K-modules. The categoy of FI-modules ove K is denoted by FI-Mod K. Fo an FI-modules V, we denote the K-module V([n]) by V n. Since the goup End FI ([n]) of endomophism of [n] is natually isomophic to the symmetic goup S n, any FI-module V detemines a sequence of K[S n ]-modules (V n ) n0 with linea maps between them especting the goup actions. Fo an FI-mophism f: [m] [n], we denote the map V(f) : V m V n by f. It is known that the categoy of functos fom any small categoy to an abelian categoy is abelian ([CEF, Remak 2.1.2], [We, A.4.3]), so FI-modules fom an abelian categoy. Moeove, notions such as kenel, cokenel, subobject, quotient object, injection, o sujection ae all defined "pointwise", meaning that a popety holds fo an FI-module V if and only if it holds fo each V n. Fo example, we say a map V W of FI-modules is an injection if and only if the maps V n W n ae injections fo all n. An FI-module V is finitely geneated if thee is a finite set S of elements in i V i so that no pope sub-fi-module of V contains S. Definition 1.1 (The FI-module M(m) [CEF, Definition 2.2.3], [CEFN, Definition 2.2]). Fo any m 0, the FI-module M(m) takes a finite set S to the fee K-module M(m) S on the set of injections [m] S. In othe wods, M(m) = K[Hom FI ([m], )]; by the Yoneda lemma, M(m) is uniquely detemined by the natual identification An FI-module V is said to be fee if Hom FI-Mod (M(m),V) = V m. V = i I M(m i ). Definition 1.2 (The FI-module M(W) [CEF, Definition 2.2.2]). Fo any K[S m ]-module W, the FI-module M(W) takes a finite set S of size n to the K[S n ]-module M(W) S given by M(W) S := K[Hom FI ([m],s)] K[Sm ]W. In othe wods, M(W) n = Ind S n S m S n m W K = K[S n ] K[Sm ] K[S n m ](W K). By the Yoneda lemma, M(W) is uniquely detemined by the natual identification Hom FI-Mod (M(W),V) = Hom Sm (W,V m ). Note hee that M(K[S m ]) is same as M(m). 5

6 Remak 1.3. It is shown in [CEF, Theoem 4.1.5] that FI-modules of the fom V = i I M(W i ) fo some K[S mi ]-modules W i ae pecisely the FI-modules with FI (see [CEF, Definition 4.1.1]) stuctue on them. Fo ou pupose, an FI -module is an FI-module admitting a decomposition as above. If an FI -module V is finitely geneated as an FI-module. Then it follows (fo example, fom the agument in Poposition 2.5) that the cohomology goups H t (S n,v n ) stabilizes and hence ae peiodic with peiod 1. We have the following chaacteization of finite geneation in tems of FI-modules M(m). Definition 1.4 (Finitely geneated FI-modules [CEF, Poposition 2.3.4], [CEFN, Definition 2.1]). Let V be an FI-module. We say that V is finitely geneated if thee exists a sujection d M(m i ) V i=1 fo some integes m i 0. We say that V is geneated in degee m if thee exists a sujection Π: M(m i ) V with all m i m. i I Hee the sum may be infinite. The data of the integes m i, i I is the degee stuctue of Π and max i I m i is the degee of Π. We need the following Noetheian popety of finitely geneated FI-modules. Theoem 1.5 (Noetheian popety [CEFN, Theoem A]). If V is a finitely-geneated FImodule ove a Noetheian ingk, andw is a sub-fi-module ofv, thenw is finitely geneated. The positive shift functos wee used extensively to pove the above Noetheian popety. Shift functos ae essential fo ou pupose as well. Definition 1.6 (Positive shift functo S +a [CEFN, Definition 2.8]). Given an FI-module V and an intege a 1, the functo S +a : FI-Mod FI-Mod is defined by (S +a V) S := V S [a] It follows that as a K[S n ]-module S +a V n = Res S n+a S n V n+a Since kenels and cokenels ae computed pointwise, S +a is an exact functo. Also note that thee is a natual isomophism S +(a+b) V = S +a S +b V. 6

7 Definition 1.7. The natual inclusion S S [a] is a mophism in the categoy FI, which induces a map X a (V) : V S +a V of FI-modules. The tosion submodule of V denoted T(V) is a sub-fi-module of V given by: T(V) = a0 kex a (V). Woking ove a Noetheian ing K, T(V) is finitely geneated by Theoem 1.5 and hence fo each a 0, (X a (V)) n is injective fo lage enough n (see [CEFN, Lemma 2.15] fo moe details). In some fotunate situations we can define a meaningful map in the opposite diection; S +a V V. An example whee this holds is when V is fee, which can be seen fom the following lemma. Lemma 1.8 ([CEFN, Poposition 2.12]). Fo any a 0 and any d 0, thee is a natual decomposition S +a M(d) = M(d) Q a (2) whee Q a is a fee FI-module finitely geneated in degee d 1. In paticula, if V is a fee FI-module then V is a diect summand of S +a V. Remak 1.9. When a = 1 we have S +1 M(m) = M(m) M(m 1) m. The decomposition of S +a M(m) can be computed using the casea = 1 and the natual isomophism S +(a+b) V = S +a S +b V. We genealize the above decomposition in Lemma 2.2 which shows in paticula that if V is an FI -module and a 0, then V is a diect summand of S +a V. 1.2 Intoduction to -filteed FI-modules In 2, we show that if V is a finitely geneated FI-module ove a Noetheian ing K then V is built out of FI-modules of the fom M(W) which, by Remak 1.9 is a membe of a stictly smalle categoy. To make it pecise we need a couple of definitions. Definition 1.10 ( -filteed FI-modules). A -filteed FI-module is a sujection Π : d M(m i ) V i=1 of FI-modules such that the filtation 0 = V 0 V 1... V d = V given by V := Π( M(m i )), i=1 7 0 d

8 has gaded pieces of the fom M(W), that is, the filtation satisfies V V 1 = M(W ) whee W ae some K[S m ]-modules. We call the filtation induced by Π, the -filtation of V. We use Π and V intechangeably if thee is an obvious sujection giving V a -filteed FI-module stuctue. We call d the length of the -filtation and the pai Ṽ := ( d i=1 M(m i),(m i ) 1id ) a cove of V. The second coodinate just keeps tack of the ode which yields the desied -filtation. We dispense with the second co-odinate wheneve it is clea fom the context and identify the cove with the fist co-odinate. The above definition implies that thee ae sujections π : M(m ) M(W ) and hence each W is a singly geneated K[S m ]-module. But this is just a matte of convenience (see Lemma 2.2). Note that NOT evey finitely geneated FI-module admits a -filtation. But if a finitely geneated FI-module V admits a filtation 0 = V 0 V 1... V d = V satisfying V V = 1 M(W ) whee W ae some K[S m ]-modules, then V admits a -filteed FI-module stuctue. The definition of a -filteed FI-module just contains exta data of a cove (this exta data is useful in poducing bounds on the peiod of the cohomology goups). Definition Let Π 1 d1 : i=1 M(m i) V and Π 2 d2 : k=1 M(n k) W be -filteed FI-modules. An FI-module map φ : V W is called sequential if thee ae subsets S [d 1 ], T [d 2 ] and an isomophism f φ : S T such that n k = m f 1 φ (k) fo each k T and the map of coves φ : d 1 i=1 M(m i) d 2 k=1 M(n k) given by L f 1 ( φ n (L)) k = (k), if k T φ 0, othewise detemines φ. In othe wods, if l V n and L = (L i ) 1id1 d 1 i=1 M(m i) n is any lift of l (that is, Π 1 n(l) = l) then φ n (L) is a lift of φ(l). The following theoem is poven in 2. Theoem A (A -filteed esolution of a finitely geneated FI-module). Let V be a finitely geneated FI-module ove a Noetheian ing K and is geneated in degee D, that is, V admits a sujection d Π : Ṽ := M(m i ) V with m i D fo each i. Then, (A) fo lage enough a, S +a V is -filteed; i=1 8

9 ι φ 0 φ 1 0 Ṽ J0 J 1 J 2... J N 0 Π Π 0 Π 1 Π 2 Π N 0 V J 0 J 1 J 2... J N 0 φ 2 φ N 1 ι φ 0 φ 1 φ 2 φ N Figue 1 (B) thee exists a commutative diagam (Figue 1) with the columns and the fist ow exact and the second ow exact in high enough degee (say n C), that is, the sequence 0 V n J 0 n J1 n... JN n 0 is exact if n C. Hee fo each 0 i N, J i is a -filteed FI-module with cove Π i : J i J i and fo each 0 i N 1, the map φ i is sequential (sequentialness is witnessed by φ i ). Moeove, N D and J i is geneated in degee at most D i. Remak If W is a pojective K[S m ]-module, then M(W) is a pojective FI-module (see [CEF, Remak 2.2.A] o [We]). This implies that if K is a field of chaacteistic 0 o of chaacteistic > D then each J i in Theoem A is a diect sum of FI-modules of the fom M(W). Thus [CEF, Theoem 1.13], which shows that a finitely geneated FI-module is unifomly epesentation stable in the sense of [CF], follows fom Theoem A because by Piei s ule FI-modules of the fom M(W) ae unifomly epesentation stable. Remak Ove a field of chaacteistic 0, Sam and Snowden showed in [SS] that FImodules of the fom M(W) ae injective in the categoy of finitely geneated FI-modules and that evey finitely geneated FI-module admits a finite injective esolution. Example 3.35 shows that in positive chaacteistic, thee ae -filteed FI-modules which ae not NOT diect sums of M(W) s. Unlike the categoy of K[S m ]-modules, pojective objects in FI-Mod may not be injective. We believe that the categoy of finitely geneated FI-modules ove a field of positive chaacteistic does not have enough injectives. Remak We believe that it is possible to obtain a good bound on the constant C in Theoem A via upcoming woks of Chuch and Ellenbeg ([ChEl]) and that of Ramos ([Ram]). Note that whenkis a field andw is ak[s m ]-module, then dim K M(W) n = ( n n m) dimk W is an intege valued polynomial in n. It is poven in [CEFN, Theoem B] that fo a finitely geneated FI-module V, dim K V n is eventually a polynomial in n. Theoem A immediately yields a stengthening of this esult. 9

10 Theoem 1.15 (Polynomiality). Suppose K is a Noetheian ing. Let V be a finitelygeneated FI-module ove K. Then thee exist finitely many intege-valued polynomials P i (T) Q[T] and nontivial K-modules W i so that fo all sufficiently lage n, [V n ] = i P i (n)[w i ] in the Gothendieck goup K 0 (K) of K. Definition Fo a finitely geneated FI-module V ove a Noetheian ing K, we define χ(v) to be the smallest nonnegative intege c such that fo each of the finitely many polynomials P i (T) in Theoem 1.15 the degee of P i (T) c. When K is a field, χ(v) is equal to the degee of the polynomial P(T) which satisfies dim K V n = P(n) fo lage enough n. We define D V to be the least numbe c such that V is geneated in degee c. Remak Note that fo a finitely geneated FI-module V, we have D V χ(v). Fo a -filteed FI-module J the equality holds: D J = χ(j). Moeove, Theoem A and its poof (see 2) implies that χ(v) = χ(j 0 ) and that J i ae geneated in degee χ(v) i. 1.3 Statements of the main theoems fo -filteed FI-modules The goal of 1.3 is to state Theoem B, which is the special case of ou main theoem whee V is a -filteed FI-module. In ode to state the theoem, we need to set up some peliminay notations and definitions. Fo any n 0, let B t (S n ) be the fee abelian goup on Sn t+1 with S n acting diagonally. Then B (S n ) Z 0 is the ba esolution of S n whee the diffeential t : B t 1 (S n ) B t (S n ) is given by t (σ 0,σ 1,...,σ t ) = t (σ 0,σ 1,..., ˆσ i,...,σ t ). i=0 We allow t to equal 1 and define B 1 (S n ) to be the tivial K[S n ]-module. We assume that K is a field of chaacteistic p and fix a -filteed FI-module Π : Ṽ := d i=1 M(m i) V ove K of length d. Let 0 = V 0 V 1... V d = V be the -filtation induced by Π. Fo each 1 d, the map Π : Ṽ := i=1 M(m i) V obtained by esticting Π to i=1 M(m i) is a cove of V and gives V a -filteed FI-module stuctue. Let D = max i [d] m i be the degee of Π. Remak Recall that the definition of a -filteed FI-module Π : Ṽ := d i=1 M(m i) V contains the data of thed-tuple(m 1,...,m d ). HenceΠgives ise to the -filteed FI-modules Π fo 1 d as above in a unique way. 10

11 With these notations, we have the following definition. Definition Let z Hom Sn (B t+1 (S n ),Ṽ n). We call l Hom Sn (B t (S n ),V n) a twisted z -cycle if l t+1 = Π n z. We say that two twisted z -cycles l 1 and l 2 ae equivalent if l 1 l 2 is a bounday (classical cobounday) in the usual sense, that is, thee exists a b Hom Sn (B t 1 (S n ),Ṽn ) such that l 1 l 2 = Π n b t. This defines an equivalence elation on twisted z -cycles. We denote the set fomed by these equivalence classes by H t,z (S n,vn ) and the equivalence class of l by [l ]. Note that when z = 0, H t (S n,vn ) := (S Ht,z n,vn ) is the classical cohomology goup and fo any z, H t,z (S n,vn ) is a Ht (S n,vn )-toso. Since the data of z is contained in the data of a twisted z -cycle, the above definition is NOT independent of the cove. The nice lift constuction that we descibe in 3.2 depends on this fact. In Definition 3.17, we descibe peiodic elements of Hom Sn (B t+1 (S n ),Ṽn ) whee a peiod is a -length sequence of nonnegative which keeps tack of peiodicity in each of the component (ecall Ṽn = i=1 M(m i)). To be able to state the main theoem fo - filteed FI-modules we need cetain maps on d length sequences which depend on the tuple m = (m 1,m 2,...,m d ) appeaing in the cove Ṽ d of V d. We fist define the opeato H. v p ((b 1 b 2 )!)+1, if b 1 > b 2 H(b 1,b 2 ) := (3) 0, othewise. Definition Suppose Π : Ṽ := d i=1 M(m i) V be a -filteed FI-module. Recall that the d-tuple m = (m 1,m 2,...,m d ) is a pat of the data of the cove Ṽ. Let (H i,d ) 1id Z d 0 be a sequence of nonnegative integes. We define integes Hi, fo i [], < d ecusively by max(h i,,h, + H(m H i, 1,m i )), if m m i := (4) H i,, othewise This ecusive definition clealy depends on m but we have suppessed the dependence hee fo claity. We define the opeatos D Π : Zd 0 Z 0, D Π : Z d 0 Zd 0 and I Π : Z d 0 Z 0 11

12 given by D Π ((Hi,d ) 1id ) := (H i, ) 1i. (5) D Π ((H i,d ) 1id ) := (H i,i ) 1id, and (6) I Π ((H i,d ) 1id ) := max 1id (Hi,i + H(m i, 0)), (7) We may eplace the subscipt Π by V when a -filteed stuctue on V is clea (o equivalently when m is clea fom the context), o emove the subscipt altogethe when it is clea which -filteed FI-module is unde discussion. Example Suppose Π: M(0) M(m) V be a -filteed FI-module of length d = 2 (see Example 3.35). Then D 2 V ((0, 0)) = (0, 0), D1 V ((0, 0)) = H(m, 0), D V((0, 0)) = ( H(m, 0), 0) and I V ((0, 0)) = H(m, 0). We also need the following map to state ou theoem. Definition 1.22 (The map R). Let U be an FI -module. By Lemma 2.2 in 2, U admits a natual pojection λ a : S +a U U which induces a map Hom Sn a (B t (S n a ),S +a U n a ) Hom Sn a (B t (S n a ),U n a ). Also we have the natual estiction map Hom Sn (B t (S n ),U n ) Hom Sn a (B t (S n a ),S +a U n a ). Let R n,a t,u : Hom S n (B t (S n ),U n ) Hom Sn a (B t (S n a ),U n a ) be the composition of these two maps. We allow ouself to shed some of the subscipts and supescipts when thee is no isk of confusion. Remak Fo a geneal FI-module, thee is no analog of the map λ a (as in Definition 1.22). In fact, [CEF, Theoem 4.1.5] implies that if thee is a natual mapλ a : S +a V V fo each a then V must admit a FI -stuctue. Remak Let Π 1 : d 1 i=1 M(m i) V and Π 2 : d 2 k=1 M(n k) W be -filteed FImodules. And let φ : V W be a sequential map (see Definition 1.11). Then R t commutes with the coesponding map of coves φ n, that is, if L Hom Sn (B t (S n ), d 1 i=1 M(m i) n ) then we have R t ( φ n L) = φ n a R t (L). With the notations of Definition 1.19, we show in Claim 3.25, that the map R n,a,zd t,v d : H t,zd (S n,v d n) H t,r t+1(z d) (S n a,v d n a) (8) 12

13 given by [l d ] [Π d n a R t ( l d )] (hee l d is any "nice lift" of l d as defined in 3.2) is welldefined if a is divisible by a sufficiently lage powe of p. We now state the main theoem fo -filteed FI-modules. Theoem B (The main theoem fo -filteed FI-modules). LetΠ d : Ṽ d := d i=1 M(m i) V d be a -filteed FI-module of length d as in Definition 1.19 and let D = max i [d] m i. Let SQ be the sequence of length d consisting of zeos. If p I V d D V d(sq) a and n a 2(t+d 1)+D, then is an isomophism. R n,a,0 t,v d : H t (S n,v d n) H t (S n a,v d n a) We also extend Theoem B to H t (S n,v d n)-tosos. Theoem C (Peiodicity of tosos). Let SQ Z d 0 and let zd be SQ-peiodic (peiodicity is defined late in Definition 3.17). If p I V d D V d(sq) a and n a 2(t+d 1)+D, then R n,a,zd t,v d : H t,zd (S n,v d n) H t,r t+1(z d) (S n a,v d n a) is an isomophism (as sets) povided H t,zd (S n,v d n) is nonempty. In Lemma 3.33, we give bounds on the peiod depending only on the degee D and the peiodicity of z d. In paticula, such a bound on the peiod in Theoem B is p 2D. In example 3.35 we show that the smallest peiod fo the cohomology goups of a -filteed FI-module of length 2 could be an abitaily lage powe of p. 1.4 Statements of the main theoem and its genealizations We keep the assumption that K is a field of positive chaacteistic. Let V be a finitely geneated FI-module ove K geneated in degee D. By Theoem A, thee exists a esolution 0 V J 0 J 1... J N 0 of V with -filteed FI-modules J i, which is exact in high enough degee; say n C. Let φ i : J i J i+1 and ι : V J 0 be the maps given by the exact sequence above. Note that by Theoem A, the maps φ i ae sequential. Fo each n 0, let E, (n) be a double complex spectal sequence suppoted in columns 0 x N, with the following data on page 0: E x,y (n) = Hom Sn (B y (S n ),J x n), (E) d x,y (n) : E x,y (n) E x+1,y (n), induced by φ x and, d x,y (n) : E x,y (n) E x,y+1 (n), induced by y+1. 13

14 In 3.7, we analyze the spectal sequence (E) and define maps R x,y (n,a) : E x,y (n) E x,y (n a) whee E x,y is the K-vecto space at the position (x,y) of the page of the vetically oiented spectal sequence (whee we take the homology with espect to vetical maps to get the fist page) as above. This leads us to ou main theoem fo finitely geneated FI-modules. Theoem D (The main theoem fo finitely geneated FI-modules). Let V be a finitely geneated FI-module geneated in degee D. Then fo n C, H t (S n,v n ) admits a filtation of length N + 1 with ( E x,y (n)) x+y=t,0xn as gaded pieces (hee N D and C ae constants as in Theoem A). And thee ae constants M t and SDt such that if p Mt a and n a max{sd t,c} then the map Rx,y (n,a) : E x,y (n) E x,y (n a) is an isomophism. In paticula, dimh t (S n,v n ) is eventually peiodic in n with peiod p Mt. We povide an algoithm to calculate the stable ange SD t and the peiod p Mt (see Remak 3.36). Lemma 3.37 povides the following estimates: M t SD t min{(t+3)d, max{2d,d(d+ 1)/2}}, 2(t+max x d x 1)+D. whee d x is the length of the -filteed FI-module J x. By Remak 1.17, we can eplace D in the bounds above by χ(v) (ecall that χ(v) D). At the expense of inceasing the peiod slightly, we constuct isomophism H t (S n,v n ) H t (S n a,v n a ) in Theoem 3.41 peseving the filtation as in the Theoem D. In 3.8, we genealize ou main theoem to a complex of finitely geneated FI-module. Conside an abitay complex of finitely geneated FI-modules 0 V 0 V 1... V x... with diffeential δ. Fo each n 0, define a double complex spectal sequences E, (n) with the following data on page 0: E x,y (n) = Hom Sn (B y (S n ),Vn), x d x,y (n) : E x,y (n) E x+1,y (n), induced by δ x and, d x,y (n) : E x,y (n) E x,y+1 (n), induced by y+1. ( E) We deduce the genealization to the main theoem by analyzing the spectal sequence ( E). Theoem E (Peiodicity of the cohomology of the FI-complexes). Let V be a complex of finitely geneated FI-modules and E, be the coesponding spectal sequence as defined M above. Let N { }. Then, if p x,y a and n a max( SD x,y,c x,y ) then the map R x,y (n,a) : E x,y (n) E x,y (n a) is an isomophism. 14

15 Equations 28 and 29 (that ae stated in 3.8) povide a ecusive way to calculate the and the stable ange max( SD x,y,c x,y ). Remak 3.44 gives an estimate on these M peiod p x,y quantities. The full stength of this theoem is used fo the application to configuation spaces. Also, see some of the inteesting consequences (Coollay 3.45 and Coollay 3.46) of Theoem E. 1.5 Statements of the esults on unodeed configuation spaces Let M be a manifold. As noted in [CEF, 6] o [CEFN, 4], the configuation space of M is a co-fi-space, that is, a contavaiant functo fom FI to topological spaces: Fo any finite set S, let Conf S (M) denote the space Inj(S,M) of injections S M. An inclusion f: S T induces a estiction map f : Conf T (M) Conf S (M) giving Conf(M) a co-fispace stuctue. When S = [n], the space of injections [n] M can be identified with the classical configuation space Conf n (M) of odeed n-tuples of distinct points in M: Conf n (M) := {(P 1,P 2,...,P n ) M n Pi P j } Since cohomology is contavaiant, the functo taking S to H m (Conf S (M),K) is an FImodule H m (Conf(M),K) ove K. Unde cetain mild conditions on M, this FI-module is finitely geneated. Theoem 1.25 ([CEFN, Theoem E]). Let K be a Noetheian ing, and let M be a connected oientable manifold of dimension 2 with the homotopy type of a finite CW complex (e.g. M compact). Then, fo any m 0, the FI-module H m (Conf(M),K) is finitely geneated. as: The classical configuation space of unodeed n-tuples of distinct points in M is defined conf n (M) := {(P 1,P 2,...,P n ) M n Pi P j }/S n. In the past, thee have been seveal investigations on the stability of the cohomology goups H t (conf n (M),K) but all of them equied eithe as assumption on the manifoldm(esticting to an open o a punctued manifold o an odd-dimensional manifold), o an assumption on the chaacteistic of the field K (esticting to F 2 o a field of chaacteistic 0), o an assumption on the bounday (esticting to manifolds with nonempty bounday); see [BCT], [Nap], [Ch], [RW], [McD], [Se], [KJ]. Fo a moe complete histoy of the subject, see [Fa]. In this pape, we assume that the manifold satisfy the mild assumptions fom Theoem 1.25 and we allow the field to be of abitay positive chaacteistic. In paticula, ou esult is new when the manifold is even-dimensional and the field is diffeent fom F 2 and unique in the sense that we show peiodicity and not stability (see Remak 1.27). Ou method is simila in spiit to Chuch s method in [Ch] (which we late updated in [CEF] and [CEFN]) 15

16 but we have to deal with nonexactness of the functo H 0 (S n,.) and that foces us to make use of the full stength of Theoem E. Now we state ou esult. Theoem F (Peiodicity of cohomology of unodeed configuation spaces). Let K be a field of chaacteistic p > 0 and let M satisfies the hypothesis of Theoem Thee exist constants M t, SD t and Ct such that dim K H t (conf n (M),K) = dim K H t (conf n a (M),K) wheneve p M t a and n a max( SD t,c t ). (See (30), (31)and (32) fo the definitions of the constants M t, SD t and C t.) Remak 4.1 in 4 implies that M t above can be taken to be (t+3)(2t+2). Remak Recent wok of Canteo and Palme ([CaPa, Coollay E]) implies that if p is an odd pime and the Eule chaacteistic χ(m) of M is nonzeo then, H t (conf n (M),K) is eventually peiodic with peiod p v p(χ(m))+1 whee v p ( ) is the p-adic valuation. Note that ou esults give bounds that ae independent of χ(m) but depend only on t. Remak The easiest example whee peiod is not 1 is the 2-sphee, whee we have [Bi, Theoem 1.11]: H 1 (conf n (S 2 ),Z) = Z/(2n 2)Z and by the univesal coefficient theoem Z/pZ, if p 2n 2 H 1 (conf n (S 2 ),Z/pZ) = 0, othewise Thus when p 2, the smallest peiod is p. 2 Theoy of -filteed FI-modules ove Noetheian ings The aim of this section is to pove Theoem A and see some of its immediate consequences. Except in Lemma 2.2, which holds ove a geneal commutative ing, K is assumed to be a Noetheian ing thoughout 2. Definition 2.1 (The set D m,n and the coset epesentatives γ f ). Fo m n, let D m,n be the set of subsets of [n] of size m. S n acts natually on D m,n. Fo each f D m,n, we have a unique element γ f S n such that γ f [m], γ f [n]\[m] ae ode peseving and γ f ([m]) = f. Then K[S n ] = f D m,n K[S m S n m ]γ 1 f as a K[S m S n m ]-module. We do not distinguish D m,n fom the set of ode peseving injections f : [m] [n]. We adopt the conventions f(0) := 0 and f(m+1) := n+1. The definition above is used extensively only in section 3 but we give it hee because of its use in the following lemma. 16

17 Lemma 2.2. Let m 0 and W be a K[S m ]-module ove a commutative ing K. Then the following hold: 1. Let 0 U 1 M(W) U 2 0 be an exact sequence of FI-modules with U 1 geneated in degee m. Then U 1 and U 2 ae FI -modules. Moe pecisely, U i = M(U i m ), fo i {1, 2}. 2. S +1 M(W) = M(W) M(Res S n S n 1 W). 3. If W is finitely geneated as a K[S m ]-module, then fo any a, S +a M(W) is -filteed with M(W) as a diect summand. Poof. 1. Poof follows fom the exactness of Ind G H. 2. It is enough to show that Res S n+1 S n Ind S n+1 S m S n+1 m W K = (Ind S n S m S n m W K) (Ind S n S m 1 S n m+1 (Res S m S m 1 W) K). Any element of the module on left may be witten uniquely as f D m,n+1 γ f a f, a f W fo each f D m,n+1. Note thatd m,n+1 = D m,n (D m,n+1 \D m,n ) andd m,n+1 \D m,n is natually isomophic to D m 1,n. This induces the decompostion γ f a f = γ f a f + f D m,n+1 f D m,n finishing the poof. f D m 1,n γ f a f 3. By the exactness of the shift functo and the natual isomophisms +(a+b) V = S +a S +b V, it is enough to show pat 3 fo the case a = 1. By pat 2, it is enough to show that M(W) is filteed. Now by exactness of Ind G H we ae educed to showing that W admits a filtation with gaded pieces which ae singly geneated K[S m ]-modules; which follows because W is finitely geneated. Remak 2.3. The above lemma implies that the shift functo S + peseves sequentialness (see Definition 1.11): if δ : V W is sequential and a 0, then S +a δ is sequential. 17

18 Poof of Theoem A. We fist show that S +a V is -filteed when a is lage enough. Ou poof is by induction on max i m i and the numbe of times max i m i appeas in the tuple (m 1,...,m d ). We will efe to this tuple as the degee stuctue of Π (see Definition 1.4). The base case is tivial because if max i m i = 0 and 0 appeas only once in the degee stuctue of Π, then fo lage enough a, S +a V is a tosion fee FI-module singly geneated in degee 0; and such an FI-module must be eithe 0 o M(0). By Lemma 1.8, S +a Ṽ = d i=1 (M(m i) Q i,a ) whee Q i,a = d i j=1 M(m i,j) and m i,j < m i fo each j. We pick an i so that m i is maximum. The natual pojection S +a Ṽ M(m i ) induces the following natual commutative diagam (Figue 2) with exact ows and columns. Hee Π a is the estiction of S +a Π to U a. The K-modules (keϕ a ) mi K[S mi ] 0 Ũ a S +a Ṽ M(m i ) 0 Π a S +a Π ϕ a 0 U a S +a V A a Figue 2 ae inceasing in a and hence must stabilizes fo a lage enough (see Poof of Theoem A in [CEFN] fo a poof that these modules ae inceasing). Fixing such an a, keϕ a is geneated in degee m i and hence by pat 1 of Lemma 2.2, A a is of the fom M(W) fo some quotient W of K[S mi ]. So we have an exact sequence: 0 U a S +a V M(W) 0 with the cadinality of m i in the degee stuctue of Π a one less than the cadinality of m i in the degee stuctue of Π (o S +a Π). Hence by induction, S +b U a is -filteed fo lage enough b. Shifting the exact sequence above by b yields: 0 S +b U a S +(a+b) V S +b M(W) 0. By pat 3 of Lemma 2.2, S +b M(W) is -filteed and hence S +a V is -filteed fo a lage, completing the fist pat of the poof. Fo the second pat we fix such an a and poceed as follows. Let Q be the cokenel of the map ι := X a (V) (Definition 1.7). Then we have a commutative diagam (Figue 3) with exact columns, exact fist ow and the second ow exact in lage enough degee; say C 1. Recall, fom Definition 1.7, that the failue of exactness in 18

19 ι 0 Ṽ S +a Ṽ Q 0 Π S +a Π Ξ 0 V S +a V Q 0 φ ι φ Figue 3 cetain finitely many degees is coming fom the tosion submodule of V (which vanishes in high enough degee). It follows fom Lemma 1.8 that Q is geneated in degee < D. By induction on degee, the theoem is tue fo Ξ: Q Q, that is, we have a commutative diagam as in Figue 4 which satisfies the assetions of theoem. 0 Q K 0 K 1 K 2... K N 0 Ξ ι Ξ 0 ψ 0 ψ 1 Ξ 1 Ξ 2 0 Q K 0 K 1 K 2... K N 0 ψ 2 ψ N 1 ι ψ 0 ψ 1 ψ 2 ψ N 1 Ξ N Figue 4 The theoem fo Π : Ṽ V then follows by concatenating the two diagams (Figue 3 and 4), setting J 0 := S +a V, J i := K i 1, φ i := ψ i 1 fo i > 0 and noting that φ 0 := ι φ is sequential. Remak 2.4. The poof of Theoem A shows that a esolution with -filteed FI-modules can be constucted fo complexes of FI-modules (in 3.8, we will need a vesion of Theoem A fo complexes of finitely geneated FI-modules). Conside an abitay complex 0 V 0 V 1... V x... of finitely geneated FI-modules with diffeential δ and assume that V x = 0 fo x > N. Note that we may constuct fee finitely geneated FI-modules Ṽ x and sujections Π x : Ṽ x V x such that the diagam in Figue 5 commutes. Moeove, the maps δ x may be assumed to be sequential: by induction on x pick finite geneating set G x of V x containing the image δ x 1 (G x 1 ). These G x define Ṽ x and make δ x sequential. Let Ṽ x be geneated 19

20 δ 0 δ 1 δ 2 0 Ṽ 0 Ṽ 1 Ṽ 2... δ 0 δ 1 δ 2 0 V 0 V 1 V Figue 5 in degee D x. Unlike Theoem A, we may not assume that D x is the smallest so that V x is geneated in degee D x. But, if D is lage enough so that each V x is geneated in degee D then we may assume that D x D fo each 0 x N. The shift functo S + peseves the sequentialness. Hence by applying the constuction in Theoem A simultaneously fo each x (that is, choosing a lage enough so that fo each x, S +a V x is -filteed and epeating the pocess with Q x := S +av x ), we obtain esolutions 0 V x J 0,x J 1,x... J N x,x 0 exact in high enough degee (say n C x ) making the diagam in Figue 6 commutes and such that the maps δ y,x, 0 y N x, 0 x N ae sequential. V δ 1,0 δ 1,1 δ 1,2 0 J 1,0 J 1,1 J 1,2... δ 0,0 δ 0,1 δ 0,2 0 J 0,0 J 0,1 J 0,2... ι 0 ι 1 ι 2 δ 0 0 V 0 V 1 V 2... δ 1 δ Figue 6 As in Theoem A, the maps φ x : J y,x J y+1,x, 0 y N x 1 0 x N given by the diagam above ae sequential, N x D x and J y,x is geneated in degee D x y. We 20

21 also have the coesponding diagam (Figue 7) on coves with exact columns and sequential ows δ 1,0 δ 1,1 δ 1,2 0 J 1,0 J 1,1 J 1,2... δ 0,0 δ 0,1 δ 0,2 0 J 0,0 J 0,1 J 0,2... ι 0 ι 1 ι 2 δ 0 δ 1 δ 2 0 Ṽ 0 Ṽ 1 Ṽ Figue 7 Poposition 2.5. Assume that K is a field with positive chaacteistic and V be a finitely geneated FI-module. Then, lim sup dimh t (S n,v n ) <. n Poof. As in the poof of Theoem A the sequence, 0 V S +a V Q 0 is exact in high enough degee, S +a V is -filteed and Q is geneated in lowe degee. Hence by induction on the degee, it is enough to show the esult fo -filteed FI-modules. By induction on the length of the -filtation, it is enough to show the esult fo V = M(W), whee W is a finitely geneated K[S m ]-module. By the Shapio s lemma and the Künneth fomula we have, H t (S n,m(w) n ) = H t (S m S n m,w K) = a+b=t H a (S m,w) H b (S n m,k). Now it is poved in [Nak] thath b (S n,k) = H b (S n 1,K) ifn > 2b. Hence dimh b (S n m,k) is bounded in n, completing the poof. 21

22 3 Peiodicity of invaiants In this section we pove ou main esults on the peiodicity of the cohomology goups H t (S n,v n ) fo a finitely geneated FI-module V. By Theoem A, V admits a esolution with -filteed FI-module. So we can expect that a spectal sequence agument should educe the poblem to showing the peiodicity when V is a -filteed FI-module. This is discussed in detail in 1.4. With the notations of the afoementioned section, let l Hom K[Sn ](B t (S n ),V d n) be a zeo cycle (a classical cocycle) then it admits "a nice lift" l Hom K[Sn ](B t (S n ),Ṽ d n ) that is "peiodic". 3.2 povides the nice lift constuction, 3.3 defines "peiodic" and 3.4 poves that "a nice lift" is "peiodic". Ou main technical esult is the following: if l Hom K[Sn ](B t (S n ),Ṽ d n) is "peiodic" and Π d n l = 0 then Π d n a R n,a t ( l) = 0 (see Definition 1.22) as long as a is divisible by a high enough powe of p. This is poved in Claim 3.23 of and 3.6 use the technical esult above to show peiodicity of the cohomology goups H t (S n,v d n ) fo a -filteed FI-module Vd. 3.7 contains a spectal sequence agument that poves ou main theoem fo finitely geneated FI-modules and 3.8 contains its genealization to complex of finitely geneated FI-modules. Thoughout K is assumed to be a field of chaacteistic p. We stat with some peliminay definitions and notations. 3.1 Peliminaies Let B(S n ) Z 0 be the ba esolution of S n ove K and fo m n, D m,n and γ f be as in Definition 2.1. Definition 3.1 (The tace maps t and T). We have ak[s m S n m ]-module isomophism K[S n ] = f D m,n K[S m S n m ]γ 1 f. The map t m : K[S n ] K[S m S n m ] is the K[S m S n m ]-module homomophism defined by γ 1 f 1, f D m,n. It induces the natual K[S m S n m ]-module homomophism T m t : B t (S n ) B t (S m S n m ) defined by (σ 0,σ 1,...,σ t ) (t m n (σ 0), t m n (σ 1),...,t m n (σ t)). We denote the natual inclusion B t (S m S n m ) B t (S n ) by ι m t. Now note that we have a commutative ladde as in Figue 8 that extends the identity map id : Z Z. This implies that thee is a commutative ladde with id m t ι m t T m t as vetical maps, that extend the zeo map 0 : Z Z. Thus thee exist S m S n m -equivaiant homotopy maps h m t : B t(s n ) B t+1 (S n ), t 1 satisfying id m t ιm t Tm t = hm t 1 t + t+1 h m t. 22

23 ... B 1 (S n ) B 0 (S n ) Z 0 ι m 1 Tm 1 ι m 0 Tm 0 id... B 1 (S n ) B 0 (S n ) Z 0 Figue 8 Given ω : {0, 1,...,t+1} {0, 1,...,t} and ε : {0, 1,...,t+1} {0, 1}, we define an S m S n m -equivaiant map h t,ω,ε : B t (S n ) B t+1 (S n ) by (σ 0,σ 1,...,σ t ) ((t m ) ε(0) (σ ω(0) ),(t m ) ε(1) (σ ω(1) ),...,(t m ) ε(t+1) (σ ω(t+1) )) whee (t m ) 0 is the identity map. The poof of the following lemma is classical and pointed out to us by Thomas Chuch. The homotopy map constucted in the poof is quite explicit but we still emphasize the desciption in tems of h t,ω,ε because it is useful in undestanding the poof of Claim 3.19 late. Lemma 3.2. Thee ae ω i, ε i and a i K, 0 i t (all the quantities hee ae independent of n) such that h t = t i=0 a ih t,ωi,ε i. Poof. One may check that h t (σ 0,σ 1,...,σ t ) = t ( 1) i (tσ 0, tσ 1,..., tσ i,σ i,σ i+1,...,σ t ) i=0 is an explicit desciption of the homotopy map. Remak 3.3. When we need to be moe pecise we add a subscipt and wite t m n, Tm t,n, h m t,n and ι m t,n. These maps satisfy the following compatibility conditions: t m n a = t m n K[Sn a ] T m t,n a = T m t,n Bt (S n a ) h m t,n a = h m t,n Bt (S n a ) ι m t,n a = ι m t,n Bt (S n a ) The following maps ae essential to descibe the nice lift constuction. Definition 3.4 (The maps and ). Fo any K[S m ]-module W and K[S n ]-module P, we denote the natual estiction isomophism Hom Sn (P,M(W) n ) = Hom Sm S n m (P,W K) with m and its invese (the adjunction) with m. A concete desciption of these maps is as follows: Recall that M(W) n = Ind S n S m S n m W K. Thus any element x M(W) n may be witten uniquely as f D m,n γ f a f, a f W (hee we identify W K with W fo 23

24 convenience). This implies that any F Hom Sn (P,M(W) n ) has a unique desciption of the fom F = f D m,n γ f F f fo some maps F f : P W. We define the map m by F m := F [m], fo any F Hom Sn (P,M(W) n ). We also have a elation F f (p) = F [m] (γ 1 f p) fo each f D m,n that let us ecove F fom F m. The map m is theefoe given by (H m )(p) = γ f H(γ 1 f p), f D m,n fo any H Hom S m S n m (P,W). When we need to be moe pecise we add supescipts and wite n m and n m. Definition 3.5 (The tansfe map T). Let V be an FI-module and m n. We define the natual S n -equivaiant map by T V,m n f D m,n γ f a f : M(V m ) n V n f D m,n f (a f ). This is the degee n piece of the natual map T V,m : M(V m ) V n of FI-modules. In paticula when 0 m i m n the map T M(m i),m n : M(M(m i ) m ) n M(m i ) n is given by (γ f f D m,n γ g a f,g ) γ f g a f,g. g D mi,m f D m,n g D mi,m 3.2 The nice lift constuction As in 1.3, conside a -filteed FI-module Π : Ṽ := d i=1 M(m i) V ove K of length d. Let 0 = V 0 V 1... V d = V be the -filtation induced by Π. Fo each 1 d, the map Π : Ṽ := i=1 M(m i) V obtained by esticting Π to i=1 M(m i) is a cove of V and gives V a -filteed FImodule stuctue. Assume that V = 1 M(W ) fo some singly geneated K[S m ]-module V W. We have natual pojection maps ψ : V M(W ), p : d i=1 M(m i) M(m ), stands fo p : d i=1 M(m i) i=1 M(m i) and π : M(m ) M(W ). Hencefoth Tn i, the tansfe map T M(m i),m n (see Definition 3.5). We have the following definition. 24

25 Definition 3.6. Fix an n 0. Let z d Hom Sn (B t+1 (S n ), d i=1 M(m i) n ) and l d Hom Sn (B t (S n ),V d n) be a twisted z d -cycle (see Definition 1.19). A nice lift of a twisted z d -cycle l d is the data of an element l d Hom Sn (B t (S n ),Ṽ d n ) satisfying Π d n l d l d (the two elements ae equal up to a cobounday) togethe with the data of the following quantities (see Figue 13) and compatibility conditions: 1. twists z Hom Sn (B t+1 (S n ), i=1 M(m i) n ) fo each [d 1]; 2. twisted z -cycles l Hom Sn (B t (S n ),Vn ) fo each [d 1]; 3. elements a Hom Sm S n m (B t (S m S n m ),K[S m ]) fo each [d] such that p n l := (a T m t +(p n z ) m h m t ) m (9) whee l := p n l d (in the special case t = 1, l d = 0); 4. elements x Hom Sm S n m (B t+1 (S n ), keπ m ) fo each [d] such that p m x = w (see Figue 11) whee w := (p n l t+1 p n z ) m. We equie x to satisfy the following compatibility condition: if e j := (σ 0,j,σ 1,j,σ 2,j,...,σ t+1,j ) B t+1 (S n ), j {1, 2} then x (e 1 ) = x (e 2 ) (10) wheneve w (e 1 ) = w (e 2 ) and t m (σ 0,1 ) = t m (σ 0,2 ); 5. elements y Hom Sn (B t+1 (S n ),Ṽn) fo each [d] obtained by composing x m withtṽ,m n = (Tn i, ) 1i. This implies Π n y = 0 (see Step 3 in and Figue 12). Since Tn, is the identity map we also have p n y = w m = p n l t+1 p n z ; (11) 6. elements c Hom Sn (B t (S n ), 1 i=1 M(m i) n ) fo each [d] such that (c t+1 (e), p n l t+1 (e)) z (e) = 0 mod keπ n (12) fo any e B t+1 (S n ) (in the special case t = 1, c = 0); 7. l 1 = Π 1 n c ; 8. z 1 = c + c whee c := p ( 1) n y and c := p ( 1) n z. 25

26 Evey twisted cycle admits a nice lift. Befoe poving the existence of a nice lift, we ecod a elation between the paametes x and y in Definition 3.6 which will be used to pove the "peiodicity" of the nice lift constuction in 3.4. Lemma 3.7. Let e B t+1 (S n ) and i. We have, (p i n y mi )(e) = g D m,n [m i ] g In paticula, both the expessions vanish if m i > m. Poof. By definition of the map we have, (p i m x m )(e) = = (p i m x m m i )(γ 1 g e) g D m,n γ g ( p i m x (γ 1 g e)) γ g ( g D m,n f D mi,m and hence by applying the tansfe map we obtain, γ f (p i m x m m i )(γ 1 f γ 1 g e)) (p i n y )(e) = (Tn i, (p i m x m ))(e) = γ g f (p i m x m m i )(γ 1 g D m,n f D mi,m f γ 1 g e) (13) The poof now follows by noting that the ode peseving maps g and f satisfy g f = [m i ] if and only if f = [m i ] and [m i ] g. Next we descibe how to constuct a nice lift. This constuction can be ead afte the poof of the main theoem in Constuction of a nice lift of a twisted cycle We keep the notations of Definition 3.6. The constuction is by downwad induction on. Assume that we have constucted a twist z Hom Sn (B t+1 (S n ), i=1 M(m i) n ) and a z -cycle l Hom Sn (B t (S n ),V n ). The following steps (Step 1 Step 4) constuct the coesponding quantities fo 1. The commutative diagam in Figue 9 summaizes the situation. Step 1. The fist step of the constuction is to analyze the pojection ψ n l of l. We stat by post-composing the elation id m t ι m t T m t = h m t 1 t + t+1 h m t 26

27 B t+1 (S n ) t+1 B t (S n ) z p n i=1 M(m i) n M(m ) n π n 1 l Π n i=1 M(m i) n M(W ) n ψ n V n V 1 n Figue 9 with (ψ n l ) m to obtain: (ψ n l ) m = (ψ n l ) m ι m t T m t +(ψ n l ) m h m t 1 t+(ψ n l ) m t+1 h m t. This beaks (ψ n l ) m into thee pats; the fist factos though the tace map hence is "nice", the second is a bounday that we can get id of while staying within the equivalence class and the thid has a desciption in tems of z because l t+1 = Π n z. Since the second tem (ψ n l ) m h m t 1 t is a bounday, it follows that the extension ((ψ n l ) m h m t 1 t) m is a bounday (because m commutes with taking t ). By pojectivity of B t 1 (S n ), ((ψ n l ) m h m t 1 t) m can be lifted to a bounday b t Hom Sn (B t (S n ),Vn ) (see Figue 10). t B t (S n ) B t 1 (S n ) b ((ψ n l ) m h m t 1 ) m V n M(W ) n 0 ψ n Figue 10 Let l = l b t. Clealy l l and we have (ψ n ( l )) m = (ψ n l ) m ι m t T m t +(ψ n l ) m t+1 h m t. 27

28 Let a Hom Sm S n m (B t (S m S n m ),K[S m ]) be a lift of (ψ l ) m ι m t. Then we have (ψ ( l )) m = π m a T m t +(π n p n z ) m h m t. We now define p n l := (a T m t +(p n z ) m h m t ) m. Note that in the special case t = 1, p n l = 0. To complete the desciption of l we still need to define p i n l fo 1 i 1. We do it by induction on in the following steps. Step 2. We constuct the paametes w and x in this step. We define w := (p n l t+1 p n z ) m. It is clea fom Step 1 that w maps to keπ m. Hence by pojectivity of B t+1 (S n ), we may find S m S n m -equivaiant map x : B t+1 (S n ) keπ m such that p m x = w (see Figue 11). keπ m i=1 M(m i) m x B t+1 (S n ) keπ m M(m ) m w Figue 11 But fo ou poofs to go though, we equie x to be a paticulaly nice choice of lift of w. To make it pecise, note that G := {(1,σ 1,σ 2,...,σ t+1 ) : σ i S n i [t+1]} is a basis of B t+1 (S n ) as a fee S n -module. Hence γ 1 f s, f D m,n, s G foms a basis of B t+1 (S n ) as a fee S m S n m -module. So by feeness of B t+1 (S n ), we may assume that x (γ 1 f 1 s 1 ) = x (γ 1 f 2 s 2 ) wheneve w (γ 1 f 1 s 1 ) = w (γ 1 f 2 s 2 ), s 1,s 2 G. This allows us to say that if e j := (σ 0,j,σ 1,j,σ 2,j,...,σ t+1,j ) B t+1 (S n ), j {1, 2} then x (e 1 ) = x (e 2 ) wheneve w (e 1 ) = w (e 2 ) and t m (σ 0,1 ) = t m (σ 0,2 ). Step 3. By extending and then using the tansfe we constuct the paamete y in this step. Note that x m maps to M(Ṽ m ) n = i=1 M(M(m i) m ) n. Composing x m with TṼ,m n = (T i, n ) 1i we obtain a map y : B t+1 (S n ) Ṽ n = i=1 M(m i) n (see commutative diagam in Figue 12). We claim that y has its image contained in keπ n Ṽ n: note that by definition of tansfe y (e) = g D m,n g (x (γ 1 g e)) fo any e in B t+1 (S n ) and the claim follows because x maps to keπ m 28