Complexes of Injective Words and Their Commutation Classes

COMPLEXES OF INJECTIV E W ORDS AND THEIR COMMUT A TION CLASSES JAKOB JONSSON AND VOLKMAR WELKER Abstract. Let S be a finite alphab et. An injective word ov er S is a w ord o ver S such that eac h letter in S appea r s at most once in the w or d. W e s tudy Bo o lean cell complexes of injective words o ver S and their comm uta tio n classes . This gener alizes work b y F ar mer and by Bj¨ orner and W a chs on the complex of all injectiv e words. Spec ific a lly , for an abstract simplicial complex ∆, we c onsider the Bo olean cell c o mplex Γ(∆) whose cells are indexed by all injective words ov er the sets forming the faces of ∆. ⊲ F or a partial or der P = ( S, ≤ P ) on S , we study the Boo lean cell complex Γ(∆ , P ) of all words from Γ(∆) whose se q uence of letters comes from a linear extension of P . ⊲ F or a graph G = ( S, E ) on vertex set S , we study the Bo o le an cell complex Γ /G (∆) whose cells are indexed b y commutation classes [ w ] o f words from Γ(∆). More precisely , [ w ] consists of all words that ca n be obtained from w by successively applying co mm utatio ns of neighbor ing letters no t joined by an e dge of G . Our main results ar e as follows: ⊲ If ∆ is shellable then so are Γ(∆ , P ) and Γ /G (∆). ⊲ If ∆ is Cohen-Mac a ulay (resp. sequentially Cohen-Macaulay) then so are Γ(∆ , P ) and Γ /G (∆). ⊲ The complex Γ(∆) is partitiona ble. 1. Introduction A w ord ω o v er a finite alphab et S is called inje ctive if no letter app ears more than once; that is ω = ω 1 · · · ω r for some ω 1 , . . . , ω r ∈ S and ω i 6 = ω j for 1 ≤ i < j ≤ r . F or n + 1 = # S w e denote by Γ n the set of all injective words on S . A w ord ω = ω 1 · · · ω r with r letters is said to b e of length r . A sub word of a w ord ω 1 · · · ω r is a word ω j 1 · · · ω j s 1991 Mathe matics Su bje ct Classific ation. Key wor ds and phr ases. injectiv e word, Boo le an cell complex, simplicial complex, Cohen Macaulay complex, shellable complex. First author supp orted by Gra duiertenkolleg ‘Combinatorics, Geometry , Co mpu- tation‘, DF G-GRK 58 8 /2. Bo th author s were supp orted by EU Research T raining Net work “ Alg ebraic Co m bina torics in E urop e”, g rant HPRN-CT-2001 -0027 2. 1 2 JAKOB JONSSON AND V O LKMAR WELKER suc h that 1 ≤ j 1 < · · · < j s ≤ r . Clearly , a sub w ord of an injectiv e w ord is injectiv e. W e order Γ n b y say ing that ρ 1 · · · ρ s  ω 1 · · · ω r if and only if ρ 1 · · · ρ s is a sub w ord of ω 1 · · · ω r . W e write c ( w ) for the con ten t { ω 1 , . . . , ω r } of the w ord w = ω 1 · · · ω r . Then for any A ⊆ c ( w ) there is a unique sub w ord v  w of w with A = c ( v ). This implies the w ell kno wn fact (see [7]) that Γ n together with the partial order  is the face poset of a Bo olean cell complex. Recall t hat a Boo lean ce ll complex is a r egula r CW-complex for which the p oset of faces of eac h cell is a Bo olean lattice. Clearly , simplicial complexes are special cases of Bo olean cell complexes. F ro m now on we will iden tif y the p oset Γ n with the Bo olean cell complex with face p oset Γ n . In particular, we also identify the injectiv e w or ds of length d + 1 with d -cells. Th us the faces of a giv en d - cell w are the cells corresp onding t o all sub w o r ds of w . The complex Γ n is a w ell-studied o b ject. F armer [7] demons tr a ted that Γ n is homotopy equiv alent to a we dge of spheres of top dimension. Bj¨ orner and W achs [4] prov ed the strong er result that Γ n is shellable. See Reiner and W ebb [14] and Hanlon and Hersh [10 ] for further r efine- men ts. Our generalizations are partly mot iv ated b y sp ecific examples of complexes of injective w ords that are used in a lg ebraic K -theory; see e.g. [9, 13, 11, 12, 15, 16 ]. W e will mak e this connection a bit more precise after Example 3.3 . All our simplicial complexes and Bo olean cell complex es are assumed to b e finite. In this pap er, w e generalize Γ n in three directions: • Give n a simplicial subcomplex ∆ on ground set S , w e define a sub complex Γ(∆) o f Γ n b y restricting to injective w o r ds w ∈ Γ n suc h that the conten t c ( w ) is a face of ∆. • Give n a partially ordered set P = ( S, ≤ P ) on the alphab et S , w e define a subcomplex of Γ n b y restricting to w ords ω 1 · · · ω r suc h that i < j whenev er ω i < P ω j . F or a simplicial complex ∆ on S w e write Γ(∆ , P ) for the set of all w ords w ∈ Γ(∆) satisfying this restriction. In particular, Γ(∆ , P ) ∼ = ∆ if P is a total or der and Γ(∆ , P ) = Γ( ∆) if P is an antic hain. • Give n a graph G = ( S, E ) on the alphab et S , w e define t he equiv alence class [ w ] of an injectiv e w o r d w ∈ Γ n as t he set of all words v that can b e o bta ined from w b y applying a sequence of commutations ss ′ → s ′ s suc h that { s, s ′ } is not an edge in E . F or a simplicial complex ∆ o v er S w e write Γ /G (∆) fo r the set of equiv alence classes [ w ] of inj ective words w with con tent c ( w ) in ∆. W e o rder Γ /G (∆) b y sa ying [ v ]  [ w ] if there are COMPLEXES OF I NJECTIVE W O RDS 3 represen tativ es v ′ ∈ [ v ] and w ′ ∈ [ w ] suc h that v ′  w ′ . In particular, if E = ∅ then Γ /G (∆) ∼ = Γ(∆). It is easy to see that Γ( ∆) and Γ(∆ , P ) a re low er o rder ideals in Γ n and therefore can b e seen as sub complexes of Γ n . This in turn implies that we can also regard them as Bo olean cell complexes. Slightly more care is needed to recognize Γ /G (∆) as a Bo olean cell complex. Lemma 1.1. Γ /G (∆) is a Bo ole an c el l c omplex. Pr o of. Clearly , if tw o w ords are in the same equiv alence class, then their conte nts m ust coincide. Also [ v ]  [ w ] implies that the conten t of v is a subse t of t he con tent of w . These facts sho w tha t for a word w of length r t here is a surjectiv e p oset map from { [ v ] | [ v ]  [ w ] } to the Bo olean lattice of subsets of an r - elemen t set. In or der to sho w that it is an isomorphism w e need to see that if v 1 and v 2 are words with the same conten t and [ v 1 ] , [ v 2 ]  [ w ] then [ v 1 ] = [ v 2 ]. W e may assume that v 1  w and v 2  w ′ for some w ′ ∈ [ w ]. Since w and w ′ are equiv alent, there is a sequence of comm utations that leads from w to w ′ . The comm utations tha t in v olve only letters from c ( v 1 ) can then b e used to mov e f r o m v 1 to v 2 . In pa r t icular, [ v 1 ] = [ v 2 ].  The following three theorems are our main results. Their pro ofs are pro vided in the subsequen t sections. The concepts f rom top ological com binatorics used to form ulate the theorems are in tro duced in t he corresp onding section. F o r further reference we refer t o the surv ey article by Bj¨ orner [3] and for particular information ab out seque ntial Cohen-Macaula yness (CM) to Bj¨ orner et al. [6]. Theorem 1.2. L et ∆ b e a shel lable s implicial c o m plex on the vertex set S . (i) L et P = ( S, ≤ P ) b e a p artial or der o n S . Then the B o ole an c el l c o mplex Γ(∆ , P ) is shel lable. (ii) L et G = ( S, E ) b e a simple gr aph on S . The n the Bo ole an c el l c o mplex Γ /G (∆) is shel lable. Using the preceding theorem for ∆ b eing a simplex and p oset fib er theorems from [5], we deriv e our second main result. Theorem 1.3. L et ∆ b e a se q uential ly homotopy CM (r esp. se quen- tial ly CM over K ) simplicial c omplex on the vertex set S . (i) If P = ( S, ≤ P ) is a p artial or der on S , then the B o o le a n c el l c o mplex Γ(∆ , P ) is se quential ly homotopy CM (r esp. se quen- tial ly CM over K ). In p articular, if ∆ is homotopy CM (r esp. CM over K ), then so is Γ(∆ , P ) . 4 JAKOB JONSSON AND V O LKMAR WELKER (ii) If G = ( S, E ) is a gr aph on vertex set S , then the Bo o le a n c el l c o mplex Γ /G (∆) is se quential ly hom otopy CM (r esp. se quen- tial ly CM over K ). In p articular, if ∆ is homotopy CM (r esp. CM over K ), then so is Γ /G ( ∆) . Our t hird result exhibits a general pro p ert y of the complexe s Γ(∆ , P ) in the case that P is the antic hain. Theorem 1.4. L et ∆ b e a simplicial c omple x on vertex set S . Then the c omplex Γ(∆) of inje ctive wor ds derive d fr om ∆ is p artitionable. 2. Auxiliar y lemmas In this section we list some lemmas that giv e more insigh t into the structure of the complexes Γ(∆ , P ) and Γ /G (∆) and also serv e as in- gredien ts for the pro o f s in la ter sections. F o r a partial order P = ( S, ≤ P ) and a se t B ⊆ S , let P | B b e the induced partial order on B . F or a linear ex tension w = a 1 a 2 · · · a r of P | { a 1 ,a 2 ,...,a r } , P + w denotes the partial or der obtained from P by adding the relations a i < a i +1 for 1 ≤ i ≤ r − 1 and t aking the transitiv e closure of the resulting set of relations. F or example, if a, b ar e incomparable in P then P + ab is the part ial order obtained from P b y adding the relation c < d for all pairs ( c, d ) satisfying c ≤ P a and b ≤ P d . F o r a Bo olean cell complex Γ and a face w ∈ Γ, let fdel Γ ( w ) denote the complex obta ined by remo ving the face w and a ll fa ces containing w . Let st Γ ( w ) b e the complex consisting of all fa ces w ′ of Γ suc h that some face of Γ con tains b oth w and w ′ . W e call fdel Γ ( w ) the deletion of w in Γ and st Γ ( w ) the star of w in Γ. If ∆ and Γ are simplicial complex es on disjoin t ground sets, then the join ∆ ∗ Γ is the simplicial complex ∆ ∗ Γ := { σ ∪ τ : σ ∈ ∆ , τ ∈ Γ } . Still restricting to simplicial complexes, w e define the link of τ in Γ to b e the simplicial complex lk Γ ( τ ) := { σ : σ ∪ τ ∈ Γ , σ ∩ τ = ∅} . Clearly , for a simplicial complex Γ w e hav e that st Γ ( τ ) = 2 τ ∗ lk Γ ( τ ). Lemma 2.1. L et P = ( S, ≤ P ) b e a p a rtial or der on S a n d le t σ and τ b e subsets of S . If w is a li n e a r extension of P | σ , then ther e exists a line ar extension w ′ of P | σ ∪ τ c o ntaining w as a subwor d. Pr o of. By a simple induction argumen t, it suffices to consider the case τ = { x } , where x ∈ S \ σ . If x is maximal in P | τ + x , then we ma y c ho ose x to b e the maximal elemen t in w ′ . Otherwise, let y b e the COMPLEXES OF I NJECTIVE W O RDS 5 leftmost elemen t in w such that x < P y . L et w ′ b e the word obtained from w b y inserting x just b efore y . Then w ′ is a linear extens ion of P | σ + x . Namely , if z < P x , then z < P y , whic h implies that z app ears b efore y and hence b efore x in w ′ .  Lemma 2.2. L et ∆ b e a simplicial c omplex on the vertex set S and let P = ( S, ≤ P ) b e a p artial or d er on S . I f w ∈ Γ(∆ , P ) , then Γ(st ∆ ( c ( w )) , P + w ) = st Γ(∆ ,P ) ( w ) Pr o of. A cell w ′ b elongs to the left- hand side if and only if c ( w ) ∪ c ( w ′ ) ∈ ∆ a nd w ′ is a linear extension of ( P + w ) | c ( w ′ ) . By Lemma 2.1, there is then a face w ′′ of Γ(∆ , P + w ) con taining w ′ suc h that c ( w ′′ ) = c ( w ) ∪ c ( w ′ ). This implies that w ′ ∈ st Γ(∆ ,P + w ) ( w ). As a consequence, Γ(st ∆ ( c ( w )) , P + w ) ⊆ st Γ(∆ ,P + w ) ( w ) = st Γ(∆ ,P ) ( w ) . Con v ersely , w ′ b elongs to the right-hand side if and only if there is a face w ′′ of Γ(∆ , P ) with conten t c ( w ) ∪ c ( w ′ ) suc h that w ′′ is a linear extension of ( P + w ) | c ( w ) ∪ c ( w ′ ) . This implies that the t wo families are iden tical.  3. Shellable comple xes A pure Bo olean cell complex is a Bo olean cell complex in which all cells that are maximal with respect to inclusion hav e the same dimen- sion. W e define the class of shellable Bo olean cell complexes a s follo ws. A finite Bo o lean cell complex Γ is called shellable if Γ satisfies one of the following t w o conditions: (i) Γ = 2 Ω for some finite set Ω. (ii) Γ is pure and t here is a cell σ in Γ that is con tained in a unique maximal cell τ suc h that fdel Γ ( σ ) is shellable. Note that w e allow Ω = ∅ and Γ = {∅} in (i) and σ = τ in (ii). Also in the situation of ( ii) we ha v e fdel Γ ( σ ) = Γ \ [ σ , τ ], where [ σ, τ ] = { ρ : σ ⊆ ρ ⊆ τ } . A simple inductiv e a r g umen t yields the follo wing. Prop osition 3.1. L et Γ b e a pur e d -dimension al Bo ole an c el l c omplex and let τ 1 , . . . , τ r b e its inc lusion max imal c el ls. Then Γ is shel lable if and only if ther e is an or de r e d p artition of Γ into intervals [ σ 1 , τ 1 ] , . . . , [ σ r , τ r ] such that for 1 ≤ k ≤ r − 1 the union S k i =1 [ σ i , τ i ] is a shel lable Bo ole a n c el l c omplex of dimension d . In p a rticular, Γ is homotopy e q uivalent to a w e d g e of d -spher es. The n umb er of spher es is given by # { j | σ j = τ j } . 6 JAKOB JONSSON AND V O LKMAR WELKER W e refer to suc h an or dered partition as describ ed in Prop osition 3.1 as a shel ling or der . Indeed the usual definition of a shellable Bo olean cell complex Γ of dimension d p o stulates tha t Γ is pure and tha t there is linear order τ 1 , . . . , τ r of its d -dimensional cells suc h that the in tersection of the complex generated b y τ 1 , . . . , τ i and the cell τ i +1 is shellable of dimen- sion d − 1 f o r all 1 ≤ i ≤ r − 1. It is easy to c hec k that the existence of suc h an ordering is equiv alen t to the existence of a shelling order in our sense. Lemma 3.2. If Γ is a pur e Bo ole an c el l c omplex and ρ is a fac e such that Γ 1 = fdel Γ ( ρ ) and Γ 2 = st Γ ( ρ ) ar e s h el lable of d i m ension d , then Γ is sh e l lable of dimension d . Pr o of. L et [ α i 1 , β i 1 ] , . . . , [ α ir i , β ir i ] b e a shelling or der of Γ i for i = 1 , 2. Note that eac h β 2 j con tains the face ρ , b ecause eac h maximal face of a star complex st Γ ( ρ ) con tains ρ . In the interv al [ α 2 j , β 2 j ], there is a unique minimal face γ 2 j con taining ρ . Namely , the in tersection of t wo suc h faces would again lie in the in terv al and contain ρ . W e claim that the a b o v e shelling order on Γ 1 , together with [ γ 21 , β 21 ] , . . . , [ γ 2 r 2 , β 2 r 2 ] , yields a shelling order on Γ = Γ 1 ∪ Γ 2 . Namely , the fa ces in [ α 2 j , β 2 j ] \ [ γ 2 j , β 2 j ] are all contained in Γ 1 . In particular, the family obtained b y remo ving [ γ 2 j , β 2 j ] , . . . , [ γ 2 r 2 , β 2 r 2 ] is a pure Boo lean cell complex for eac h j ∈ [ r ]. By an induction argumen t, starting with Γ 1 , we henc e obtain that Γ 1 ∪ Γ 2 is shellable.  Pr o of of The or em 1.2 (i) . If P is a linear order, then Γ := Γ(∆ , P ) is isomorphic to ∆ and hence shellable. Otherwise, let a ∈ S b e maximal suc h that a is incomparable to some other elemen t in S (with respect to the order on P ). Let b ∈ S \ a b e minimal suc h that a and b are incomparable. Note that if c < P b , then c < P a b y minimalit y of b . Analogously , if a < P c , then b < P c . This implies that if w is a face of Γ suc h that { a, b } 6⊆ c ( w ), then w is a linear extension o f ( P + ba ) c ( w ) . Namely , b y Lemma 2.1, there is a linear exte nsion w ′ of P | c ( w ) ∪{ a,b } suc h that w is a sub w ord o f w ′ . Suppose tha t a app ear s b efore b in w ′ . Then all elemen ts c b et w een a and b in w ′ are incomparable to a and b with resp ect to P b y the ab ov e prop erties. In particular, w e ma y insert a just b efore b or insert b just after a and obtain a word with the desired prop erties. COMPLEXES OF I NJECTIVE W O RDS 7 As a consequence, we hav e that Γ(∆ , P + ba ) = fdel Γ ( ab ) . In pa rticular, if ab is not in ∆, then Γ coincides with Γ(∆ , P + ba ). By induction on P , we obtain that Γ(∆ , P ) is shellable in t his case. If ab ∈ ∆, then define Γ 1 = Γ(∆ , P + ba ) = fdel Γ ( ab ); Γ 2 = Γ(st ∆ ( ab ) , P + ab ) = st Γ ( ab ); the very last equality is a consequence of Lemma 2 .2 . Note that Γ = Γ 1 ∪ Γ 2 . By induction on P , w e ha v e that Γ 1 is shellable. Moreo ve r, since st ∆ ( ab ) = 2 { a,b } ∗ lk ∆ ( ab ) ∼ = Cone 2 (lk ∆ ( ab )) , st ∆ ( ab ) is shellable; shellabilit y of simplicial complexes is closed under links and cones. Induction on P yields that Γ 2 is shellable. Using Lemma 3.2, w e deduce that Γ = Γ 1 ∪ Γ 2 is shellable, whic h concludes the pro of.  Example 3.3. Let ∆ = ∆ M b e the simplicial complex of indep endent sets of a matro id M of rank n . Is is w ell-know n that ∆ M is shellable (see [2] fo r this fact a nd further bac kground on matroids). Thu s Γ (∆ M ) is shellable a nd hence its homology is concen trated in top dimension. As a consequen ce, rank Z e H n − 1 (Γ(∆ M ); Z ) = X F ∈ ∆ M ( − 1) n − # F # F ! . (3.1) This clearly is a matroid inv arian t. If ∆ M is the f ull simplex, then already F armer’s results [7] sho w that the left-ha nd side of (3 .1) equals the n umber of fixed p oint free p erm uta tions o n # M letters. W e ha ve not b een able to r ecognize the n umerical v alue in (3.1) for other ma- troids M . Do es there exist a ‘nice’ class of combin a torial ob jects coun ted b y this v alue? A sp ecial case of the pr eceding example a lso app ears in [16] and [15, Lemma 2 .1 ]. There the fo llo wing situation is considered . F or finite dimensional vec to r spaces V and W let ∆ V ,W b e the simplicial complex of all collections { ( v 0 , w 0 ) , . . . , ( v l , w l ) } of pairs ( v i , w i ) ∈ V × W , 0 ≤ i ≤ l , suc h that v 0 , . . . , v l are linearly indep enden t. One easily chec ks that if V a nd W a r e v ector spaces o ver a finite field then ∆ V ,W indeed is the set of indep endent sets of a matroid. Now v an der Kallen’s result [16], whic h sa ys that the homolog y of Γ(∆ V ,W ) is concen trated in top dimension, is a sp ecial case of Example 3.3. 8 JAKOB JONSSON AND V O LKMAR WELKER Before w e pro cee d w e list a few other app earances of complexes Γ(∆) in algebraic K -theory , ev en though they do in general not corresp ond to matroids or shellable ∆. In connection with work on Grassmann homology [9] and in [15], for a finite dimensional v ector space V and a n um b er s ≤ dim V the complex Γ( ∆ V ,s ) app ears for the simplicial complex ∆ V ,s of all collections { v 0 , . . . , v l } of v ectors from V suc h tha t an y subset of size ≤ s is linearly indep enden t. Ag a in it is crucial that homology v anishes except for the top degree (see [15, Lemma 2.2] for the case s = dim V ). In [11] the same v anishing of the homology of the ‘classical’ complex Γ n is applied. Finally , in [13] sev eral class es of Bo olean cell complexes are studied. F or example, for a given ring R the complex Γ(∆ R ) is studied for the simplicial complex ∆ R of all subsets { x 0 , . . . , x r } of R suc h that the ideal generated by x 0 , . . . , x n is R . Again v anishing of homology in lo w dimensions is applied in algebraic K -theory . All these examples hav e in common that they emerge in the following w ay . Giv en a group G one searc hes for a (c hain) complex with free action of G . This is ac hieve d b y considering the cellular c hain complex of Γ(∆) for a G - in v arian t simplicial complex ov er a ground set with free G -action. F o r the pro of o f Theorem 1.2 (ii) it will turn out to b e pro fitable to co de classes [ w ] b y acyclic o r ientations. Let ∆ b e a simplicial complex on the v ertex set S and let G = ( S, E ) b e a simple graph on the same v ertex set. T o w ∈ Γ(∆) we assign a directed gra ph D w = ( c ( w ) , E w ) with v ertex set c ( w ) and with a directed edge from a to b whenev er { a, b } ∈ E and a precedes b in the w o r d w . Lemma 3.4. L et ∆ b e a simpl i c ial c omplex on gr ound set S and let G = ( S, E ) b e a simp le gr aph. The n the fol lowing hold: (i) F or w ∈ Γ(∆) the dir e cte d gr aph D w is acyclic. (ii) F or [ w ] ∈ Γ /G (∆) and w ′ ∈ [ w ] we have D w = D w ′ . In p artic- ular, the map [ w ] 7→ D w is wel l defin e d for [ w ] ∈ Γ /G (∆) . (iii) F or e ach fac e σ ∈ ∆ , the map [ w ] 7→ D [ w ] pr o vides a bije ction b e twe en fac es of Γ /G (∆) with c o n tent σ and acyclic orien tations of the induc e d sub gr aph G | σ . (iv) The map [ w ] 7→ D w is a n isomorphism of p a rtial ly or der e d sets b e twe en Γ /G (∆) and the s e t of acyclic o ri e n tations of induc e d sub gr aphs G | σ for σ ∈ ∆ or der e d by inclusion of v e rtex and e dge sets. Pr o of. (i) Since edges in D w are directed from left to rig ht in w , the gra ph D w cannot hav e any directed cycles. (ii) D [ w ] is w ell-defined, b ecause if e = ab ∈ G a nd a app ears b efore b in some represen tativ e w , then a app ears b efore b in ev ery COMPLEXES OF I NJECTIVE W O RDS 9 represen tativ e. This is b ecause an y seque nce of commutations of neigh b oring letters transfor ming w into a w ord in whic h b app ears b efore a m ust con tain a step in whic h a and b are transp osed, whic h is fo rbidden. (iii) T o pro v e tha t the map is surjectiv e, simply note that eve ry acyclic digraph D on gro und set σ admits a linear extension w , i.e., a linear o rdering of σ such that a ll edges of D go from smaller to larger v ertices. Then clearly D = D w . T o prov e that the map is injectiv e, supp ose that [ w ] and [ w ′ ] yield the same acyclic orien tation D w = D w ′ . Let b b e the first elemen t of w and write w = bγ a nd w ′ = a 1 a 2 · · · a r bγ ′ , where γ and γ ′ denote w or ds and a 1 , . . . , a r letters. By construction, D w con tains no edge directed to b . Since D w = D w ′ this implies that b is not adjacent to an y a i , 1 ≤ i ≤ r , in G . In particular, we ma y apply a sequence of comm utations on w ′ to obtain the w ord w ′′ = ba 1 a 2 · · · a r γ ′ . By a simple induction argument, w e ma y transform a 1 a 2 · · · a r γ ′ in to γ via a sequenc e of commutations, whic h yields that w and w ′′ , and hence w and w ′ , belong to the same comm utation class [ w ]. (iv) By (iii), it remains to v erify that [ w ′ ]  [ w ] if and o nly if D w ′ is the subgraph of D w induced on c ( w ′ ). But this f act is immediate from the definition of D w .  Pr o of of The or em 1.2 (ii) . By Lemma 3.4, w e ma y iden tify a g iven face [ w ] o f Γ with the acyclic o r ientation D w of G | c ( w ) induced by [ w ]. Fix a linear order o n S . F or v ertices i and j of a digra ph D , w e write i D − → j if there is a directed path from i to j . By con v en tio n, i D − → i for all i . In the following w e set up functions on v ertex sets o f digraphs and an order relation on these functions that will later be the ke y ingredien t in the definition of the shelling. F o r a digraph D o n v ertex set ρ , define a function δ D = δ : ρ → ρ b y δ ( i ) = min { j | i D − → j } . Note that δ ( i ) ≤ i and δ 2 ( i ) = δ ( i ) for a ll i . Since δ 2 = δ , it is clear that δ ( i ) = i whenev er # δ − 1 ( { i } ) = 1. F or tw o functions δ 1 , δ 2 : ρ → ρ , sa y that δ 1 > δ 2 if δ 1 ( x ) ≥ δ 2 ( x ) fo r all x ∈ ρ with strict inequalit y for some x . Claim 1: L et D b e an acyclic o rien tation of G | ρ and let x ∈ ρ . If δ − 1 D ( { x } ) = { x } then t he restriction o f δ D to ρ \ { x } coincides with δ D \{ x } . 10 JAKOB JONSSON AND V O LKMAR WELKER ⊳ Pro of: Supp ose that w e ha v e a path from a ve rtex y 6 = x to x . By construction, δ D ( y ) = a fo r some a < x . Since there is no pat h from x to a it follow s that there is a path from y to a in D \ { x } and hence that δ D \{ x } ( y ) = a . ⊲ Claim 2: Let D be an acyclic orien ta tion of G | ρ and let x ∈ S \ ρ . Then there is a unique a cyclic orien tation D ′ of G | ρ + x con taining D suc h t ha t the restriction of δ D ′ to ρ coincides with δ D and such that δ D ′ ( x ) = x . The digraph D ′ has the pr o p ert y that δ D ′ > δ D ′′ for all other digraphs D ′′ on ρ + x con taining D . ⊳ Pro of: Let A b e the subset of ρ consisting of all elemen ts a suc h that δ D ( a ) < x . Consider a n acyclic orientation D ′′ of G | ρ + x con taining D . Let e = xb b e an edge in G | ρ + x . If b ∈ A and e is directed from x to b , then δ D ′′ ( x ) ≤ δ D ( b ) < x . If b ∈ ρ \ A and e is directed from b to x , then δ D ′′ ( b ) ≤ x < δ D ( b ). Th us for the conditions in Claim 2 to hold, w e mus t direct e from b to x whenev er b ∈ A a nd from x to b whenev er b ∈ ρ \ A . F or the particular face D ′ with this prop ert y , one easily c hec ks that the conditions are indeed satisfied. The final statemen t in Claim 2 follo ws immediately . ⊲ T o sho w that Γ /G (∆) is shellable, it suffices to v erify the following claim. Claim 3: Let τ b e a maximal face of ∆ and let σ ⊆ τ . Then the family Γ σ , τ := { D w ∈ Γ /G (∆) | σ ⊆ c ( w ) ⊆ τ } admits a partition in to in terv a ls (3.2) [ D 1 | σ 1 , D 1 ] , . . . , [ D r | σ r , D r ] suc h that D i | σ i is not a sub digraph of D j unless i ≤ j a nd suc h that eac h D i is an acyclic orientation of G | τ . Here, D i | σ i is the induced sub digraph of D i on the v ertex set σ i . Before w e pro cee d to the pro of of Claim 3 w e provide the argumen ts that show the sufficiency of Claim 3 for shellability of Γ /G (∆). F ro m the shellabilit y of ∆ we deduce from Prop osition 3 .1 that there is a maximal face τ a nd a fa ce σ ⊆ τ suc h that ∆ \ [ σ , τ ] is shellable. Since Γ /G ( ∆) \ Γ σ , τ = Γ /G ( ∆ \ [ σ, τ ]), Claim 3 implies by inductiv e applications of Prop osition 3.1 tha t Γ /G (∆) is shellable. ⊳ Pro of of Claim 3: Let D 1 , . . . , D r b e the acyclic orien tations of G | τ ordered suc h that i < j whenev er δ D i > δ D j . F or an y D i , let X i b e the set of eleme nts x ∈ τ \ σ suc h that δ − 1 D i ( { x } ) = { x } . Define σ i = τ \ X i . W e claim that the in terv als [ D i | σ i , D i ] yield the desired partition. First, rep eated application of (i) yields that δ D i | σ i is the restriction of δ D i to σ i . Moreo v er, rep eated application of (ii) yields that δ D i ( x ) ≥ δ D j ( x ) for all x ∈ τ whenev er D i | σ i is a sub digraph of D j and that the COMPLEXES OF I NJECTIVE W O RDS 11 inequalit y is strict for some x , a nd hence i < j , if D i 6 = D j . By a similar argumen t, o ne obtains that an y digraph D i | ρ suc h that σ i ⊆ ρ ⊆ τ has the same prop ert y . In particular, w e obtain the desired claim. No w, let [ σ, τ ] b e the last interv al in the shelling o rder of ∆. By induction, w e kno w that Γ 0 = Γ /G (∆ \ [ σ , τ ]) is shellable. Supp ose that we ha ve an ordered pa rtition o f the f orm (3.2 ) of the remaining family { D ∈ Γ /G (∆) \ σ ⊆ V ( D ) ⊆ τ } with prop erties as ab ov e. F o r 1 ≤ i ≤ r , define Γ i = Γ i − 1 ∪ [ D i | σ i , D i ]. W e claim that eac h Γ i defines a pure Bo olean cell complex; b y def- inition and induction on i , this will imply that eac h Γ i is a shellable Bo olean cell complex. By assumption the claim is true for i = 0. As- sume that i > 0. All maximal cells of Γ i ha v e the same dimension dim ∆, b ecause this is true in Γ 0 , and Γ i is the union of Γ 0 and a se- quence o f in terv als in whic h eac h top elemen t has dimension dim ∆. It remains t o pro v e that all subfaces of D i b elong t o Γ i . Let D i | ρ b e suc h a subface. If D i | ρ b elongs to Γ 0 , then w e are done. Otherwis e, D i | ρ ∈ [ D j | σ j , D j ] for some 1 ≤ j ≤ r . By construction, j ≤ i , whic h implies that D i | ρ ∈ Γ i as desired. ⊲  Example 3.5. Let ∆ b e the simplicial complex on ground set S = { 1 , · · · , 5 } with maximal fa ces 1234 , 2346 , 3456. The ordered partition [ ∅ , 1234] , [6 , 2 346] , [5 , 234 5] defines a shelling order of ∆. Let G b e the g r a ph with v ertex set S and edge set { 12 , 13 , 24 , 34 , 4 6 , 56 } . T able 3.5 pro vides a shelling order of Γ /G (∆) constructed as in the pro o f of Theorem 1 .2 (ii) from the g iven shelling order on ∆ with the natura l order on S . In the ta ble, eac h acyclic orien tation D i is represen ted b y its lexicographically smallest represen tativ e. The function δ D i is represen ted a s a w ord a 1 a 2 a 3 a 4 a 5 a 6 , where a i = δ D i ( i ) if i ∈ V ( D i ) and a i = ∗ otherwise. Underlined v alues k hav e the prop erty that δ − 1 ( { k } ) = { k } . An analysis of the pro of of Theorem 1 .2 (ii) allo ws us to describ e the rank of the ho mo lo gy groups of Γ /G (∆) for shellable ∆. Corollary 3.6. L et ∆ b e a shel lable d -dim ensional simplicial c omplex on gr o und set S with shel ling or der [ σ 1 , τ 1 ] , . . . , [ σ r , τ r ] . Fix a line ar or d er < on S . The n the r ank of the unique non-van ishing r e duc e d homolo gy gr oup e H d (Γ /G (∆); Z ) of Γ /G (∆) e quals the numb er of p airs ( τ i , D ) wher e 1 ≤ i ≤ r and D is an acyclic orientation of G | τ i such that f o r al l x ∈ τ i \ σ i ther e is a y ∈ τ i \ { x } such that one of the fol lowing c on d itions holds. 12 JAKOB JONSSON AND V O LKMAR WELKER T able 1. Shelling order for the complex Γ /G ( ∆), where ∆ and G a r e defined in Example 3.5. i D i δ D i D i [ σ i ] i D i δ D i D i [ σ i ] 1 1234 1234** ∅ 16 2364 *234*4 64 2 1243 1233* * 43 17 2436 *233*6 436 3 1423 1232* * 42 18 2643 *233*3 643 4 1342 1222* * 342 19 4236 *232*6 426 5 3124 1 214* * 3 1 20 6423 * 232*2 642 6 3142 1 2 12** 3142 21 3426 *222*6 3426 7 4312 1 211* * 4 31 22 3642 * 222*2 3642 8 2134 1 1 34** 21 23 3456 * *3456 5 9 2143 1 1 33** 2143 24 3465 **3455 65 10 4213 1131** 421 25 3645 **3454 645 11 2314 1114** 231 26 3564 **3444 564 12 2431 1111** 2431 27 4356 **3356 435 13 3421 1111** 3421 28 4365 **3355 4365 14 4231 1111** 4231 29 6435 **3353 6435 15 2346 *234*6 6 30 5643 **3333 5643 (C1) y < x and ther e is a dir e cte d p ath fr om x to y in D . (C2) y > x and ther e is a dir e cte d p ath fr om y to x in D and for n o z < x ther e is a dir e cte d p a th fr om y to z in D . Pr o of. F rom the pro of of Theorem 1.2 (ii) and Prop o sition 3.1 we de- duce that rank Z e H d (Γ /G (∆); Z ) is giv en b y the n um b er of pairs ( τ i , D ) where 1 ≤ i ≤ r and D is an acyclic orien tation of G | τ i suc h that for all x ∈ τ i \ σ i w e ha ve δ − 1 D ( { x } ) 6 = { x } . W e distinguish t w o cases: (1) δ − 1 D ( { x } ) = ∅ . In this case there is a y < x for whic h there is a directed pat h fro m x to y in D . (2) # δ − 1 D ( { x } ) ≥ 2. In this case there is a y > x for whic h there is a directed path f r om y to x in D and for no z < y there is a directed pat h fro m y to z in D . It is easy to see that (1 ) and (2) are equiv alen t to (C1) and (C2), resp ectiv ely .  Example 3.7. Let G b e a graph on the set S = { 1 , . . . , n } and ∆ n = 2 S the full simplex. W e consider the natural order on S . By Corollary 3.6 the rank of the top ho mology group of Γ /G ( 2 S ) is equal to the n umber of acyclic orien tatio ns D of G suc h that for eac h v ertex x ∈ S there is a y ∈ S satisfying at least one of (C1) and ( C2) fro m Corollary 3.6. COMPLEXES OF I NJECTIVE W O RDS 13 (i) F or the complete graph G = K n , b y the work of F armer [7] the homolog y rank is known to b e the n um b er of fix-p oint free p erm utations. It is an in teresting question whether there ex- ists a simple bijection betw een suc h p erm utatio ns and acyclic orien tations of K n satisfying the conditions of Corollary 3.6. (ii) Now w e consider the graph G on v ertex set S with all edges presen t except fo r { 1 , 2 } , { 2 , 3 } , . . . , { n − 1 , n } , { 1 , n } . T o a v oid trivialities consider only n ≥ 4 . Computer calculations for 4 ≤ n ≤ 9 suggest that rank Z e H n − 1 (Γ /G (∆ n ) , Z ) − ( − 1) n = a n , where a n is the num b er of w ays to arra nge n non-attacking kings o n an n × n ch essb oard with tw o sides iden tified to form a cylinder, with o ne king in eac h row and one king in eac h column. Is this indeed t r ue for all n ≥ 4 ? Is there a nice bijectiv e pro of ? Note that the left-hand side equals the a bsolute v alue o f the unreduced Euler c hara cteristic of Γ /G ( ∆ n ). W e refer the reader to Abramson a nd Moser [1 ] for more information on the n um b er a n . F or small v alues of n , w e ha ve that a 4 = 0, a 5 = 1 0 , a 6 = 60, a 7 = 4 6 2, a 8 = 3 9 20, a 9 = 3 6 954, and a 10 = 382740 . 4. Cohen-M aca ula y and s equentiall y Cohen-M aca ula y complexes F o r the formulation of the results of this section w e need to review some fa cts ab out (sequen tial) Cohen-Macaula yness. R ecall ( see e.g. [6]) that a simplicial complex ∆ is called seque ntially homotopy Cohen- Macaula y (SHCM for short) if for all r ≥ 0 and all σ ∈ ∆ the sub- complex (lk ∆ ( σ )) h r i generated by all maximal faces of dimension ≥ r in lk ∆ ( σ ) is ( r − 1)-connected. F or K a field or K = Z a simplicial complex ∆ is called sequen tially Cohen-Macaulay o ver K (SCM / K for short) if for a ll r ≥ 0 and all σ ∈ ∆ the subcomplex (lk ∆ ( σ )) h r i generated by all maximal faces of dimension ≥ r in lk ∆ ( σ ) has v anishing reduced simplicial ho mology in dimensions 0 through ( r − 1 ). In order to define SHCM, SCM / K , HCM and CM / K for partia lly ordered sets w e need to introduce the order complex. F or a partially ordered set Q = ( M , ≤ Q ) on gr o und set M w e denote b y ∆( Q ) = { m 0 < Q · · · < m l | m i ∈ M , l ≥ − 1 } its order complex. If Q is the face poset of a Bo olean cell complex Γ then ∆( Q ) is the barycen t ric sub division o f Γ. 14 JAKOB JONSSON AND V O LKMAR WELKER W e call a partially o r dered set Q = ( M , ≤ Q ) on ground set M SHCM (resp. SCM / K , HCM, CM / K ) if ∆( Q ) is SHCM (resp. SCM / K , HCM, CM / K ). In particular, w e call a Bo olean cell complex Γ SHCM (resp. SCM / K , HCM, CM / K ) if its barycen tric sub division ∆(Γ) is SHCM (resp. SCM / K , HCM, CM / K ). A partially o rdered set Q = ( M , ≤ Q ) is called pure if a ll inclusion- wise maximal faces of ∆( Q ) ha v e the same dimension. F o r m ∈ M , w e denote by Q ≤ m the subp oset of Q on ground set M ≤ m = { m ′ ∈ M | m ′ ≤ Q m } . W e call Q semipure if the p oset Q ≤ m is pure for all m ∈ M . The ra nk of an elemen t m ∈ M is the dimension of the simplicial complex ∆( Q ≤ m ). Note that if Q is a pure partially ordered set then the concepts SHCM and HCM (resp. SCM / K and CM / K ) coincide. It is w ell-kno wn and easy t o pro v e t ha t shellable Bo olean cell complexes are HCM and hence also CM / K . F o r simplicial complexes, it is w ell-know n that the properties of being SHCM, SCM / K , HCM, and CM / K are preserv ed under barycen tric sub division. The k ey ingredien ts to the pro of of Theorem 1.3 are Theorem 1.2 and the follo wing results from [5]. Prop osition 4.1 ([5, Theorem 5.1]) . L et R = ( N , ≤ R ) an d Q = ( M , ≤ Q ) b e semipur e p artial ly or der e d sets and let f : R → Q b e a surje c tive and r ank-pr eserving map of p artial ly or der e d sets. (i) Assume that for al l m ∈ M the fib er ∆( f − 1 ( Q ≤ m )) is HCM . If Q is SHCM , then so is R . (ii) L et K b e a field or K = Z and assume that for al l m ∈ M the fib er ∆( f − 1 ( Q ≤ m )) is SCM / K . I f Q is SCM / K , then so is R . Pr o of of The or em 1.3. (i) Consider the map φ : Γ(∆ , P ) → ∆ that sends an injectiv e w ord ω 1 · · · ω r in Γ(∆ , P ) to φ ( ω 1 · · · ω r ) := { ω 1 , . . . , ω r } ∈ ∆ . Clearly , φ is a monotone map if w e consider Γ(∆ , P ) and ∆ as p osets o rdered b y the sub w ord order and inclusion resp ective ly . Surjectivit y is ob vious as w ell. Since the rank of a w ord from Γ(∆ , P ) is giv en by one less than the cardinality of its conten t and since the rank of an elemen t of ∆ is a g ain one less than its cardinality the map is rank preserving. No w for a simplex σ ∈ ∆ w e study the preimage φ − 1 (∆ ≤ σ ), whic h consists of all ω 1 · · · ω r ∈ Γ(∆ , P ) for whic h { ω 1 , . . . , ω r } ⊆ σ . Hence, if w e again den o te by P | σ the r estriction of P to σ we can iden tify φ − 1 (∆ ≤ σ ) with the complex Γ(2 σ , P ). Since the full simplex COMPLEXES OF I NJECTIVE W O RDS 15 2 σ is shellable, Γ (2 σ , P ) is a shellable Bo olean cell complex b y Theorem 1 .2 (i). Therefore, Γ(2 σ , P ) is HCM (CM / K ). Th us b y Prop osition 4.1 it fo llows that Γ(∆ , P ) is SHCM (resp. SCM / K ) if ∆ is. (ii) Consider the map φ : Γ /G (∆) → ∆ that sends a class [ ω 1 · · · ω r ] in Γ(∆ , P ) to φ ([ ω 1 · · · ω r ]) := { ω 1 , . . . , ω r } ∈ ∆ . As in the first case, w e arrive at the conclusion that φ is a rank preserving, surjectiv e, and monot o ne map. The preimage φ − 1 (∆ ≤ σ ) of a simplex σ ∈ ∆ consists of all [ ω 1 · · · ω r ] ∈ Γ /G (∆) for whic h { ω 1 , . . . , ω r } ⊆ σ . As a conse- quence, w e can iden tify φ − 1 (∆ ≤ σ ) with the complex Γ /G | σ (2 σ ), where G | σ is the induced subgraph of G on the set σ . Since 2 σ is shellable, so is Γ / G | σ (2 σ ) by Theorem 1.2 (ii) . Therefore, Γ /G | σ (2 σ ) is HCM (CM / K ). Th us b y Prop osition 4.1 it follows that the p oset Γ /G (∆) is SHCM (resp. SCM / K ) if ∆ is.  5. Complexes of injective words are p ar t itionable A cell complex Γ is p artitionable if Γ admits a partition in to pa ir wise disjoin t in terv a ls [ σ i , τ i ] suc h that eac h τ i is maximal in Γ. An y shellable Bo olean cell complex is partitionable, but the conv erse is not true in general. In fact, somewhat surprisingly , f or P equal to the an tichain A = ( S, ≤ A ), a ll complexes of injectiv e w ords are partitiona ble: Pr o of of The or em 1.4. Let V be the v ertex set of ∆ and define a to tal order  on V . With the v ertices in eac h face of ∆ ar ranged in increasing order from left to right, this induce s a lexicographic order on the faces. Sp ecifically , let σ ≤ τ if and only if either τ is a prefix of σ or if σ is lexicographically smaller than τ . F o r example, for t he complex on the v ertex set [4] (naturally ordered) with maximal fa ces 12 3 and 234, w e ha v e that 123 ≺ 12 ≺ 13 ≺ 1 ≺ 234 ≺ 23 ≺ 24 ≺ 2 ≺ 34 ≺ 3 ≺ 4 ≺ ∅ . F o r a w ord w = w 0 · · · w r , recall that c ( w ) = { w 0 , . . . , w r } . W rite Γ := Γ(∆ , A ). Define a function f : Γ → ∆ b y f ( w ) = c ( w ) ∪ f 0 ( w ), where f 0 ( w ) is minimal with resp ect t o  among all fa ces of lk ∆ ( c ( w )). Note that f ( w ) is necessarily a maximal face o f ∆. Define another function g : Γ → Γ by letting g ( w ) b e the shortest prefix v of w suc h that f ( v ) = f ( w ). Let Γ v b e the family of faces w suc h that g ( w ) = v . It is clear that the families Γ v constitute a partition of Γ. 16 JAKOB JONSSON AND V O LKMAR WELKER No w, consider a nonempty family Γ v . W e claim that Γ v = { v w | c ( w ) ⊆ f 0 ( v ) } . Namely , ev ery member of Γ v certainly b elongs t o the set in the r ig h t- hand side. Moreo v er, if v w b elong s to this set, then f 0 ( v ) = c ( w ) ∪ f 0 ( v w ). Namely , supp ose tha t some face σ of lk ∆ ( c ( v w )) is smaller than f 0 ( v w ). Then c ( w ) ∪ σ is smaller than c ( w ) ∪ f 0 ( v w ), whic h is a con tradiction. As a conclusion, w e ma y write Γ v = v · Γ ′ = { v σ | σ ∈ Γ ′ } , where Γ ′ is the complex of injectiv e words deriv ed f r om the f ull simplex on t he vertex set f 0 ( v ). Since Γ ′ is shellable, w e may par tition Γ ′ in to in terv als [ u , w ] such that each top cell w is maximal in Γ ′ . This induces a partition of Γ v in to inte rv als [ v u, v w ] suc h that eac h top cell v w is maximal in Γ v and hence in Γ.  The h -p olynomial h (Γ; t ) := P i h i t i of a Bo olean cell complex o f dimension d is defined b y X i h i t i = X i f i t i (1 − t ) d +1 − i ⇐ ⇒ X i f i t i = X i h i t i (1 + t ) d +1 − i , where f i is the n um b er of cells o f dimension i − 1 in Γ. Corollary 5.1. L et ∆ b e a pur e simplicia l c omplex. Then al l c o effi- cients of the h -p olynomial of Γ(∆) ar e nonne gative. Pr o of. By Theorem 1.4, we may par t it ion Γ(∆) into a disjoin t union of in terv als [ σ i , τ i ] suc h that eac h τ i has maxim um dimension d . It follows that h (Γ(∆); t ) = P t # σ i .  A partition in to in terv a ls of a Bo olean cell complex Γ induces a matc hing of cells suc h that the only unmatc hed cells in the complex are the ones tha t form singleton interv als [ τ , τ ] in the partit ion. Sp ecifically , consider the gra ph on the set of fa ces of Γ, where w e ha v e an edge b et we en tw o fa ces σ and τ whenev er σ < τ and there is no face γ f o r whic h σ < γ < τ . Th us this graph is the graph of the Hasse dia g ram o f Γ. No w, for t w o faces σ < τ of Γ, eac h in terv al [ σ , τ ] is a Bo olean lattice and therefore the asso ciated Hasse diagram has a p erfect matching if and only if σ 6 = τ . In particular, this sho ws that on the Hasse diagram of a partitionable Bo olean cell complex there is a mat ching whose only unmatc hed faces are the ones corresponding to one-elemen t in terv a ls. In Discrete Morse theory [8], matchin g s of the Ha sse diagram o f the face p oset of a regular CW-complex are used to determine the top ological structure of the complex. Ho w eve r, in general, discrete Morse theory COMPLEXES OF I NJECTIVE W O RDS 17 [8] do es not apply to the matc hings constructed ab ov e. Na mely , a matc hing relev an t to discrete Morse theory has to satisfy the additional assumption that if one directs all edges from the matc hing upw ard b y dimension and all o ther edges down w ard, then the resulting directed graph mus t b e acyclic. F o r examp le, consider the complex with maximal faces 12 and 3 4. The induced order o f the faces is 12 ≺ 1 ≺ 2 ≺ 34 ≺ 3 ≺ 4 ≺ ∅ , whic h yields Γ ∅ = { ∅ , 1 , 2 , 12 , 21 } , Γ 3 = { 3 , 34 } and Γ 4 = { 4 , 43 } . W e obtain a partition consisting o f the four in terv als [ ∅ , 12], [21 , 21], [3 , 34] and [4 , 43], whic h yields a matc hing including the pair s { 3 , 34 } and { 4 , 43 } . This is illegal in terms of discrete Morse theory . Reference s [1] M. Abramso n and W. O. J. Moser . Permutations without ris ing or falling ω - sequences. Ann. Math. Stat. 3 8 (1 9 67) 1 2 45-12 54. [2] A. Bj¨ orner. The homology a nd shella bilit y of matro ids and g eometric la ttices . In: Matr oid applic ations , Encycl. Math. Appl. 40 (19 92) pp. 226-2 83. Cam- bridge Universit y Press . Cambridge, 199 2 .. [3] A. Bj¨ orner, T op olo gical Methods, In: Handb o ok of Combinatorics, V ol II . R. Graham, M. Gr¨ otschel and L. Lov´ asz, (Eds). pp. 1 819-1 872. North-Holland, Amsterdam. 199 5. [4] A. Bj¨ orner and M. W achs, On lexicographica lly shella ble po sets. T r ans. Amer. Math. So c. 277 (1983) 323 –341, [5] A. Bj¨ orner, M. W a chs and V. W elker. Poset fiber theorems , T r ans. Amer. Math. So c. 357 (2005) 187 7-189 9. [6] A. Bj¨ orner, M. W achs a nd V. W elker. Sequentially Cohen-Maca ulay complexes and p osets. Israel J. Math., to app ear , Pr eprint 2007. [7] F. D. F ar mer. Cellular homology for po sets. Math. Jap an 23 (1978 / 79), 607– 613. [8] R. F or ma n, Mor se theor y for cell complexes. Ad v. Math. 134 (1998) 90–14 5. [9] W. Gerdes. Affine Grassmann homology and the ho mo logy o f general linear groups. Duke Math. J. 63 (1991 ) 85 -103. [10] P . Hanlon and P . Hersh. A Ho dge Deco mp os ition for the Complex of Injectiv e W ords. Pacific J. Math. 214 (200 4) 109 –125 . [11] M. Kerz. The complex of w or ds and Nak aok a stabilit y . Homolo gy, Homotopy and Applic ations 7 (20 0 5) 77-85. [12] K.P . Knudson, Homolo gy of Line ar Gr oups - Prog ress in Mathematics 193 , Birkh¨ auser, Basel, 2001 . [13] B. Mirzaii and W. v an der Kallen. Homology stability for unitar y groups. Do c. Math. 7 (2002 ) 143– 166. [14] V. Reiner and P . W ebb, The com bina to rics of the bar resolution in group cohomolog y , J. Pu r e Appl. Algebr a 190 (200 4) 291 –327 . 18 JAKOB JONSSON AND V O LKMAR WELKER [15] A.A. Suslin, Homology of Gl n , characteristic classes, a nd Milnor K -theor y . In Alge br aic K -The ory, Numb er The ory, Ge ometry and A n alysis , pp. 357–3 75. Lecture Notes in Math. 1046 . Springer -V erlag. Heidelb erg. 1984. [16] W. v an der Kallen. Homology stability for linear groups. Invent. Math. 6 0 (1980) 269 -295. Dep ar tment of Ma thema tics, KTH, 1 0044 Stockholm, Sweden E-mail add r ess : j akobj@ math. kth.se F a chbereich Ma thema tik und Informa tik, Philipps-Universit ¨ at Mar- burg , 35032 Marburg, G ermany E-mail add r ess : w elker@ mathe matik.uni-marburg.de