Applications of Klee s Dehn-Sommerville relations.

Transcription

1 Applcatons of Klee s Dehn-Sommervlle relatons. Isabella Novk Department of Mathematcs, Box Unversty of Washngton, Seattle, WA , USA, novk@math.washngton.eu E Swartz Department of Mathematcs, Cornell Unversty, Ithaca NY, , USA, ebs@cornell.eu May 7, 008 Abstract We use Klee s Dehn-Sommervlle relatons an other results on face numbers of homology manfols wthout bounary to () prove Kala s conjecture provng lower bouns on the f-vectors of an even-mensonal manfol wth all but the mle Bett number vanshng, () verfy Kühnel s conjecture that gves an upper boun on the mle Bett number of a k-mensonal manfol n terms of k an the number of vertces, an () partally prove Kühnel s conjecture provng upper bouns on other Bett numbers of o- an even-mensonal manfols. For manfols wth bounary, we erve an extenson of Klee s Dehn-Sommervlle relatons an strengthen Kala s result on the number of ther eges. 1 Introucton In ths paper we stuy face numbers of trangulate manfols (an, more generally, homology manfols) wth an wthout bounary. Here we scuss our results eferrng most of efntons to subsequent sectons. Our startng pont s a beautful theorem known as the Dehn-Sommervlle relatons. It asserts that the upper half of the face vector of a trangulate manfol wthout bounary s etermne by ts Euler characterstc together wth the lower half of the face vector. In ths generalty the theorem s ue to Vc Klee [8]. Research partally supporte by Alfre P. Sloan Research Fellowshp an NSF grant DMS Research partally supporte by NSF grant DMS

2 Perhaps the most elegant way to present the Dehn-Sommervlle relatons s va the h-vector of a manfol. The entres of ths vector are certan (alternatng) lnear combnatons of the face numbers. On the level of h-vectors, the Dehn-Sommervlle relatons for trangulate spheres an o-mensonal manfols merely state that the h-vector of these complexes s symmetrc. In the case of spheres, the components of the h-vector are also known to be postve as they equal mensons of algebracally etermne nonzero vector spaces [0, Chapter ]. Motvate by Dehn-Sommervlle relatons together wth several commutatve algebra results on Stanley-Resner rngs of trangulate manfols, Kala suggeste [15, Secton 7] a mofcaton of the h-vector, the h -vector, as the correct h-vector for (orentable) manfols wthout bounary. The h -vector of orentable manfols (both o-mensonal an even-mensonal) has snce been shown to be symmetrc [15] an nonnegatve [16]. Our frst result s an extenson of Klee s Dehn-Sommervlle relatons to manfols wth bounary. Specfcally, we show that for a trangulate manfol wth a fxe bounary Γ, the upper half of the h-vector s etermne by the Euler characterstc, ts lower half, an the h-vector of Γ. Ths result s not entrely new. In the language of f-vectors t was frst worke out by Maconal [1], an then rescovere by Klan [7], an Chen an Yan []. However ts h-vector form appears to be absent from the lterature. We then use ths result to efne a sutable verson of the h -vector for manfols wth bounary as well as show that t s symmetrc an nonnegatve. Our next result concerns new nequaltes on the face numbers an Bett numbers of manfols wthout bounary. Kala conjecture (prvate communcaton) that the face numbers of a k-mensonal manfol wth all but the mle Bett number vanshng are smultaneously mnmze by the face numbers of a certan neghborly k mensonal manfol. We verfy ths conjecture. We also prove a part of a conjecture by Kühnel [11, Conjecture 18] that proves an upper boun on the mle Bett number of a kmensonal manfol n terms of k an the number of vertces. Both results turn out to be a smple consequence of the Dehn-Sommervlle relatons an results from [16]. Kühnel further conjecture [11, Conjecture 18] an upper boun on the -th Bett number (for all ) of a ( 1)-mensonal manfol wth n vertces n terms of,, an n. We prove that ths conjecture s mple by the g-conjecture for spheres. In partcular, Kühnel s conjecture hols for manfols all of whose vertex lnks are polytopal. In the last secton we return to scussng manfols wth bounary. Here we erve a strengthenng of Kala s theorem [6, Theorem 1.3] that proves a lower boun on the number of eges of a manfol n terms of ts menson, total number of vertces, an the number of nteror vertces. Our new boun also epens on the Bett numbers of the bounary. The structure of the paper s as follows. In Secton we revew necessary backgroun materal. In Secton 3 we erve the Dehn-Sommervlle relatons an efne the h -vector for manfols wth bounary. In Secton 4 we eal wth Kala s an Kühnel s conjectures. Fnally, n Secton 5 we prove a new lower boun on the number of eges of manfols wth bounary.

3 Smplcal complexes an face numbers In ths secton we revew necessary backgroun materal on smplcal complexes, Dehn- Sommervlle relatons, an Stanley-Resner rngs of homology manfols. We refer our reaers to [0, Chapter ] an the recent paper [16] for more etals on the subject. Recall that a smplcal complex on the vertex set [n] = {1,,..., n} s a collecton of subsets of [n] that s close uner ncluson an contans all sngletons {} for [n]. The elements of are calle faces. The maxmal faces (wth respect to ncluson) are calle facets. The menson of a face F s m F := F 1 an the menson of s the maxmal menson of ts faces. For a smplcal complex an ts face F, the lnk of F n, lk (F ), s the subcomplex of efne by lk (F ) = lk (F ) := {G G F = an G F }. In partcular, the lnk of the empty face s the complex tself. A basc combnatoral nvarant of a smplcal complex on the vertex set [n] s ts f-vector, f( ) = (f 1, f 0,..., f 1 ). Here, 1 = m an f enotes the number of -mensonal faces of. Thus f 1 = 1 (there s only one empty face) an f 0 = n. An nvarant that contans the same nformaton as the f-vector, but sometmes s more convenent to work wth, s the h-vector of, h( ) = (h 0, h 1,..., h ) whose entres are efne by the followng relaton: h λ = =0 f 1 λ (1 λ). (1) =0 A central object of ths paper s a homology manfol (over a fel k), that s, a ( 1)- mensonal pure smplcal complex such that for all F, the reuce smplcal homology H (lk F ; k) vanshes f < F 1 an s somorphc to k or 0 f = F 1. A complex s pure f all of ts facets have the same menson. The bounary faces of are those faces F such that H F 1 (lk F ; k) = 0. When has no bounary faces, we wrte = an s calle a homology manfol wthout bounary. Otherwse, s the set of bounary faces together wth the empty set. We wll assume that s a ( )-mensonal homology manfol wthout bounary. Uner certan contons ths assumpton s superfluous, see, for nstance [13]. As emonstrate by the suspenson of the real projectve plane whose bounary woul be the two suspenson ponts for any fel whose characterstc s not two, some atonal assumpton s requre. We say that s orentable f the par (, ) satsfes the usual Poncaré-Lefschetz ualty assocate wth orentable compact manfols wth bounary. The prototypcal example of a homology manfol (wth or wthout bounary) s a trangulaton of a topologcal manfol (wth or wthout bounary). A beautful theorem ue to Klee [8] asserts that f s a homology manfol wthout bounary, then the f-numbers of satsfy lnear relatons known as the Dehn- Sommervlle relatons: h h = ( 1) ( ) (( 1) 1 χ( ) 1 ) for all 0. () 3

4 Here χ( ) := 1 = 1 ( 1) f s the reuce Euler characterstc of. Proofs of several results n ths paper rely heavly on Klee s formula () an ts varatons, whle other results are concerne wth ervng analogs of ths formula for manfols wth bounary. In aton to the Dehn-Sommervlle relatons we explot several results on the Stanley- Resner rngs of homology manfols. If s a smplcal complex on [n], then ts Stanley- Resner rng (also calle the face rng) s k[ ] := k[x 1,..., x n ]/I, where I = (x 1 x x k : { 1 < < < k } / ). (Here an throughout the paper k s an nfnte fel of an arbtrary characterstc.) Snce I s a monomal eal, the rng k[ ] s grae, an we enote by k[ ] ts th homogeneous component. The Hlbert seres of k[ ], F (k[ ], λ) := =0 m k k[ ] λ, has the followng propertes. Theorem.1 (Stanley) Let be a ( 1)-mensonal smplcal complex. Then F (k[ ], λ) = =0 h λ (1 λ). Theorem. (Schenzel) Let be a ( 1)-mensonal homology manfol, an let θ 1,..., θ k[ ] 1 be such that k[ ]/Θ := k[ ]/(θ 1,, θ ) s a fnte-mensonal vector space over k. Then F (k[ ]/Θ, λ) = where β j 1 := m k Hj 1 ( ; k). ( h ( ) + =0 ( ) 1 j=1 ( 1) j 1 β j 1 ( ) Theorem.1 can be foun n [0, Theorem II.1.4], whle Theorem. s from [18]. In vew of Theorem., for a ( 1)-mensonal homology manfol, efne h ( ) := h ( ) + ) λ, ( ) 1 ( 1) j 1 β j 1 ( ). (3) We remark that f k = then a set of lnear forms {θ 1,..., θ } satsfyng the assumptons of Theorem. always exsts, e.g., choosng generc θ 1,..., θ oes the job. The followng theorem summarzes several results on the h -numbers of homology manfols that wll be neee later on. For 0 < m = ( x ) := x(x 1) (x + 1)/! where 0 < x R, efne m <> := ( x+1 +1). Also set 0 <> := 0. Theorem.3 Let be a ( 1)-mensonal homology manfol. Then j=1 1. h 0 = 1, h 1 = f 0, an for all 1 ( h ) β 1 an h +1 4 ( h ( ) ) <> β 1.

5 . Moreover, f s a homology manfol wthout bounary that s orentable over k,.e., β 1 ( ) = β 0 ( ) + 1, then ( ) h h = (β β 1 ) for all 0. (4) Part 1 of ths theorem was recently prove n [16] (see Theorems 3.5 an 4.3 there). Part s a smple varaton of Klee s Dehn-Sommervlle relatons, see [15, Lemma 5.1]. It s obtane by combnng equatons () an (3) wth Poncaré ualty for homology manfols. Eq. () mples that all homology spheres an o-mensonal manfols wthout bounary satsfy h = h for all. Whle ths symmetry fals for even-mensonal manfols wth χ 1, Theorem.3 together wth Poncaré ualty suggests we conser the followng mofcaton of the h-vector an yels the followng algebrac verson of (). Proposton.4 Let be a ( 1)-mensonal homology manfol wthout bounary. Assume further that s connecte an orentable over k. Let h := h an h ( ) := h ( ) ( ) β 1 ( ) = h Then h 0 an h ( ) = h ( ) for all 0. ( ) ( 1) j β j 1, for 0 1. In vew of Proposton.4 an results of [17] that nterpret h -numbers as mensons of homogeneous components of a Gorensten rng, h can be regare as the correct h-vector for orentable homology manfols wthout bounary. What s the analog of h for manfols wth bounary? We eal wth ths queston n the followng secton. j=1 3 Dehn-Sommervlle for manfols wth bounary Klee s equatons () generate a complete set of lnear relatons satsfe by the h-vectors of homology manfols wth empty bounary. More generally, one can fx a (non-empty) homology manfol Γ an ask for the set of all lnear relatons satsfe by the h-vectors of homology manfols whose bounary s Γ. Dervng such relatons an efnng what seems to be the correct verson of the h -vector s the goal of ths secton. We have the followng verson of Dehn-Sommervlle relatons: Theorem 3.1 Let be a ( 1)-mensonal homology manfol wth bounary. Then ( ) h ( ) h ( ) = ( 1) 1 χ( ) g ( ) for all 0, where g ( ) := h ( ) h 1 ( ). 5

6 Proof: Wrte f := f ( ) an h := h ( ). Let f b := f ( ), an efne h b an g b n a smlar way. Also let f := f ( ) f b be the nteror f-vector, an let h be efne from f accorng to Eq. (1). Wth ths notaton, we obtan from [0, Corollary II.7.] that ( 1) F (k[ ], 1/λ) = ( 1) 1 f χ( ) + 1λ (1 λ). Substtutng Theorem.1 n the above formula yels ( 1) =0 h λ (λ 1) = ( 1) 1 χ( ) + whch s equvalent to =1 f 1λ (1 λ) = ( 1) 1 χ( ) + (1 λ) =1 (h h )λ = ( 1) 1 χ( )(1 λ). =0 =0 h λ (1 λ), Subtractng =0 gb λ from both ses an notng that h + g b = h, mples the result. Whle Theorem 3.1 appears to be new, f-vector forms of the same equalty have appeare before. Chen an Yan gave a generalzaton whch apples to more general stratfe spaces []. However, we beleve that the frst place where an equvalent formula appears s ue to Maconal [1]. We now turn to fnng the rght efnton of h for orentable homology manfols wth bounary. Recall that a connecte ( 1)-mensonal homology manfol s orentable over k f H 1 (, ; k) = k. By Poncaré-Lefschetz ualty, f s such a manfol, then H 1 (, ) = H ( ). Wrte β 1 (, ) to enote m H 1 (, ). We start by expressng g ( ) n terms of ts Bett an h -numbers. Substtutng Eq. (3) n g ( ) = h ( ) h 1 ( ) an recallng that m( ) =, we obtan g ( ) = [ h ( ) h 1( ) + ( ) ] 1 β ( ) + 1 ( ) 1 j=1 ( 1) j β j 1 ( ). (5) Theorem 3. Let be a ( 1)-mensonal homology manfol wth nonempty bounary. If s orentable, then for all 0 <, ( ) ( ) h ( ) β 1 ( ) = h ( ) g ( ) m Im (H 1 ( ) ψ H 1 (, )), where g ( ) := h ( ) h 1( ) + ( 1 1) β ( ) an ψ s the map n the long exact sequence of the par (, ). 6

7 Proof: If = 0, then both ses are equal to 0. For 0 < <, usng Eq. (3) an Theorem 3.1, we obtan ( ) h ( ) β 1 ( ) ( = h ( ) g ( ) + ( 1) ) [ χ( ) ] 1 + ( 1) j β j 1 ( ) = h ( ) g ( ) + ( 1) 1 ( = h ( ) g ( ) + ( 1) ( ) 1 ) [ j= +1 j=1 ( 1) j β j (, ), j=0 ( 1) j 1 β j 1 ( ) where the last step s by Poncaré-Lefschetz ualty. Substtutng equatons (3) an (5) n the last expresson then yels, ( ) h ( ) β 1 ( ) ( ) 1 = h ( ) g ( ) ( 1) j 1 [β j (, ) β j 1 ( ) + β j 1 ( )]. j=0 The result follows, snce by long exact homology sequence of the par (, ), the last summan equals ( ) m Im (H 1 ( ) H 1 (, )). Theorem 3. suggests the followng efnton of the h -vector an shows (together wth theorem.3) that t s symmetrc an non-negatve. Defnton 3.3 For a ( 1)-mensonal orentable homology manfol wth a nonempty bounary, efne { h h ( ) = ( ) g ( ) ( ) m Im (H 1 ( ) H 1 (, )) for / h ( ) ( ) β 1 ( ) for > /. Note that n the case of the empty bounary an <, ths efnton agrees wth the one gven n Proposton.4. 4 Manfols wthout bounary: Kala s an Kühnel s conjectures In ths secton we settle a conjecture of Kala that proves lower bouns for the face numbers of even-mensonal homology manfols wth all Bett numbers but the mle one vanshng. We also partally settle a conjecture by Kühnel on the Bett numbers of 7 ]

8 homology manfols. Throughout ths secton, enotes a ( 1)-mensonal orentable homology manfol wthout bounary. Note that f k s a fel of characterstc two, then ths class nclues all trangulate topologcal manfols wthout bounary. We start by scussng even-mensonal manfols. The followng result was conjecture by Kühnel [11, Conjecture 18]. Theorem 4.1 Let be a k-mensonal orentable homology manfol wth n vertces. Then ( ) ( ) k + 1 n k β k ( ). k k + 1 Moreover, f equalty s attane then β = 0 for all < k. Proof: Choose a nonnegatve real number x such that ( ) ( ) k + 1 x h k β k 1 =. k k It exsts snce accorng to Theorem.3, h k ( ) k+1 βk 1 0. Moreover, the same theorem k mples that h k+1 ( x+1 k+1). Thus ( ) ( k + 1 by (4) k + 1 β k = h k+1 h k + k k ) β k 1 ( ) x + 1 k + 1 ( ) ( ) x x =. k k + 1 Fnally, snce h 1 = n k 1, another applcaton of Theorem.3 shows that h k ( ) n k k, hence x n k, an ( ) k+1 βk ( ) n k k k+1, as requre. Furthermore, equalty mples that h = ( ) n k = (h 1 ) <> for all k + 1, whch by Theorem.3 s possble only f β = 0 for all < k. Theorem 4.1 mples that f β k 1, then n k k+1, or equvalently, n 3k+3. In other wors, havng a non-vanshng mle Bett number requres at least 3k + 3 vertces. (Ths result was orgnally prove by Brehm an Kühnel for PL-trangulatons [1].) Moreover, f such a homology manfol, M k, has exactly 3k + 3 vertces, then ( ) k β k (M k ) = 1, β (M k ) = 0 for < k, an h (M k ) = h (M k ) = for k + 1. In partcular, the face numbers of M k (whether t exsts or not) are unquely etermne by Eqs. (1) an (). These face numbers turn out to be mnmal n the followng sense (as was conjecture by Gl Kala, personal communcaton): Theorem 4. Let be a k-mensonal orentable homology manfol wth β k 0 beng the only non vanshng Bett number out of all β l, l k. Then f 1 ( ) f 1 (M k ) for all 1 k

9 Proof: Substtutng β l = 0, l < k, n Theorem. an Eq. (), we obtan that ( ) k + 1 h j ( ) = h j( ) an h k+j+1 ( ) = h k j ( ) + ( 1) j β k ( ) for 0 j k. k j Eq. (1) then mples ( ) k + 1 j f 1 ( ) = h k + 1 j( ), f k, an (6) j=0 k [( ) ( )] k + 1 j j f 1 ( ) = + h k + 1 k + 1 j( ) j=0 [ k 1 ( )( ) ] k j k β k ( ) ( 1) j f k + 1. (7) k + 1 k j j=0 Snce () the same formulas apply to the f-numbers of M k, () the coeffcents of the h -numbers n Eqs. (6) an (7) are nonnegatve, an () β k ( ) 1 = β k (M k ), to complete the proof t only remans to show that h ( ) h (M k ) for all k an that the coeffcent of β k n Eq. (7) s nonnegatve for all k + 1. The latter asserton follows by notng that the sequence a j = ( k j k + 1 )( k + 1 k j ), 0 j k 1 s ecreasng (nee, a j /a j+1 = (k + + j)/( k 1 j) > 1), an hence a 0 a ( 1) k 1 a k 1 0. To verfy the former asserton, we use the same trck as n the proof of Theorem 4.1. Let 0 x R be such that h k ( ) = ( ) x k. Then accorng to Theorem.3, h k+1 ( ) ( ) x+1 k+1 whle h k+1 ( ) h k ( ) = ( ) k+1 βk ( ) k+1 k+1 k+1. Thus we have ( ) ( ) ( ) ( ) k + 1 x + 1 x x h k + 1 k+1( ) h k( ) =. k + 1 k k + 1 Hence x k + 1, an so h k ( ) ( ) k+1 k. Applyng Theorem.3 once agan, we nfer that h ( ) ( ) k+1+ = h (M k ) for all k. In aton to Theorem 4.1, Kühnel conjecture (see [11, Conjecture 18]) that a ( 1)- mensonal manfol wth n vertces satsfes ( ) +1 βj ( ) ( ) n +j 1 j+1 j+1 for all 0 j / 1. The case of j = 0 merely says that every connecte component of has at least + 1 vertces. The case of j = 1 s equvalent to Kala s lower boun conjecture [6, Conjecture 14.1] that was recently settle n [16, Theorem 5.]. For other values of j we have the followng partal result. We recall that a ( 1)-mensonal homology sphere Γ s sa to have the har Lefschetz property f for a generc choce of θ 1,..., θ, ω k[γ] 1, the map ω k[γ]/(θ 1,..., θ ) k[γ]/(θ 1,..., θ ) 9

10 s an somorphsm of k-spaces for all /. It s a result of Stanley [19] that n the case of char k = 0 all smplcal polytopes have ths property, an t s the celebrate g-conjecture that all homology spheres o. Theorem 4.3 Let be a ( 1)-mensonal orentable homology manfol wth n vertces. If for every vertex v of the lnk of v has the har Lefschetz property (e.g., char k = 0 an all vertex lnks are polytopal spheres), then ( ) ( ) + 1 n + j 1 β j ( ) for all 0 j j + 1 j If equalty s attane for some j = j 0, then β = 0 for all j 0, 0 / 1. Proof: Snce all vertex lnks of have the har Lefschetz property, Theorem 4.6 of [1] mples that for a suffcently generc choce of θ 1,..., θ, ω k[ ] 1 an every j / 1, the lnear map ω k[ ]/(θ 1,..., θ ) j 1 k[ ]/(θ 1,..., θ ) j s surjectve. The mensons of the spaces nvolve are h j 1 an h j, respectvely (see Theorem (.). Also, by [16, Cor. 3.6], the menson of the kernel of ths map s at least j 1) β j. Therefore, ( ) h j h j 1 β j for all j / 1. (8) j 1 Apply Poncaré ualty an Eq. (4) to rewrte ths nequalty n the form ( ) ( ) h j + (β j β j 1 ) h j+1 β j, j j + 1 or, equvalently, ( ) [ ( ] + 1 β j h j+1 h j )β j 1. (9) j + 1 j Let 0 x R be such that h j+1 = ( ) x+1 j+1. Then by Theorem.3, h j ( ) βj 1 ( x j j), an so the rght-han-se of (9) s ( ) x j+1. Also, snce h 1 = n, h j+1 ( ) n +j j+1, an hence x n + j 1. Thus ( ) +1 βj ( ) ( x j+1 j+1 n +j 1 ) j+1, as requre. If equalty occurs for some j = j 0, then x = n + j 0 1, an we obtan that h +1 = ( ) n + +1 = (h ) <> for all j 0. By Theorem.3 ths can happen only f β 1 = 0 for all j 0. Moreover n ths case, h +1 = ( ) n + +1 for all j0, hence s (j 0 + 1)- neghborly (that s, every set of j vertces of s a face of ). What about β for > j 0? To prove that all these Bett numbers vansh as well, note that for equalty ( ) +1 βj0 j 0 = ( n +j 0 ) 1 +1 j 0 +1 to happen, the nequalty n (8) shoul hol as equalty for j = j 0. The same argument as n the proof of [16, Theorem 5.] then shows that h j0 (lk v) = h j0 +1(lk v) for every vertex v of. Snce, by our assumptons, all vertex 10

11 lnks of satsfy the g-conjecture, an snce s (j 0 +1)-neghborly, we conclue that for every vertex v, h (lk v) = h j0 (lk v) for all j 0 ( 1)/, an that h(lk v) = h(lk w) for all vertces v an w of. Ths nformaton about lnks turns out to be enough to compute the entre h-vector of. Inee, t follows from [5, Remark 4.3] that ( ) r 1 h r ( ) = ( 1) r r 1 ( 1 )!! + ( 1) n h (lk v). r ( r)!r! =0 Hence ( ) [ ] + 1 r 1 g r+1 ( ) = h r+1 h r = ( 1) r+1 ( 1) r h (lk v) + n r + 1 ( ) ( ) + n h r(lk v) (r + 1) ( ), +1 r+1 an snce h j0 (lk v) = h j0 +1(lk v) =, we nfer that for all j r ( 1)/, [ ] g r+1 ) + g r 1 ) = n h j0 (lk v) ( r + 1) ( 1 ) + (r + 1) ( ) r ( ) = r 1 r+1 r ( +1 r+1 Therefore, ( +1 r ( +1 r+1 ( +1 j 0 +1 =0 ( +1 j 0 +1 ( 1) r j g 0 r+1 ) = g j 0 +1 ) = h j 0 +1 h j0 ) = β j0. Substtutng ths result n Eq. (9) wth j = j an usng that all β for < j 0 vansh, yels ( [ +1 j 0 +1) βj0 +1 h j0 + + ( ) ] [ βj0 j 0 h + j0 +1 ( ) ] βj0 j 0 = g +1 j0 + + ( +1 j 0 +) βj0 = 0, an so β j0 +1 = 0. Assumng by nucton that β j0 +1 =... = β r 1 = 0, a smlar computaton usng Eq. (9) wth j = r then mples that β r = 0 for all j 0 < r ( 1)/. Kala conjecture [6, Conj. 14.] that f s a ( 1)-mensonal manfol wthout bounary, then h j+1( ) h j ( ) ( j) βj ( ). Ths s an mmeate consequence of Eq. (9). Thus Kala s conjecture hols for all manfols whose vertex lnks have the har Lefschetz property. 5 Rgty nequalty for manfols wth bounary In ths secton we return to our scusson of the face numbers of homology manfols wth nonempty bounary. The goal here s to strengthen Kala s result [6, Theorem 1.3] assertng that f s a ( 1)-mensonal manfol wth bounary an 3, then h ( ) f 0 ( ), where as n Secton 3, f 0 ( ) enotes the number of nteror vertces of. Our man result s Theorem 5.1 If s a connecte ( 1)-mensonal homology manfol wth nonempty orentable bounary an 5, then ( ) h ( ) f0 ( ) + β 1 ( ) + β 0 ( ). (10) 11

12 If = 4 an the characterstc of k s two, then h ( ) f 0 ( ) + 3 β 1 ( ) + 4 β 0 ( ). Snce the bounary of a 3-manfol wth bounary s a collecton of close surfaces, usng a fel whose characterstc s two maxmzes the relevant Bett numbers, so we have restrcte ourselves to ths case. Before begnnng the proof of ths theorem we establsh some prelmnary results pertanng to rgty n characterstc p > 0. In characterstc zero the cone lemma, glung lemma, an Proposton 5.5 follow easly from the work of Kala [6] an Lee [10]. Defnton 5. A ( 1)-mensonal complex s k-rg f for generc θ 1,..., θ +1 lnear forms an 1 +1, multplcaton θ : k[ ]/(θ 1,..., θ 1 ) 1 k[ ]/(θ 1,..., θ 1 ) s njectve. It follows from Proposton 5.5 below that f s k-rg, then the k-menson of k[ ]/(θ 1,..., θ ) s h ( ) an of k[ ]/(θ 1,..., θ +1 ) s g ( ). In fact, t s not har to see (but we wll not use t here) that the converse hols as well. Lemma 5.3 (Cone lemma) If s k-rg, then the cone on, C( ), s k-rg. Proof: Observe that k[c( )] = k[ ] k k[x 0 ]. Hence for any θ 0 of the form x 0 + n =1 α x, θ 0 s a non-zero-vsor on k[c( )] 1 an the quotent rng k[c( )]/(θ 0 ) s somorphc to k[ ]. The asserton follows. Lemma 5.4 (Glung lemma) If 1 an are ( 1)-mensonal k-rg complexes an there are at least vertces n 1, then 1 s k-rg. Proof: Set = 1. Snce l (l = 1, ) s a subcomplex of, there s a natural surjecton k[ ] k[ ] l. Conser the followng commutatve square. k[ ]/(θ 1,..., θ 1 ) k[ 1 ]/(θ 1,..., θ 1 ) 0 θ θ (11) k[ ]/(θ 1,..., θ 1 ) 1 k[ 1 ]/(θ 1,..., θ 1 ) 1 0. Here, θ s the mage of θ n k[ 1 ]. Suppose ω s n the kernel of the left-han vertcal map. Then ts mage n k[ 1 ]/(θ 1,..., θ 1 ) 1 must be n the kernel of the rght-han vertcal map, an hence zero when restrcte to k[ 1 ]/(θ 1,..., θ 1 ) 1. Smlarly, ω s zero when restrcte to k[ ]/(θ 1,..., θ 1 ) 1. But, f there are at least vertces n 1, then ω = 0 n k[ ]/(θ 1,..., θ 1 ) 1. 1

13 Proposton 5.5 Let 1,..., b be k-rg ( 1)-mensonal complexes wth sjont sets of vertces. If =, then for generc lnear forms Θ = (θ 1,..., θ ) an ω, ( ) ] m k (k[ ]/Θ) = h ( )+ (b 1) an m k ker [(k[ ]/Θ) ω 1 (k[ ]/Θ) = (b 1). Proof: Suppose that w s n the kernel of θ 1 : k[ ] 1 k[ ]. Usng a commutatve square analogous to (11), we see that restrcte to each vertex set w s zero. Hence w = 0. Therefore, m k (k[ ]/(θ 1 )) = m k k[ ] m k k[ ] 1 = (f 1 + f 0 ) f 0 = f 1. Now replace k[ ] wth k[ ]/(θ 1 ) n (11) an conser multplcaton by θ. The same argument shows that any w n the kernel must restrct to a multple of θ 1 on the vertex set of each j. The menson of the space of such w n (k[ ]/(θ 1 )) 1 s b 1. Thus, m k (k[ ]/(θ 1, θ )) = f 1 (f 0 1) + (b 1). Contnung wth ths reasonng we see that for each the menson of the kernel of multplcaton by θ on (k[ ]/(θ 1,..., θ 1 )) 1 s ( 1)(b 1). Hence, for, ( ) ( ) 1 1 m k (k[ ]/(θ 1,..., θ 1 )) = f 1 ( )f (b 1). Settng = + 1 fnshes the proof. Proof of Theorem 5.1: Frst we conser the stuaton when 5. Let Γ be the smplcal complex obtane from by conng off each component of the bounary of. Specfcally, let c 1,..., c b be the components of the bounary of. We ntrouce new vertces n + 1,..., n + b an set Σ = ((n + 1) c 1 ) ((n + b) c b ) an Γ = Σ. Then Γ s a ( 1)-mensonal pseuomanfol that s k-rg. The proof s by nucton on. Any homeomorphc to S s k-rg. Ths follows from [14, Cor. 3.5]. So the cone lemma mples that the close star of a vertex n a three-mensonal k-homology sphere s k-rg. Now, usng the glung lemma we can take the unon wth close stars of other vertces untl we see that an arbtrary three-mensonal k-homology sphere s k-rg. Snce for every vertex v, the lnk of v n Γ s a k-homology sphere, nucton on mples that ths lnk s k-rg. Hence the close star of v n Γ s k-rg for all v. Takng the unon of the close stars of the noncone ponts usng the glung lemma shows that Γ s k-rg. Observe that f 0 (Γ) = f 0 ( ) + b an f 1 (Γ) = f 1 ( ) + f 0 ( ). Thus h (Γ) = h ( ) + h 1 ( ) ( 1)β 0 ( ). For Σ we have f 0 (Σ) = f 0 ( )+b an f 1 (Σ) = f 1 ( )+f 0 ( ). Hence, h (Σ) = h ( ) ( 1)β 0 ( ). 13

14 Conser the face rngs k[γ] an k[σ], an let θ 1,..., θ, ω k[γ] 1 be generc lnear forms. Snce Σ s a subcomplex of Γ, there s a natural surjecton φ : k[γ] k[σ]. Let θ enote the mage of θ uner φ, an conser k(γ) := k[γ]/(θ 1,..., θ ) an k(σ) := k[σ]/(θ 1,..., θ ). Then φ nuces a surjecton k(γ) k(σ). Denotng by I k(γ) ts kernel, we obtan the followng commutatve agram whose rows are exact: 0 I k(γ) k(σ) 0 ω ω ω 0 I 1 k(γ) 1 k(σ) 1 0. (1) Snce Γ s k-rg, m k(γ) = h (Γ) an the mle vertcal map s an njecton. Hence the left vertcal map s also an njecton. By the cone lemma an the argument whch prove that Γ s k-rg, each of the b components of Σ s k-rg. Proposton 5.5 says that m k k(σ) = h (Σ) + ( ) β0 ( ). By Proposton 5.5, the menson of the kernel of the rght vertcal map s β 0 ( ). Applyng the snake lemma, we fn that the menson of the cokernel of ω : I 1 I s at least β 0 ( ) an thus m I 1 + β 0 ( ) m I. What are the mensons of I 1 an I? From exactness of rows, we nfer that an m I 1 = m k(γ) 1 m k(σ) 1 = (f 0 ( ) + b ) (f 0 ( ) + b ) = f 0 ( ), (13) m I = m k(γ) m k(σ) = ( ) = h (Γ) h (Σ) β 0 ( ) ( ) = h ( ) g ( ) β 0 ( ) [( ) ( ) ] ( ) h ( ) β 1 ( ) β 0 ( ) β 0 ( ) ( ) = h ( ) β 1 ( ), (14) where the penultmate step follows from [16, Theorem 5.] apple to connecte components of an from the observaton that for a ( )-mensonal complex, ts g - number equals the sum of the g -numbers of ts connecte components mnus ( ) β0 ( ). Comparng the rght-han-ses of (13) an (14) an usng m I 1 + β 0 m I, mples the result. Two mofcatons are necessary when = 4. Frst, each component of the bounary of s a close surface, so the Dehn-Sommervlle relatons tell us that the g of each component s 3β 1. Secon, to show that Σ s k-rg the nucton must begn wth any close surface nstea of just S. In hs thess [4], Fogelsanger prove that any trangulaton of a close surface s genercally 3-rg n the graph-theoretc sense. Fogelsanger use 14

15 three propertes of generc 3-rgty: a cone lemma, a glung lemma, an a result of Whteley s concernng vertex splttng []. Our cone lemma an glung lemma cover the frst two. Whteley s vertex splttng result, combne wth [10, Theorem 10] ue to Carl Lee, s characterstc nepenent. Hence, Fogelsanger s proof shows that a trangulaton of a close surface s k-rg. Now we gve a seres of examples that show that for any 5, β 1, β 0 an f 0, Theorem 5.1 s optmal. We recall a famly of complexes ntrouce by Kühnel an Lassman. Theorem 5.6 [9] For every 4 an n 1 there exsts a complex M (n) wth n vertces such that M (n) s a B -bunle over the crcle. In partcular, M (n) s a manfol wth bounary. Depenng on the party of n an the bounary of M (n) s ether S S 1 or the nonorentable S -bunle over the crcle. Hence, for 5, the frst Bett number of M (n) s one for any fel. When = 4 an the characterstc of k s, then β 1 ( M 4 (n)) =. h (M (n)) = ( ). All of the vertces are on the bounary of M (n). The lnk of every vertex s combnatorally equvalent to a stacke polytope. Evently, M (n) for 5 s an example of equalty n Theorem 5.1 wth β 1 ( ) = 1 an f0 = 0. For spaces wth β 1 ( ) > 1, begn wth two sjont copes of M (n). Choose two ( )-faces on ther respectve bounares an a bjecton between ther vertces. Now entfy these vertces an assocate faces accorng to the chosen bjecton. The resultng space has no nteror vertces an s a manfol wth bounary whose bounary s topologcally the connecte sum of two copes of the bounary of M (n). Thus the frst Bett number s now two. Drect computaton shows that h of the new space s ( ). Repeatng ths operaton of connecte sum along the bounary b tmes wth M (n) prouces an example of equalty n Theorem 5.1 wth β 1 ( ) = b an f0 = 0. To construct wth f0 = m > 0 smply take a complex wth f0 = 0 an subve a facet m tmes. Each such subvson ncreases h an f0 by one whle leavng the topologcal type of the complex unchange. To prouce spaces wth β 0 ( ) > 0, begn wth any of the above examples. It s possble to subve a facet tmes so that there s now a facet wth nteror vertces. See [3] for the algorthm. Removng the open facet leaves a manfol whose bounary has two components, the orgnal an the bounary of the smplex. The new space wll have the same number of nteror vertces an ts h wll have ncrease by. In menson three, the same constructons lea to examples of equalty n Theorem 5.1 wth arbtrary f, β 0, an even β 1. Snce the bounary of a three-mensonal manfol must have even Euler characterstc, ths s the best we can hope for. 15

16 All of the complexes constructe usng the proceures have the property that the lnk of every bounary vertex s combnatorally a stacke polytope, an the lnk of every nteror vertex s a stacke sphere. Conjecture 5.7 If s a connecte ( 1)-mensonal homology manfol wth nonempty orentable bounary an 4, then equalty occurs n Theorem 5.1 f an only f all of the lnks of are combnatorally equvalent to stacke polytopes or stacke spheres. References [1] U. Brehm an W. Kühnel, Combnatoral manfols wth few vertces, Topology 6 (1987), [] B.F. Chen an M. Yan, Lnear Contons on the Number of Faces of Manfols wth Bounary, Av. n Appl. Math. 19 (1997), [3] J. Chestnut, J. Sapr an E. Swartz, Enumeratve propertes of trangulatons of sphere bunles over S 1, European J. Combnatorcs 9 (008), [4] A. Fogelsanger, The generc rgty of mnmal cycles, Ph.. thess, Cornell Unversty, [5] P. Hersh an I. Novk, A short smplcal h-vector an the upper boun theorem, Dscrete Comput. Geom. 8 (00), [6] G. Kala, Rgty an the lower boun theorem I, Invent. Math. 88 (1987), [7] D. Klan, Dehn-Sommervlle relatons for trangulate manfols, unpublshe manuscrpt avalable at [8] V. Klee, A combnatoral analogue of Poncaré s ualty theorem, Canaan J. Math. 16 (1964), [9] W. Kühnel an G. Lassmann, Permute fference cycles an trangulate sphere bunles, Dsc. Math. 16 (1996), [10] C. W. Lee, Generalze stress an motons, n Polytopes: abstract, convex an computatonal (Scarborough, ON, 1993), 49 71, NATO Av. Sc. Inst. Ser. C Math. Phys. Sc., 440, Kluwer Aca. Publ., Dorrecht, [11] F. H. Lutz, Trangulate Manfols wth Few Vertces: Combnatoral Manfols, arxv:math/ [1] I. G. Maconal, Polynomals assocate wth fnte cell-complexes, J. Lonon Math. Soc. () 4 (1971),

17 [13] W. J. R. Mtchell, Defnng the bounary of a homology manfol, Proc. Amer. Math. Soc. 110 (1990), [14] S. Mura, Algebrac shftng of strongly ege ecomposable spheres, ArXv: [15] I. Novk, Upper boun theorems for homology manfols, Israel J. Math. 108 (1998), [16] I. Novk an E. Swartz, Socles of Buchsbaum moules, complexes an posets, arxv: [17] I. Novk an E. Swartz, Gorensten rngs through face rngs of manfols, n preparaton. [18] P. Schenzel, On the number of faces of smplcal complexes an the purty of Frobenus, Math. Z. 178 (1981), [19] R. Stanley, The number of faces of a smplcal convex polytope, Av. n Math. 35 (1980), [0] R. Stanley, Combnatorcs an Commutatve Algebra, Boston Basel Berln: Brkhäuser, [1] E. Swartz, Face enumeraton - from spheres to manfols, J. Eur. Math. Soc., to appear, math.co/ [] W. Whteley, Vertex splttng n sostatc frameworks, Struc. Top. 16 (1989),