1.., M, M.,... : M. M?, RP 2 6, 2 S 1 S 1 7 ( 1 ).,, RP 2, S 1 S 1 6, 7., : RP 2 6 S 1 S 1 7,., 19., 4
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1 1.., M, M.,... : M. M?, RP 6, S 1 S 1 7 ( 1 ).,, RP, S 1 S 1 6, 7., : RP 6 S 1 S 1 7,., 19., RP 3 S 1 S 1 S 1., i., 10. h. h.,., h, h.,,,..
2 . (, )., i f i ( ). f 0 ( ), f 1 ( )., β i ( ; F) F H i ( ; F). ( ) M f min 0 (M) = min{f 0 ( ) : M }...1. M, f min 0 (M)..1..,., 1 RP S 1 S 1., RP 6, S 1 S 1 7,., Heawood.. (Heawood [He]). M v, ( ) v 3 (1) 3( χ(m))., χ(m) M... v f 1 ( ) + f ( ) = χ( ), f 1 ( ) = 3f ( )..,, 3., f 1 ( ) ( n ), f ( ) v χ(m) = 1 3 f 1( ) 1 ( ) v 3. v χ(m) 3( 1 v ) (, v 3 ) 3( χ(m)). Heawood RP, S 1 S 1. χ(rp ) = 1, χ(s 1 S 1 ) =,. ( ) f0 min (RP ) 3 3, f min 0 (RP ) 6. ( ) f0 min (S 1 S 1 ) 3 6, f min 0 (S 1 S 1 ) 7.., f0 min. S g = (S 1 S 1 ) #g g, N g = (RP ) #g g,, M #g M g., Heawood..3 (Jungerman and Ringel [Ri, JR] 1955, 1980). M S, N, N 3, (1) v M v.
3 , M,,.3., 1955 Ringel, 1980 Jungerman Ringel., Heawood Jungerman Ringel,, 1... M S, N, N 3, f0 min (M) = min { ( ) v 3 v N : 3( χ(m)) }., 3 S, N, N 3, f min 0 (S ) = 10, f min 0 (N ) = 8, f min 0 (B 3 ) = 9 (. )... ( )..,,,..,. M, f0 min (M). ( ) f0 min (M) ( ). ( ) f0 min (M) ( ). ( ), ( )., ( ),. ( ) ( ). ( ) Heawood, 3. ( ) 1987 Brehm Kühnel [BK1].. S n f0 min (S n ) = n +,,?..5 (Brehm Kühnel 1987). n f 0 ( ) 3 n + 3., RP, CP, HP, OP , n =,, 8, 16. ( f min 0 ). n =, RP 6. n =,, CP 9 [KB]. 1..,. link PL. PL. 3, [EK] combinatorial manifold.
4 n = 8, , HP Brehm Kühnel,..7 (Brehm Kühnel 1987). M n, i < n H i(m) 0, f min 0 (M) n + i..8. i = n., n i = n 3 n +, 3 n S i S j. f0 min (S i S j ) i + j + (, i j ),. f0 min (S 3 S ) = 1. f0 min (S 3 S 3 ) = 13. f0 min (S S ) = 11. f0 min (S d 1 S 1 ) d d + 3, d d +. f0 min (S d 1 S 1 ) d d + 3, d d +., S d 1 S 1 S 1 S d 1 5. S S i + j +, 10 S S [KL] 11., S d 1 S 1, S d 1 S 1 ( ) [BD, CSS]. (i, j), f0 min (S i S j ). ([KN], i + j +.),.5.7, Heawood. (.10.).10 (Kühnel [Kü] 1990). M v ( ) v 10(χ(M) ) ,. (S S ) # 1, K3 16, χ((s S ) # ) = 6, χ(k3) =. f0 min ((S S ) # ) = 1. f0 min (K3) =. 3 HP.,. Pontrjagin? 5 S 1 S d 1 [Ste].
5 .3. ( )..,.7.., F..1 (Novik Swartz [NS3], M [Mu] 6 ). k M v ( ) ( ) v k k + 1 β k (M; F). k + 1 k + 1, F = Z/Z, k = 1 Heawood, k = Kühnel., f0 min...13 (Novik Swartz [NS1], Bagchi [Ba], Datta-M [DM], M [Mu] 7 ). n 3. n M v ( ) ( ) v n 1 n + β 1 (M; F). M (S n 1 S 1 ) #β 1( ) (S n 1 S 1 ) #β 1( )., H 1 (M) 0 f min 0 (M) d + 3,.7 i = 1..1., S 1 S n 1., Lutz Slanke Swartz [LSS] f0 min (S S 1 ) #b f0 min (S S 1 ) #b. b =,..., 8, 10, 11, 1 (. [LSS, Table 1].).13 tight,,. [DS] f0 min ((S n 1 S 1 ) #n +5n+6 ) = n + 5n + 5 (d: ). [DS] f0 min ((S n 1 S 1 ) #n +5n+6 ) = n + 5n + 5 (d: ). [BDSS] f0 min ((S S 1 ) #99 ) = 9, f0 min ((S S 1 ) #08 ) = 69, f0 min ((S S 1 ) #357 ) = 89, f0 min ((S S 1 ) #56 ) = 109, f0 min ((S 3 S 1 ) #13 ) = 71, f0 min ((S 3 S 1 ) #3 ) = 101, f0 min ((S S 1 ) #390 ) = 97., n ( ).13 ( )..13 i 1.15 (Kühnel). 1 r < n. n M v, ( ) ( ) v n + r n + β r (M; F). r + 1 r [NS3]. [Mu]. 7 [NS1]. 3 [Ba, DM], [Mu].
6 . Lutz [Lu] h.1.13,, h. h,. h,, n, f( ) = (f 0 ( ), f 1 ( ),..., f n ( )) f., f. f Dehn Sommerville... M, f 0 ( ) f 1 ( ) + f ( ) = χ(m) 3f 1 ( ) = f ( ). ( ). f 1, f f 0, f( ) = ( f 0 ( ), 3(f 0 ( ) χ(m)), (f 0 ( ) χ(m)) )., f( ) , f( ) f 0 ( ), f 1 ( ), f ( ), f 3 ( ), f 0 ( ) f 1 ( ) + f ( ) f 3 ( ) = 0 f ( ) = f 3 ( ), f( ) = ( f 0 ( ), f 1 ( ), f 1 ( ) f 0 ( ), f 1 ( ) ), f( ) f 0 f 1.,, 3,? f, f,., h. (n 1), h h( ) = (h 0 ( ), h 1 ( ),..., h n ( )) Z n+1 h i ( ) = i j=0 ( 1) i j ( n j i j ) f j 1 ( ) ( f 1 ( ) = 1 )., f i 1 ( ) = i ( n j ) j=0 hj ( ), i j f( ) h( ). 8 f min 0 (RP 3 ) = 11, f0 min (RP ) = 16 f0 min (RP 5 ).. 0 (S 1 S 1 S 1 ) f min
7 , f, f,. 3.1 (Dehn Sommerville [Kl]). (n 1),. ( ) n h i ( ) = h n i ( ) + ( 1) n i (χ( ) χ(s n 1 )) (i = 0, 1,..., n). i 3... S, f( ) = (6, 1, 8). h h( ) = (1, 3, 3, 1) h i ( ) = h 3 i ( ) Dehn Sommerville.,.. F, V, S = F[x v : v V ]., 1., I I = (x v1 x v x vk : {v 1,..., v k } V, {v 1,..., v k } ) S. F[ ] = S/I ( F ). (n 1), F[ ] Krull n., F, n θ 1,..., θ n, F[ ]/(θ 1,..., θ n )F[ ] ( 0 ). θ 1,..., θ n F[ ] (linear system of parameters) , F[ ] = F[x 1, x,..., x 6 ]/(x 1 x, x 3 x, x 5 x 6 ).. x 1 x, x 3 x, x 5 x 6 F[ ]., F[ ]/(x 1 x, x 3 x, x 5 x 6 ) = F[x 1, x 3, x 5 ]/(x 1, x 3, x 5). ( R, R = R 0 R 1 R R 3, dim F R 0 = 1, dim F R 1 = 3, dim F R = 3, dim F R 3 = 1.),. n S n, Dehn Sommerville h i ( ) = h n i ( ). 10 R = s i=0 R i ( s ) (Poincaré duality algebra), R s = F, R i R s i R s i = 0, 1,..., s perfect pairing 11, R i R i. R R i = Rs i R i Hom F (R s i, R s )
8 , Dehn Sommerville ([Sta] ). 3. ( ( )). (n 1), Θ = θ 1,..., θ n F[ ].. (1) dim F (F[ ]/ΘF[ ]) k = h k ( ) for k = 0, 1,..., n. () F[ ]/ΘF[ ] n.,. 3., Cohen-Macaulay,,.,. (n 1), h - h ( ) = (h 0( ), h 1( ),..., h n( )) 1. { h i ( ) = h i ( ) ( n i) ( i j=1 β j 1( ; F)), (i n ) h n ( ) ( ) n i ( n 1 j=1 β j 1( ; F)), (i = n ). (, h n( ) = β n ( ; F) ), S 1 S 1 7 f( ) = (7, 1, 1), h( ) = (1,, 10, 1). β 1 (S 1 S 1 ) = h ( ) = h( ) (0, 0, 3, 1) = (1,,, 1). (!) ( 3.1 ). 3.6 (h - Dehn Sommerville [No]). (n 1),. h i ( ) = h n i( ) (i = 0, 1,..., n). Krull n S- W Θ = θ 1,..., θ n, W Σ(Θ; W ) 13. n Σ(Θ; W ) = ΘW + (θ 1,..., ˆθ i,..., θ n )W : W θ i. i=1, W W S f, W : W f = {m W : fm W }. Buchsbaum [Go].,, 3. ΘF[ ] Σ(Θ, F[ ]),. 1 h. h i = h i + ( n i) βi 1 (i n), h n = h n. 13 ΘW + n. 5 1
9 3.7 ( ( ) [NS1, NS]). (n 1) M, Θ = θ 1,..., θ n F[ ]. (1) dim F (F[ ]/Σ(Θ; F[ ])) k = h k ( ) for k = 0, 1,..., n. () M char(f) = (F[ ]/Σ(Θ; F[ ])) n. 3.8.,. F = Z/Z, RP 6. ( ) x1 x I = x 3, x 1 x x 5, x 1 x 3 x 6, x 1 x x 5, x 1 x x 6 x x 3 x, x x x 6, x x 5 x 6, x 3 x x 5, x 3 x 5 x 6, Θ = (x 1 + x 3 + x 5, x + x 3 + x 5, x + x 5 + x 6 ) 1 F[ ]. Σ(Θ, R) I Θ, x 1 x 3 + x x 5 + x 3 x 5 + x 1 x 6 + x 3 x 6 + x 5 x 6, X = x 1 x + x 3 x 5 + x 1 x 6 + x 3 x 6 + x 5 x 6,. x x + x x 5 + x 3 x 5 + x x 6 + x 3 x 6 + x x 6 + x 5 x ( ) F[x 1,..., x 6 ]/(I + (Θ) + (X)) = F[x, y, z]/(x + xy + yz, y + xz + yz, z + xy + xz) M, M. Γ, f i (, Γ) Γ i, β i (, Γ; F) = dim F Hi (, Γ; F)., h(, Γ), h (, Γ). 3.9 (Dehn Sommerville [MN] (essentially [Gr])). (n 1) h i ( ) = h n i(, ) (i = 0, 1,..., n)., Γ F[, Γ]. F[, Γ] = I Γ /I 3.10 ([MNY]). (n 1), R = F[ ], W = F[, ], Θ = θ 1,..., θ n R. M char(f) =,. (1) dim F (R/Σ(Θ; R)) k = h k ( ) for k = 0, 1,..., n. () dim F (W/Σ(Θ; W )) k = h k (, ) for k = 0, 1,..., n. (3), (R/Σ(Θ; R)) i (W/Σ(Θ; W )) n i (W/Σ(Θ; W )) n perfect pairing
10 , (W/Σ(Θ; W )) n = F, R/Σ(Θ; R) W/Σ(Θ; W ) W/Σ(Θ; W ) well-defined [MNY]. 3..?,., M k. M v, ( ) ( v k k+1 k+1 ) βk (M; F). R = F[ ], Θ R k+1,. (1) dim F (R/ΘR) i dim F (R/Σ(Θ, R)) i = ( ) k+1 βi 1 (M; F), i () f f Σ(Θ, R) = 0., R/ΘR w, () (R/(Θ, w)r) k+1 dim F (R/ΘR) k+1 dim F (R/Σ(Θ, R)) k, (R/Σ(Θ, R)) k = (R/Σ(Θ, R))k+1 (1)., F[ ] n, Θ, w F k + ( n k k + 1 ) = dim F (F[x 1,..., x n k ]) k+1 dim F (R/(Θ, w)r) k+1 ( ) k + 1 β k (M; F). k + 1 (.1 k + 1 F-.).13.,. 3.1 ([NS1, Ba, DM, Mu]). n 3. n M v,. ( ) n + f 1 ( ) (n + 1)v + (β 1 (M; F) 1)., M (S n 1 S 1 ) #β 1( ) (S n 1 S 1 ) #β 1( ).,.,,, f 1 ( ) f 1 ( ) ( v ) , M. R = F[ ]. Θ = θ 1,..., θ n+1 w w : (R/ΘR) n (R/ΘR) n 1 [Sw]., (1), () ( ) n + 1 dim F (R/(Σ(Θ; R))) n 1 + β n 1 (M; F) dim F R/(Σ(Θ; R))) n 1, h 1 h. f.
11 3.5.., [KN, Sw]., 3 [LSS] 1., h. [Mu1]. References [Ba] B. Bagchi, The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds, European J. Combin. 51 (016), [BD] B. Bagchi and B. Datta, Minimal triangulations of sphere bundles over the circle, J. Combin. Theory, Ser. A 115 (008), [BK1] U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 6 (1987), [BK] U. Brehm and W. Kühnel, 15-vertex triangulations of an 8-manifold, Math. Ann. 9 (199), [BDSS] B.A. Burton, B. Datta, N. Singh and J. Spreer, A construction principle for tight and minimal triangulations of manifolds, arxiv: [CK] M. Casella and W. Kühnel, A triangulated K3 surface with the minimum number of vertices, Topology 0 (001), [CSS] J. Chestnut, J. Sapir and E. Swartz, Enumerative properties of triangulations of spherical bundles over S 1, European J. Combin. 9 (008), [DM] B. Datta and S. Murai, On stacked triangulated manifolds, arxiv: [DS] B. Datta and N. Singh, An infinite family of tight triangulations of manifolds, J. Combin. Theory, Ser. A 10 (013), [EK] J. Eells and N.H. Kuiper, Manifolds which are like projective planes, Publ. Math. I.H.E.S. 1 (196), 181. [Go] S. Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Alg. 85 (1983), [Gr] H-G. Gräbe, Generalized Dehn-Sommerville equations and an upper bound theorem, Beiträge Algebra Geom., 5 (1987), [He] P.J. Heawood, Map-color theorem, Quart. J. Pure Apple. Math. (1890), [JR] M. Jungerman and G. Ringel, Minimal triangulations on orientable surfaces, Acta. Math. 15 (1980), [Kü1] W. Kühnel, Higher dimensional analogues of Császár s torus, Results. Math. 9 (1986), [Kü] W. Kühnel, Triangulations of manifolds with few vertices, In: Advances in Differential Geometry and Topology (F. Tricerri, ed.), 59 11, World Scientific, [KB] W. Kühnel and T.F. Banchoff, The 9-vertex complex projective plane. Math. Intelligencer 5 (1983),11. [KL] W. Kühnel, and G. Lassmann, The unique 3-neighborly -manifold with fewvertices, J. Combin. Theory, Series A 35 (1983), [KN] S. Klee and I. Novik, Face enumeration on simplicial complexes, arxiv: , 015. [Kl] V. Klee, A combinatorial analogue of Poincaré s duality theorem, Canad. J. Math. 16 (196), [Lu] F. Lutz, Triangulated Manifolds with Few Vertices: Combinatorial Manifolds, arxiv:math/ [LSS] F. Lutz, T. Sulanke and E. Swartz, f-vectors of 3-manifolds, Electron. J. Combin. 16 (009), Research Paper 13, 33 pp. [Mu1],, 56. [Mu] S. Murai, Tight combinatorial manifolds and graded Betti numbers, Collect. Math. 66 (015), Tight triangulation.
12 [MN] S. Murai and I. Novik, Face numbers of manifolds with boundary, Int. Math. Res. Not., to appear, arxiv: [MNY] S. Murai, I. Novik and K. Yoshida, A duality in Buchsbaum rings and triangulated manifolds, arxiv: [No] I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 108 (1998), 5 8. [NS1] I. Novik and E. Swartz, Socles of Buchsbaum modules, complexes and posets, Adv. Math. (009), [NS] I. Novik and E. Swartz, Gorenstein rings through face rings of manifolds, Compos. Math. 15 (009), [NS3] I. Novik and E. Swartz, Applications of Klee s Dehn-Sommerville relations, Discrete Comput. Geom. (009), [Ri] G. Ringel, Wie man die geschlossenen nichtorientierbaren Flächen in möglichst wenig Dreiecke zerlegen kann, Math. Ann. 130 (1955), [Sc] P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), [Sta] R.P. Stanley, Combinatorics and commutative algebra, Second edition, Progr. Math., vol. 1, Birkhäuser, Boston, [Ste] N.E. Steenrod, The classification of sphere bundles, Ann. of Math. 5 (19) [Sw1] E. Swartz, Face enumeration: from spheres to manifolds, J. Eur. Math. Soc. 11 (009), 9 85 [Sw] E. Swartz, Thirty-five years and counting, arxiv , 01. Satoshi Murai, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, , Japan
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