Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/mypapers/keldysh100ru.pdf Äàòà èçìåíåíèÿ: Sat Feb 19 16:41:26 2005 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:04:38 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 13

. . , . . . ­ . , . f - , . , f - h-, , ­ . , , .

1. . . . 1980- , . [13]. , , . , [14] ( ­ ) . [9] . , (unfoldings) . ­ 3- .
1


2

. . , . .

-, . [11]. - 2n- M n- T , . - . Q = M /T , - M . , , Q ( ). , . (, , ) - . , [11], - M Z[Q] Q (, Q ). [11], - , Q . . 2 , . ( ), , , . , ( 2.4). 3 f - h- . f - h-. ( 3.1 3.2). f - . f - f - , f - f - . ­ ( 3.4, 3.5 3.6). 4 , X




3

, ¨ X K K ( 4.1). 5 , [14]. ( 5.3, 5.9, 5.10) ( 5.4 5.5). 6 , S n - 1 m ZS T m , - ZK (. [6] [2, . 7]). , ZS - ( 6.3). X k , T k X . , S n - 1 m ZS m - n ( 6.4). [7] ( 6.5). . (G. Lupton), , . . , 4.1.

2. M K = { } M, K ( ) K . K () K . . K : dim = | | - 1. . , Rn . ( ) P , Rn , , P P P . P K , K P , K P . K


4

. . , . .

, |K |. . S . ord(S ) x1 < x2 < . . . < xk , xi S . , ord(S ) . K K ord(K \ ) ( ) K . S , ^ S 0 [^, ] = { S : ^ } 0 0 ( ) . S , rk ^ = 0 rk = k , [^, ] 0 0 (k - 1)-. dim S = maxS rk - 1. S 1. k k . , . . , X , eq i eq ( ), , i eq q i Dq X ( ), Dq eq . (, , , i .) . (C) eq i er r < q . j (W) Y X , eq Y eq . i i , . S . S \ ^ , 0 [^, ], 0 S . , , . |S |. ,




5

S K , |S | |K |. . 2.1. , (n - 1)- ( ) . n > 1. . , , . , , . : S1 S2 , S1 S2 . , (.. ). , . () . , S S ord(S \ ^ . 0) , S . , . . , , S S , . X , q X , . 2.2. . 1 . , , .


6

. . , . .

'$ '$ '$ s s s s s s &% &% &%

)

)

)

. 1. . 2.3. X , . . X S , , ­ . . , eq i q eq , i ep eq j i p- q . S , X |S |. . , (., , [12]). X . X , q q X X . . . "" "" . , , X X X . , X . ( , ., , [12]).




7

2.4. X X , X . . 2.3, , X . , .. - x, y q - eq X . , q . X , eq q X X q - q . q , x, y q . , , . , . 3. f - ­ X (n - 1). fi i- . f (X ) = (f0 , . . . , fn-1 ) f - X . , f-1 = 1. h- X (h0 , h1 , . . . , hn ), (3.1) h0 tn + . . . + hn
-1

t + hn = (t - 1)n + f0 (t - 1)n

-1

+ . . . + fn-1 .

, f - h- , ,
k n

(3.2) hk =
i=0

(-1)

k-i n-i n-k

fi-1 ,

f

n-1-k

=
q =k

q k

hn

-q

,

k = 0, . . . , n.

, h0 = 1 hn = (-1)n (1 - f0 + f1 + . . . + (-1)n fn-1 ) = (-1)n (1 - (X )). f - , X = S (.. ). S1 S2 S1 S2 , 1 2 , 1 S1 , 2 S2 . : 1 2 1 2 , 1 1 2 2 . , S1 S2 . ( ) |S1 S2 | ( ) |S1 | |S2 |. S , .


8

. . , . .

S1 S2 , 1 S1 2 S2 ( , , 1 2 ). S1 #1 ,2 S2 , S1 S2 1 2 = 1 = 2 , S1 S2 . , 1 2 , S1 # S2 . f - h- S1 # S2 f h- S1 S2 . dim S1 = dim S2 = n - 1, fi (S1 # S2 ) = fi (S1 ) + fi (S2 ) - fn
-1 n i+1

,

i = 0, 1, . . . , n - 2;

(S1 # S2 ) = fn

-1

(S1 ) + fn

-1

(S2 ) - 2.

(3.2) , h0 (S1 # S2 ) = 1; (3.3) hi (S1 # S2 ) = hi (S1 ) + hi (S2 ), i = 1, 2, . . . , n - 1; hn (S1 # S2 ) = hn (S1 ) + hn (S2 ) - 1. dim S1 = n1 - 1 dim S2 = n2 - 1, S1 S2
n1 -1

fk (S1 S2 ) =
i=-1

fi (S1 )fk

-i-1

(S2 ),

k = -1, 0, . . . , n1 + n2 - 1.

h(S ; t) = h0 + h1 t + . . . + hn tn , (3.1) (3.4) h(S1 S2 ; t) = h(S1 ; t)h(S2 ; t).

f - h- . B = (bij ),
i

0

i, j

n - 1;
+1

bij =
k=0

(-1)

k i+1 k

(i - k + 1)j

.

, bij = 0 i > j (.. B ) bii = (i + 1)!. , B . 3.1. S (n - 1)- S . f - S S :
n-1

fi (S ) =
j =i

bij fj (S ),

i = 0, . . . , n - 1;




9

.. f (S ) = B f (S ). . j - j , bij i- (j ) , - j . fi (K ) = n=i1 bij fj (K ). , bij = bij . j , , bij : bij = (j + 1)bi-
1,j -1

+

j +1 2

bi-

1,j -2

+ ... +

j +1 j -i+1

bi-1

,i-1

.

, bij , bij . D = (dpq ),
p

0

p, q
n+1 k

n;
-q

dpq = (

(-1)k 1).
n

(p - k )q (p - k + 1)n

k=0 00 =

3.2. h- S S : hp (S ) =
q =0

dpq hq (S ),

p = 0, . . . , n;

h(S ) = Dh(S ). , D . . 3.1, (3.1) , , , [3]. f f-1 = 1 B , D = C -1 B C , C h- f - ( (3.1)). D. , B D f - h- . , K (n - 1)- , .. |K | S n-1 , h- : = (3.5) hi = hn-i , i = 0, . . . , n. ( f -) ­. , h0 = 1 , h0 = hn , . (3.1). ­, , [2].


10

. . , . .

, [10]. 3.3. ­ , f . . [10] , f -. , h- . , h- (h0 , h1 , . . . , hn ) n - (, h0 = 1 ). 2 , , n + 1 2 h-. Kj := j n-j , j = 0, 1 . . . , n , j 2 j - . h ( j ) = 1+t +. . . +tj , (3.4) , 1 - tj +1 1 - tn-j +1 · . 1-t 1-t , h (Kj +1 ) - h (Kj ) = tj +1 + , j = 0, 1, . . . , n - 1. , h (Kj ), j = 0, 1, . . . , n , 2 2 . h (Kj ) = 3.4. S (n - 1) , h- h(S ) ­ (3.5). . S . 3.2, h (S ) = Dh (S ), ¨ h (S ) , S . , D ( ¨ ) ( , dpq = dn+1-p,n+1-q , .. D ). , D 3.2. , ­ Rn+1 ( h0 , . . . , hn ) W k = n + 1 ( 2 n 2 h0 = 1). D. W e1 , . . . , ek , Dei W i. W h- (. 3.3). , Dei , i = 1, . . . , k , ­. , W D-. , h (S ) = D-1 h (S ) ­. . ­ ( ) [13, (3.40)].




11

[2] ­ . , h- (n - 1)- M : (3.6) hn
-i

- hi = (-1)i (M ) - (S = (-1) (hn - 1)
i n i

n-1

)

n i

=

,

i = 0, 1, . . . , n.

, M = S n-1 n , (3.5). , 3.4, . 3.5. S (n - 1) M . h- h(S ) = (h0 , . . . , hn ) (3.6). . S S D () 3.2 , h (S ) = Dh (S ). A k = n n+1 ( h , . . . , h ), 0 n 2 R . S , , 3.4, A D. A (.. n + 1 ), 2 h- M . . h- n + 1 2 (n - 1)- j n-j , W , hi = hn-i (. 3.3). K # (j n-j ), K M . h- A ( (3.3)). 3.6. 3.4 3.5 . . 2.4. 3.7. . 2 D2 . 0-, 1- 2- . . D2 C, 2 , Re z 0 Im z 0. z z 4 . . 2 ) . , , , 2- . f -


12

. . , . .

(1, 1, 2), h- (1, -2, 2, 1) h1 = h2 ­. h0 = h3 , , , .
'$ '$ s s s

&% &%

)

)

. 2. D2 . , . . 2 ) , . 4. . (.. ) K . K K . , , = . X K . p : X K K , : 1) K p-1 ( ) Ui ( ): p-1 ( ) =
i

Ui ( ),


i = 1, . . . , I ( );

2) p : Ui ( ) i. , Ui ( ), K i, X . 4.1. X , ¨ X K K . . X M. KS M . , M KS , X




13

. X KS , M, . K p : X K . X Ui ( ), K . , , X , , Ui ( ) . , p ( K ). = U i ( ). , p : . , , p : . , ; , p(x1 ) = p(x2 ) = y x1 , x2 . x1 x2 - : [0, 1] X , (0) = x1 , (1) = x2 (s) Ui ( ) 0 < s < 1. p : [0, 1] K y , . , F : [0, 1] â [0, 1] K , F (s, 1) = p (s), F (s, 0) = y F (s, t) 0 < s < 1 0 < t < 1. i = p-1 F ([0, 1], 1 ) i (, 1 = ). i X ( p-1 F ((0, 1), 1 ) ) i . =
N iN

i .

i , . , , p() = y , . . 5. , . , . [15], ­. [14], . , , [11], . [2, 4].


14

. . , . .

, K M [m] = {1, . . . , m}. Z[v1 , . . . , vm ] m . ­ ( ) K - Z[K ] = Z[v1 , . . . , vm ]/IK , IK , ¨ vi1 . . . vik , {i1 , . . . , ik } K . , . Z[K ] , . , K (.. , ). , K . , (.. , ) . , , ¨ . Z[v : K \ ], K . ¨ , deg v = 2| |. , v 1. ¨ " " Z[K ]. 5.1. Z[v : K \ ]/I Z[K ], = I , ¨ v v - v v .
v

.

= 0 -

. , v i vi . S . , S , ( ). ; , , = . ¨ Z[v : S \ ^ , deg v = 2 rk . 0] v^ = 1. 0




15

S - Z[S ] := Z[v : S ]/IS , IS , ¨ (5.1) v v - v


·

v .



, = , v v = 0 Z[S ]. , IS v v , , S , , . . 5.2. S , 2.1 n = 2. , S . 1 ( ), , 1 2 , 2 ( ), , 1 2 . Z[S ] = Z[v1 , v2 , v1 , v2 ]/(v1 + v deg v

1



2

= v1 v2 , v1 v2 = 0), = 4.

= deg v2 = 2, deg v1 = deg v

2

[2, . 3], : K1 K2 : Z[K2 ] Z[K1 ], .. . . 5.3. : S1 S2 . : Z[v : S2 \ ^ Z[v : S1 \ ^ 0] 0] v v ,

S1 , ( ) = dim = dim . Z[S2 ] Z[S1 ] ( ). . , (IS2 ) IS1 . S n. , , k[S ] k ( k = Z k = Q). t1 , . . . , tn k[S ] , k[S ] ¨ k[t1 , . . . , tn ]-. , t1 , . . . , tk k n , dim k[S ]/(t1 , . . . , tk ) = n - k , dim .


16

. . , . .

, , . t1 , . . . , tk k[S ] , k[S ] k[t1 , . . . , tk ]-. , . k[S ] ­ ( S , , ­ ), n = dim k[S ] = dim S + 1. ­ [14]. S s : k[S ] k[S ]/(v : ). dim = k - 1 {i1 , . . . , ik } . , s k[vi1 , . . . , vik ] k . [15, Lemma III.2.4] (. [6, Th. 7.2]). 5.4. t = (t1 , . . . , tn ) k[S ] , S s (t) k[vi : i ]. . , t . s -: k[S ]/(t ) - k[vi : i ]/s (t ). t , dim k[S ]/(t ) = 0, .. dimk k[S ]/(t ) < . , k[vi : i ]/s (t ) < . s (t ) k . S s (t ) k[vi : i ]. dimk
S

k[vi : i ]/s (t ) < .

, s : k[S ] S k[vi : i ] [2, 4.8]. , dimk k[S ]/(t ) < (. [5, Lemma 4.7.1]). , t . , , s (t ) S . , S ( n - 1), t1 , . . . , tn , ¨ (n - 1)- ¨ ( k). KS , S 4.1.




17

, k[KS ] k[S ], . 5.5. Q[S ] . . S , Q[S ] , (., , [5, Th. 1.5.17]). 5.4 , Q[KS ] Q[S ]. 5.6. 5.2 v1 , v2 , v1 v2 x2 - (v1 v2 )x = 0. Z[S ] , , S ( ). (. 6 ). , K , Z[K ] , C n (m) m 2n 8 . [6, Ex. 1.22] ( [2, . 6.33]). 5.7. v1 , . . . , vm Z[S ], S . Z[S ] (5.2) (1 + v1 ) · . . . · (1 + vm ) =
S

v .

. ver S . (5.1) Z[S ] (5.3) vi1 · . . . · vik =
: ver ={i1 ,...,ik }

v .

S , . Z[S ] Nm -, mdeg v = 2 iver ei , ei Nm i- . , mdeg vi = 2ei . , (5.2) (5.3). 5.1 , (5.3) IS , S . , , S Z[S ] Nm - Z[v : S ] (1 + v1 ) · . . . · (1 + vm ) = S v .


18

. . , . .

5.8. S , 2- 1 2 . 1 1 2 2 , , S . Z[S ] v1 v = v1 v2 v = v2 , (5.2) v1 v + v2 v = v1 + v2 . PS = S v [4] 1930 . (, , ). : S Z[S ]? . 5.9. F : Z[K2 ] Z[K1 ] 0 Nm - , , F K1 K2 . . F 0 Nm - . K1 K2 . F (PK1 ) = PK2 , (5.2). . ( ), " Nm - " " ". , F : Z[K2 ] Z[K1 ] , F (PK1 ) = PK2 . , . , . . 5.10. : S1 S2 , (PS2 ) = PS1 . , (PS2 ) = PS1 . 6. K m . [2, §5.2] cc(K ) cone K K . cone(m-1 ) (m - 1)- I m cone K




19

cone(m-1 ) . ( K m-1 .) , [2, §7.2] - ZK , ZK - - (D2 )m -- I
m

,

cc(K ) - - --

m- . , dim K = n - 1, ZK (m + n)- T m . [6] . ZK K 7­8 [2]. , , K (n - 1)- , ZK (m + n)- (., , [2, 7.13]). 4.1 , ZK , . S m . cone S cone K I m , ¨ p : S K 4.1, . ZS T m , : ZS - - ZK - - (D2 )m -- -- . cone S - - cone K - - I m -- -- , 6.1. S (n - 1)- (. 2.1). ZS (D2 )n . , ZS S 2n . = T n [11] -, [6]. ZS . 6.2. () T m ZS T ver T m , S . X K1 , K2 , .. , |K1 | |K2 | X . = = ( ), K , K1 , K2 .


20

. . , . .

(n - 1)- S n-1 ( P L- ), n . P L- S S . 6.3. S n-1 m . Z (m + n) S S . . , S . ( ) Fv v S , Fv := starS v = { S : v S }. k k . S P L-, i- i- . v S Uv cone(S ), S , v . {Uv } cone(S ), Uv Rn + . cone(S ) (. [2, . 6.13]). , ZS = -1 (cone(S )) {-1 (Uv )}, R2n â T m-n . , ZS . T k X , . X k , T k X . trk(X ). S [6, §7.1], . 6.4. S n - 1 m . trk ZS m - n. . Q[S ] t1 , . . . , tn 5.5. ti = i1 v1 + . . . + im vm , i = 1, . . . , n; (ij ) : Qm Qn . , , k k , , Zm Zn , . 5.4 , S |Zver : Zver Zn Zver Zm . N T m , : Zm Zn . N (m - n)- , N




21

T ver T m . 6.2 , N ZS , . ( [7]): X dim H (X ; Q) 2trk(
X)

.

. . 6.4 , , dim H (ZS ; Q) H (ZK ; Q) =
[m]

2m-n . H (K ; Q),

K [1] ,

K K , [m]. ( [2, C. 8.8]). 6.5. , K dim
[m]

H (K ; Q)

2

m-n

.

. , 6.4, . 6.6. T m (m - n), ZS , , Z[S ] . k , T m k , ZS , s(S ) S ( . [2, §7.1]). , - T m ZK (, m > n T ver ). , s(K ) 1 ZK . 6.1, S .
[1] . . , . . . , . . , 2000, . 55:5, . 3­106. [2] . . , . . , . .: - , 2004. [3] . . , . . , . . , . .: , 1981.


22

. . , . .

[4] J. W. Alexander, The combinatorial theory of complexes, Annals of Math. 31 (1930), 292­320. [5] W. Bruns and J. Herzog, Cohen­Macaulay rings, revised edition, Cambridge Studies in Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1998. [6] M. W. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions,. Duke Math. J. 62 (1991), 417­451. [7] S. Halperin, Rational homotopy and torus actions, in: Aspects of Topology, London Math. Soc. Lecture Notes 93 (1985), pp. 1­20. [8] A. Hattori, M. Masuda, Theory of multi-fans, Osaka J. Math. 40 (2003), 1­68. [9] I. Izmestiev, M. Joswig, Branched covering, triangulations, and 3-manifolds, preprint, 2001, arXiv:math.GT/0108202. [10] V. Klee, A combinatorial analogue of Poincar´ duality theorem, Canad. J. Math. 16 e's (1964), 517­531. [11] M. Masuda, T. Panov, On the cohomology of torus manifolds, preprint, 2003, arXiv:math.AT/0306100. [12] S. V. Matveev, Algorithmic Topology and Classification of 3-manifolds, SpringerVerlag, New York, 2003. [13] R. P. Stanley, Enumerative combinatorics, Vol. 1, Wadsworth and Brooks/Cole, Monterey, California, 1986. [ : . , , .: , 1990.] [14] R. P. Stanley, f -vectors and h-vectors of simplicial posets, J. Pure Appl. Algebra 71 (1991), 319­331. [15] R. P. Stanley, Combinatorics and Commutative Algebra, second edition, Progress in Math., vol. 41, Birkh¨ auser, Boston, 1996 (first edition: 1983). . . . E-mail address : buchstab@mendeleevo.ru . . . , - E-mail address : tpanov@mech.math.msu.su