On certain multi-variable rational identities derived from the rigidity of signature of manifolds

Transcription

1 J. Math. Aal. Appl. 453 (207) Cotets lists available at ScieceDirect Joural of Mathematical Aalysis ad Applicatios O certai multi-variable ratioal idetities derived from the rigidity of sigature of maifolds Victor J.W. Guo a,, Fei Ha b a School of Mathematical Scieces, Huaiyi Normal Uiversity, Huai a, Jiagsu , People s Republic of Chia b Departmet of Mathematics, Natioal Uiversity of Sigapore, Block S7, 0 Lower Ket Ridge Road, Sigapore 9076, Sigapore a r t i c l e i f o a b s t r a c t Article history: Received 4 Jauary 207 Available olie 5 April 207 Submitted by M.J. Schlosser Keywords: Sigature Rigidity Fixed poit theorem q-lucas theorem Sog derives certai multi-variable ratioal idetities by studyig torus actios o some homogeeous maifolds ad applyig the Atiyah Bott Segal Siger Lefschetz fixed poit theorem. I this paper, we give a direct proof of these ratioal idetities by usig the q-lucas theorem. Moreover, we also give a similar ew ratioal idetity. 207 Elsevier Ic. All rights reserved.. Itroductio Let M be a 4m dimesioal closed (compact ad without boudary) orieted smooth maifold. Let H 2m (M, R) deote the middle cohomology group of M with real coefficiets. Oe ca itroduce a biliear form B(x, y) = x y, [M], x,y V = H 2m (M,R). This is a symmetric biliear form o V. By the Poicare duality, it is o-degeerate. Let p + ad p be the umber of positive ad egative eigevalues of B respectively. Defie σ(m) =p + p. Whe the dimesio of M is ot divisible by 4, we defie σ(m) to be 0. As cohomology is a homotopy ivariat of M ad σ(m) is determied by cohomology, it is a homotopy ivariat. Moreover, Thom [5] has show that σ(m) is also a bordism ivariat of M. The iteger σ(m) * Correspodig author. addresses: jwguo@hytc.edu.c (V.J.W. Guo), mathaf@us.edu.sg (F. Ha) X/ 207 Elsevier Ic. All rights reserved.

2 V.J.W. Guo, F. Ha / J. Math. Aal. Appl. 453 (207) is called the sigature of M. This topological umber plays a sigificat role i the geometry ad topology of maifolds. To cite some examples, it was used to costruct 4dimesioal topological maifolds, which do ot admit smooth structures; it was used to costruct Milor s 7 dimesioal exotic sphere, i.e., smooth maifolds that are homeomorphic, but ot diffeomorphic, to S 7 ; it appears i surgery theory, which provides importat tools for classificatio of high-dimesioal maifolds. The sigature has profoud ad rich liks to various mathematical theories. The Hirzebruch sigature theorem [8] asserts that σ(m) is equal to the L-geus of M, which is costructed as a polyomial of x tah x Potryagi classes i a way associated to the power series ad therefore relates sigature to the theory of characteristic classes. Moreover, by equippig the maifold M with a Riemaia metric g, oe fids that σ(m) is equal to the idex of a first-order elliptic differetial operator d s, called the sigature operator of the Riemaia maifold (M, g) ad therefore relates the sigature to differetial geometry ad aalysis o maifolds. The celebrated Atiyah Siger idex theorem asserts that the idex of the operator d s is equal to the L-geus of M, makig the Hirzebruch sigature theorem a corollary of the Atiyah Siger idex theorem. I this paper, we study some liks of sigature to combiatorics. More precisely, we study some combiatorics derived from the sigature of certai homogeeous maifolds. Let G k (C )deote the Grassmaia maifold of k dimesioal complex liear subspaces of C. The G k (C )is compact ad of complex dimesio k( k) or real dimesio 2k( k). It is a homogeeous maifold, which ca be idetified with U()/U(k) U( k). The sigature of G k (C )is kow to be 0 if k( k) isodd, σ(g k (C )) = ( k 2 if k( k) iseve. There are several approaches to compute the sigature of G k (C ). The first method is usig the Hodge theory (see [8]). The secod method is usig the Schubert calculus (see [5]). Other methods use various kids of fixed poit formulas ad the rigidity of the sigature. Let M be a compact smooth maifold with a actio of a coected Lie group G. Let E ad F be two vector budles o M with the lifted G-actios. Let P :Γ(E) Γ(F )be a elliptic operator commutig with the G-actio. The equivariat idex of P is defied to be id(p, h) =tr(h KerP ) tr(h CokerP ), h G, which is a class fuctio o G. The elliptic operator P is called rigid if id(p, h) is a costat idepedet of h G. The sigature operator d s is rigid ad therefore σ(m) =id(d s, id) = id(d s,h), h G. The fixed poit theorems the express the right-had side of the above idetity by the data o the fixed poit sets of h, ad therefore provide tools to compute σ(m). I [2], a self-mappig f : G k (C ) G k (C ), which is homotopic to the idetity map, is costructed ad the the Atiyah Bott Lefschetz fixed poit formula [2] is applied to the sigature complex (i this case also called the Atiyah Siger G-sigature theorem [3]) to get σ(g k (C )). I [9], Hirzebruch ad Slodowy cosider ivolutios o homogeeous maifolds ad express the sigature of the homogeeous maifold as the sigature of self-itersectio submaifold of the ivolutio whe it is homotopic to the idetity. This provides aother method to compute σ(g k (C )). I [3], Sog cosiders the torus actio o homogeeous maifold G/H with the torus beig the commo maximal torus of G ad H, ad aalyzes the fixed poits of this actio ad writes dow the sigature of G/H by the data

3 362 V.J.W. Guo, F. Ha / J. Math. Aal. Appl. 453 (207) o the fixed poit sets usig the Atiyah Bott Segal Siger Lefschetz fixed poit theorem. Whe applied to G k (C ), the author writes dow σ(g k (C )) = k!( k)! σ S i k<j x σ(i) + x σ(j) x σ(i) x σ(j), (.) where x,..., x are idetermiates, S is the set of all permutatios of the set [] := {,..., }, ad k<. Combiig the kow result of σ(g k (C )), the author cocludes the followig result i the case where is eve: Theorem.. Let x,..., x be idetermiates ad let k<. The 0 if 0 ad k (mod 2), x σ(i) + x σ(j) = ( k!( k)! x σ S i k<j σ(i) x σ(j) k 2 otherwise. (.2) The complex quadric Q is the complex hypersurface i CP 2 which is defied by the equatio z z 2 2 =0, where z,..., z 2 are homogeeous coordiates o CP 2. The complex quadric Q is a homogeeous maifold, which ca be idetified with SO(2)/SO(2) SO(2 2). Similarly, i [3], the followig is writte dow σ(q )= 2 ( )! σ B 2 i (x σ() + x σ(i) )(x σ() x σ(i) +) (x σ() x σ(i) )(x σ() x σ(i) ), (.3) where x,..., x, x,..., x are idetermiates ad B is the -th hyperoctahedral group cosistig of all siged permutatios of []. Combiig the kow result of σ(q )(=0or 2 for eve or odd respectively, see [9]), the author cocludes Theorem.2. Let x,..., x be idetermiates. The 2 ( )! σ B 2 i (x σ() + x σ(i) )(x σ() x σ(i) +) (x σ() x σ(i) )(x σ() x σ(i) ) = { if is odd, 0 if is eve. (.4) I this paper, we shall give a direct proof of Theorems. ad.2. Combiig (.) ad (.3), our proof ca be viewed as aother approach to compute the sigature of complex Grassmaias ad complex quadrics. Actually, motivated by more geeral rigidity, the Witte rigidity [4,0,4], Sog [3] also derives some deep idetities ivolvig theta fuctios. We will study those idetities i a forthcomig project. Ispired by the proof of Theorems. ad.2, we shall also give a ew ratioal idetity as follows. Theorem.3. Let x,..., x be idetermiates ad let k<. The k!( k)! σ S i k<j (x σ(i) + x σ(j) )(x σ(i) x σ(j) + ) (x σ(i) x σ(j) )(x σ(i) x σ(j) ) 0 if 0 ad k (mod 2), = ( k 2 otherwise. (.5)

4 V.J.W. Guo, F. Ha / J. Math. Aal. Appl. 453 (207) I the ext sectio, we will give a proof of Theorems..3. Our proof will use a combiatorial iterpretatio of the q-biomial coefficiets ad a special case of the q-lucas theorem. 2. Proof of the idetities I this sectio, we give direct proof of the idetities metioed i the itroductio. Proof of Theorem.. For ay σ S, let X σ = {σ(),..., σ(k)}. The, for σ, τ S, i k<j x σ(i) + x σ(j) x σ(i) x σ(j) = i k<j x τ(i) + x τ(j) x τ(i) x τ(j) if ad oly if X σ = X τ. It is clear that there are k!( k)! differet permutatios σ havig the same X σ. Hece, the left-had side of (.2) ca be simply writte as X [] j []\X x i + x j x i x j. (2.) For ay o-empty subset X of [], deote by iv(x, ) the umber of pairs (i, j) such that i X, j [] \ X ad i > j. Cosider the followig polyomial u<v = X [] (x u x v ) ( ) iv(x,) X [] j []\X j []\X x i + x j x i x j (x i + x j ) (x i x j ) i,j []\X (x i x j ). (2.2) Let x r = x s for some r<s ad suppose r X, s / X, ad X = X \{r} {s}. The (x i x j ) iv(x,)=iv(x, )+s r +, (x i + x j )= (x i + x j ), j []\X i,j []\X j []\X (x i x j )=( ) s r (x i x j ) i,j []\X (x i x j ), which meas that the o-zero summads i (2.2) cacel pairwise, ad so the polyomial (2.2) reduces to 0. Thus, we have proved that, for ay r<s, the term (x r r s )is a factor of (2.2), ad therefore (2.) is a costat idepedet of x,..., x. Now, takig x i = q i ( i ) ad lettig q 0, we see that the sum (2.) is equal to X [] j []\X q i + q j lim q 0 q i q j = X [] The proof the follows from the idetity (see [, Theorem 3.6]) ( ) iv(x,).

5 364 V.J.W. Guo, F. Ha / J. Math. Aal. Appl. 453 (207) X [] q iv(x,) = [ ] := k q (q; q) (q; q) k (q; q) k, (2.3) where (q; q) = ( q) ( q ), ad the q-lucas theorem (see, for example, [6,7,]): [ ] 0 if 0adk (mod 2), = ( k k 2 otherwise. (2.4) Proof of Theorem.2. We shall prove ( )! σ S 2 i { (x σ() + x σ(i) )(x σ() x σ(i) +), if is odd, (x σ() x σ(i) )(x σ() x σ(i) ) = 0, if is eve. (2.5) For σ, τ S, we have (x σ() + x σ(i) )(x σ() x σ(i) +) (x σ() x σ(i) )(x σ() x σ(i) ) = (x τ() + x τ(i) )(x τ() x τ(i) +) (x τ() x τ(i) )(x τ() x τ(i) ) if ad oly if τ() = σ(). It is clear that there are ( )! differet τ s havig the same τ(). Therefore, the left-had side of (2.5) ca be simply writte as i= j []\{i} (x i + x j )(x i x j + ) (x i x j )(x i x j ). (2.6) Similarly to the proof of Theorem., we ca show that (2.6) is also a costat idepedet of x,..., x. The, lettig x i = q i ad q 0, oe sees that (2.6) equals ( ) i, i= as claimed. Sice B is the wreath product of S ad Z 2, the idetity (.4) follows immediately from (2.5). Proof of Theorem.3. The proof is similar to that of Theorem.. Firstly, the left-had side of (.5) ca be simplified as X [] j []\X (x i + x j )(x i x j +) (x i x j )(x i x j ). Secodly, for ay o-empty subset X of [], let iv(x, ) be defied as before. Cosider the followig polyomial u<v = X [] (x u x v )(x u x v ) ( ) iv(x,) j []\X X [] j []\X (x i + x j )(x i x j +) (x i x j )(x i x j ) (x i + x j )(x i x j +) (x i x j )(x i x j ) i,j []\X (x i x j )(x i x j ). (2.7)

6 V.J.W. Guo, F. Ha / J. Math. Aal. Appl. 453 (207) Let x r = x s (or x r x s =) for some r<s. Suppose that r X, s / X, ad X = X \{r} {s}. The iv(x,)=iv(x, )+s r +, j []\X (x i + x j )(x i x j +)= (x i x j )(x i x j ) =( ) s r j []\X i,j []\X (x i x j )(x i x j ) (x i + x j )(x i x j +), (x i x j )(x i x j ) i,j []\X (x i x j )(x i x j ), which implies that the o-zero summads i (2.7) cacel pairwise, ad therefore the polyomial (2.7) reduces to 0. Namely, we have show that, for ay r<s, the term (x r r s )(x r x s ) is a factor of (2.7). This proves that the expressio (2.) is a costat idepedet of x,..., x. Fially, takig x i = q i ( i ) ad lettig q, we kow that (2.) is equal to X [] j []\X (q i + q j )(q i+j + ) lim q (q i q j )(q i+j ) = X [] ( ) iv(x,), where X =[] \ X. The proof the follows from (2.3) ad (2.4). Ackowledgmets The first author was partially sposored by the Natural Sciece Foudatio of Jiagsu Provice (grat BK206304) ad the Qig La Project of Educatio Committee of Jiagsu Provice. The secod author was partially supported by Sigapore AcRF (grat R ). Refereces [] G.E. Adrews, The Theory of Partitios, Cambridge Uiversity Press, Cambridge, 998. [2] M.F. Atiyah, R. Bott, A Lefschetz fixed poit formula for elliptic complexes II, A. of Math. 37 (968) [3] M.F. Atiyah, I.M. Siger, The idex of elliptic operators II, A. of Math. 87 (968) [4] R. Bott, C. Taubes, O the rigidity theorems of Witte, J. Amer. Math. Soc. 2 (989) [5] F. Coolly, T. Nagao, The itersectio pairig o a homogeeous Kähler maifold, Michiga Math. J. 24 (977) [6] J. Désarméie, U aalogue des cogrueces de Kummer pour les q-ombres d Euler, Europea J. Combi. 3 (982) [7] V.J.W. Guo, J. Zeg, Some arithmetic properties of the q-euler umbers ad q-salié umbers, Europea J. Combi. 27 (2006) [8] F. Hirzebruch, Topological Methods i Algebraic Geometry, Spriger-Verlag, Berli, Heidelberg, New York, 966. [9] F. Hirzebruch, P. Slodowy, Elliptic geera, ivolutios, ad homogeeous spi maifolds, Geom. Dedicata 35 (990) [0] K. Liu, O elliptic geera ad theta-fuctios, Topology 35 (996) [] G. Olive, Geeralized powers, Amer. Math. Mothly 72 (965) [2] P. Shaaha, O the sigature of Grassmaias, Pacific J. Math. 84 (979) [3] R. Sog, Elliptic geera of homogeeous spi maifolds ad theta fuctios idetities, Sci. Chia Ser. A 48 (2005) [4] C.H. Taubes, S actios ad elliptic geera, Comm. Math. Phys. 22 (989) [5] R. Thom, Espaces fibré e sphères et carrés de Steerod, A. Sci. Éc. Norm. Supér. 69 (952)