Toric Cohomological Rigidity of Simple Convex Polytopes

Transcription

1 Toric Cohomological Rigidity of Simple Convex Polytopes Dong Youp Suh (KAIST) The Second East Asian Conference on Algebraic Topology National University of Singapore December 15-19, / 28

2 This talk is base on the joint work with Taras Panov (Moscow State Univ.) Suyoung Choi (KAIST). You can see the paper on the web arxiv: / 28

3 Simple polytope A (convex) polytope is the convex hull of a finite set of points in some R n. The dimension of a polytope is defined obviously. Two polytopes are combinatorially equivalent if there is a bijection between their sets of faces that preserves the inclusion relation. 3/ 28

4 Simple polytope A (convex) polytope is the convex hull of a finite set of points in some R n. The dimension of a polytope is defined obviously. Two polytopes are combinatorially equivalent if there is a bijection between their sets of faces that preserves the inclusion relation. An n-dim. polytope is simple if each vertex is the intersection of exactly n facets (faces of dimension n 1). simple not simple 3/ 28

5 Quasitoric manifold A quasitoric manifold M over an n-dim simple convex polytope P is a 2n-dim closed manifold with a locally standard action of T n = (S 1 ) n together with a map π : M P such that π 1 (p) = an orbit in M for all p P. The standard action of T n on C n is defined by (t 1,...,t n )(z 1,...,z n ) = (t 1 z 1,...,t n z n ). 4/ 28

6 Quasitoric manifold A quasitoric manifold M over an n-dim simple convex polytope P is a 2n-dim closed manifold with a locally standard action of T n = (S 1 ) n together with a map π : M P such that π 1 (p) = an orbit in M for all p P. The standard action of T n on C n is defined by (t 1,...,t n )(z 1,...,z n ) = (t 1 z 1,...,t n z n ). Combinatorial structure of P v.s. fixed point data of M: π 1 (F i ) = a component of M S1 i π 1 (F i F j ) = a component of M <S1 i,s1 j >. π 1 (vertex) = a fixed point M Tn where F i is a facet of P and Si 1 is a circle subgroup of T n. 4/ 28

7 Example 1 CP n with the following T n action is a (quasi)toric manifold over an n-simplex n : (t 1,..., t n )[z 0 : z 1 : : z n ] = [z 0 : t 1 z 1 : : z n ] 2 CP n i is a (quasi)toric manifold over n i. 3 Hirzebruch surfaces are (quasi)toric manifolds over a square. 4 Bott towers are (quasi)toric manifolds over cubes I n. (Hirzebruch sufaces are special cases of Bott towers.) 5 CP 2 CP 2 is a quasitoric manifold but not a toric manifold. Bott tower is defined as follows: B m π m B m 1 π m 1 π 2 B 1 π 1 B 0 = {a point}, where B i = P(ξ C) for i = 1,..., m and ξ is a C-line bundle over B i 1. 5/ 28

8 Characteristic Function For a quasitoric manifold manifold π : M P the characteristic function is defined as follows: Let F = {F 1,..., F m } denote the set facets of P. Define λ: F Z n such that 1 λ(f i ) is a primitive vector corresponding to the circle subgroup of T n fixing π 1 (F i ) M, and 2 if F i1,..., F in are intersecting at a vertex, then {λ(f i1 ),...,λ(f in )} forms a basis of Z n. 6/ 28

9 Characteristic Function For a quasitoric manifold manifold π : M P the characteristic function is defined as follows: Let F = {F 1,..., F m } denote the set facets of P. Define λ: F Z n such that 1 λ(f i ) is a primitive vector corresponding to the circle subgroup of T n fixing π 1 (F i ) M, and 2 if F i1,..., F in are intersecting at a vertex, then {λ(f i1 ),...,λ(f in )} forms a basis of Z n. Conversely, for a simple convex polytope with a function λ satisfying above two conditions we can construct a quasitoric manifold M λ whose characteristic function is λ. 6/ 28

10 Topological definition of cohomological rigidity It well-known that the cohomolgy group of a quasitoric manifold contains all the information of the face numbers of the underlying polytope. Question For a fixed polytope P let M be any quasitoric manifold over P, and let N be another quasitoric manifold over Q such that H (M) = H (N) as rings. Does this imply that P and Q are combinatorially equivalent? If the answer is yes, we say P is cohomologically rigid. 7/ 28

11 Topological definition of cohomological rigidity It well-known that the cohomolgy group of a quasitoric manifold contains all the information of the face numbers of the underlying polytope. Question For a fixed polytope P let M be any quasitoric manifold over P, and let N be another quasitoric manifold over Q such that H (M) = H (N) as rings. Does this imply that P and Q are combinatorially equivalent? If the answer is yes, we say P is cohomologically rigid. Theorem (Masuda-Panov, 2007) I n is cohomologically rigid. Question Which polytopes are cohomologically rigid? 7/ 28

12 Example of non-rigid simple polytopes Not all polytopes are rigid! 8/ 28

13 Example of non-rigid simple polytopes Not all polytopes are rigid! vc(p, v) = the vertex cut of P at v. v w 2 P 1 P 2 := vc(p 1, v) w 1 w 3 M 1 = CP 2 CP 1 π P 1 M 2 = blow up of M 1 at π 1 (v) π P 2 8/ 28

14 P 3,i := vc(p 2, w i ) M 3,i = blow up of M 2 at π 1 (w i ) P 3,i M 3,i = (CP 2 CP 1 ) CP 3 CP 3 9/ 28

15 Main results Theorem The following polytopes are cohomologically rigid. 1 Every polygon, i.e., 2-dimensional polytope. 2 Every trianlge-free n-dim simple polytope with facet numbers 2n Any product of simplices, i.e., n i. 4 Any vertex-cut of a product of simplices. 5 Dodecahedron. 10/ 28

16 Main results Theorem The following polytopes are cohomologically rigid. 1 Every polygon, i.e., 2-dimensional polytope. 2 Every trianlge-free n-dim simple polytope with facet numbers 2n Any product of simplices, i.e., n i. 4 Any vertex-cut of a product of simplices. 5 Dodecahedron. We also have rigidity results on some low dimensional polytopes. 10/ 28

17 Rigid 3-polytopes up to 9 facets / 28

18 Rigid 3-polytopes up to 9 facets / 28

19 Rigid 3-polytopes up to 9 facets / 28

20 Rigid 3-polytopes up to 9 facets / 28

21 Rigid 3-polytopes up to 9 facets / 28

22 Rigid 3-polytopes up to 9 facets (ii) / 28

23 Moment angle complex Note that if P n is an n-dim simple convex polytope with m facets (i.e., codim 1 faces), then the dual of its boundary K = ( P) is an (n 1)-dim simplicial complex on the set [m] = {1,..., m}. 12/ 28

24 Moment angle complex Note that if P n is an n-dim simple convex polytope with m facets (i.e., codim 1 faces), then the dual of its boundary K = ( P) is an (n 1)-dim simplicial complex on the set [m] = {1,..., m}. Let K be a simplicial complex on [m]. For a simplex σ = {i 1,...,i k } [m] of K let B σ = {(z 1,..., z m ) (D 2 ) m C m z i = 1 if i / σ} = (D 2 ) σ (S 1 ) m σ The moment angle complex Z K of a simplicial complex K is defined be Z k = σ K B σ C m. 12/ 28

25 Moment angle complex Note that if P n is an n-dim simple convex polytope with m facets (i.e., codim 1 faces), then the dual of its boundary K = ( P) is an (n 1)-dim simplicial complex on the set [m] = {1,..., m}. Let K be a simplicial complex on [m]. For a simplex σ = {i 1,...,i k } [m] of K let B σ = {(z 1,..., z m ) (D 2 ) m C m z i = 1 if i / σ} = (D 2 ) σ (S 1 ) m σ The moment angle complex Z K of a simplicial complex K is defined be Z k = σ K B σ C m. The moment angle complex (manifold) Z P of a simple convex polytope P is defined to be Z K where K = ( P). 12/ 28

26 Moment angle complex, quasitoric manifolds, and polyopes Let P be a simple convex n-polytope with m facets. Let λ : F Z n be a characteristic function. Let Z P denote the moment angle complex of P. Then n m integer matrix (λ(f 1 ),...,λ(f m )) defines a linear map Λ : Z m Z n. 13/ 28

27 Moment angle complex, quasitoric manifolds, and polyopes Let P be a simple convex n-polytope with m facets. Let λ : F Z n be a characteristic function. Let Z P denote the moment angle complex of P. Then n m integer matrix (λ(f 1 ),...,λ(f m )) defines a linear map Λ : Z m Z n. Note that there exist the canonical action of T m on Z P C m. Proposition 1 The action of the subgroup K corresponding to ker Λ = T m n on Z P is free. 2 The residual T n = T m /K action on M := Z P /K makes M a quasitoric manifold over P with λ as its characteristic function. 3 Hence Z P /T m = M/T n = P. 13/ 28

28 Their cohomology rings Definition (face ring of a polytope) For a simple convex polytope P with m facets F 1...,F m, its face ring or Stanley-Reisner ring is R[P] := R[x 1,..., x m ]/ < x i1 x ik F i1 F ik = > where R is a commutative ring with 1, and all x i s have degree 2. The quotienting ideal is called the Stanley-Reisner ideal of P. 14/ 28

29 Their cohomology rings Definition (face ring of a polytope) For a simple convex polytope P with m facets F 1...,F m, its face ring or Stanley-Reisner ring is R[P] := R[x 1,..., x m ]/ < x i1 x ik F i1 F ik = > where R is a commutative ring with 1, and all x i s have degree 2. The quotienting ideal is called the Stanley-Reisner ideal of P. Theorem (Bruns-Gubeladze, 1996) Let P and Q be two n-simple polytopes, and let R be any commutative ring with 1. R(P) = R(Q) as R-algebras P Q 14/ 28

30 Theorem (cohomology ring of a quasitoric manifold) Let M be a quasitoric manifold over P with the characteristic function given by the n m integer matrix (λ(f 1 ),...,λ(f m )) = (λ ij ). Then the cohomology ring of M is H (M) = Z[P]/ < m λ ij x j i = 1,..., n >. j=1 15/ 28

31 Theorem (cohomology ring of a quasitoric manifold) Let M be a quasitoric manifold over P with the characteristic function given by the n m integer matrix (λ(f 1 ),...,λ(f m )) = (λ ij ). Then the cohomology ring of M is H (M) = Z[P]/ < m λ ij x j i = 1,..., n >. j=1 Note that the sequence m j=1 λ ijx j for i = 1,..., n is a length n regular sequence of Z[P]. 15/ 28

32 Cohomology of moment angle complex Theorem (Buchstaber-Panov) Let k be a field. The cohomolgy of moment angle complex Z P of P is H (Z P, k) = Tor k[x 1,...,x m] (k[p], k). 16/ 28

33 Cohomology of moment angle complex Theorem (Buchstaber-Panov) Let k be a field. The cohomolgy of moment angle complex Z P of P is H (Z P, k) = Tor k[x 1,...,x m] (k[p], k). Proposition (Choi-Panov-S, 2008) P, P : n-dim simple polytopes. J (resp. J ) : an ideal of k[p] (resp. k[p ]) generated by a length n regular sequence. Then k[p]/j = k[p ]/J as graded rings H (Z P : k) = H (Z P : k). Corollary M P, N Q : 2n-dim quasitoric manifolds. Then H (M : k) = H (N : k) H (Z P : k) = H (Z Q : k). 16/ 28

34 Algebraic definition of cohomological rigidity Question When does H (Z P, k) = H (Z Q, k) imply P Q? Definition P is (cohomologically) rigid if whenever there is another polytope Q with H (Z P, k) = H (Z Q, k), there is a combinatorial equivalence P Q. 17/ 28

35 Algebraic definition of cohomological rigidity Question When does H (Z P, k) = H (Z Q, k) imply P Q? Definition P is (cohomologically) rigid if whenever there is another polytope Q with H (Z P, k) = H (Z Q, k), there is a combinatorial equivalence P Q. Compare this question with the following question. Question (Cohomological rigidity of quasitoric manifolds) Let M and N be quasitoric manifolds. Does H (M) = H (N) imply homeomorphism M = N? 17/ 28

36 Tor-algebra A : finitely generated graded k-algebra for a field k. M, N : finitely generated graded A-modules 18/ 28

37 Tor-algebra A : finitely generated graded k-algebra for a field k. M, N : finitely generated graded A-modules A free resolution [R, d] of M is an exact sequence of degree preserving maps between finitely generated free graded A-modules: 0 R n d R n+1 d Take A N to get d R 0 d M 0, 0 R n A N d 1 R n+1 A N d 1 d 1 R 0 A N 0 Then Tor i A (M, N) is defined to be Tor i A (M, N) := H i (R A N). 18/ 28

38 Tor-algebra A : finitely generated graded k-algebra for a field k. M, N : finitely generated graded A-modules A free resolution [R, d] of M is an exact sequence of degree preserving maps between finitely generated free graded A-modules: 0 R n d R n+1 d Take A N to get d R 0 d M 0, 0 R n A N d 1 R n+1 A N d 1 d 1 R 0 A N 0 Then Tor i A (M, N) is defined to be Tor i A (M, N) := H i (R A N). Since everything is graded, Tor i A (M, N) = j Tor i,j A (M, N) 18/ 28

39 Bigraded Betti numbers Definition The bigraded Betti number β i,j (P) of P is defined by β i,j (P) = dim k Tor i,j k[x 1,...,x m] (k[p], k). 19/ 28

40 Bigraded Betti numbers Definition The bigraded Betti number β i,j (P) of P is defined by β i,j (P) = dim k Tor i,j k[x 1,...,x m] (k[p], k). Note that bigraded Betti numbers are combinatorial invariants of P! 19/ 28

41 Bigraded Betti numbers Definition The bigraded Betti number β i,j (P) of P is defined by β i,j (P) = dim k Tor i,j k[x 1,...,x m] (k[p], k). Note that bigraded Betti numbers are combinatorial invariants of P! Since the the degree of any element of K[P] is even, we only have β i,2j. 19/ 28

42 Bigraded Betti numbers Definition The bigraded Betti number β i,j (P) of P is defined by β i,j (P) = dim k Tor i,j k[x 1,...,x m] (k[p], k). Note that bigraded Betti numbers are combinatorial invariants of P! Since the the degree of any element of K[P] is even, we only have β i,2j. It can also be seen that β 1,2j (P) =the number of degree 2j elements in a minimal basis of the Stanley-Reisner ideal of k[p]. 19/ 28

43 Bigraded Betti numbers Definition The bigraded Betti number β i,j (P) of P is defined by β i,j (P) = dim k Tor i,j k[x 1,...,x m] (k[p], k). Note that bigraded Betti numbers are combinatorial invariants of P! Since the the degree of any element of K[P] is even, we only have β i,2j. It can also be seen that β 1,2j (P) =the number of degree 2j elements in a minimal basis of the Stanley-Reisner ideal of k[p]. To show uniqueness of the bigraded Betti numbers of P is one of the ways to decide the rigidity of P. 19/ 28

44 Computation of bigraded Betti numbers Theorem (M.Hochster, 1977) Let P be a simple convex polytope with facets F 1,..., F m. For a subset σ {1,..., m} let P σ = i σ F i P. Then we have β i,2j (P) = dim H j i 1 (P σ ). σ =j Here dim H j i 1 ( ) = 1 by convention. 20/ 28

45 Computation of bigraded Betti numbers Theorem (M.Hochster, 1977) Let P be a simple convex polytope with facets F 1,..., F m. For a subset σ {1,..., m} let P σ = i σ F i P. Then we have β i,2j (P) = dim H j i 1 (P σ ). σ =j Here dim H j i 1 ( ) = 1 by convention. Example β 1,4 (P) = of pairs of non-intersecting facets of P. β 1,6 (P) = of 3-belts of facets of P. 20/ 28

46 Cohomological rigidity of quasitoric manifolds Recall the question Question (Cohomological rigidity of quasitoric manifolds) Let M and N be quasitoric manifolds. Does H (M) = H (N) imply homeomorphism M = N? 21/ 28

47 Cohomological rigidity of quasitoric manifolds Recall the question Question (Cohomological rigidity of quasitoric manifolds) Let M and N be quasitoric manifolds. Does H (M) = H (N) imply homeomorphism M = N? Theorem (Masuda-Panov, 2007) Let M be a quasitoric manifold such that H (M) = H ( CP 1 ). Then M = CP 1 (homeomorphism). 21/ 28

48 Cohomological rigidity of quasitoric manifolds Recall the question Question (Cohomological rigidity of quasitoric manifolds) Let M and N be quasitoric manifolds. Does H (M) = H (N) imply homeomorphism M = N? Theorem (Masuda-Panov, 2007) Let M be a quasitoric manifold such that H (M) = H ( CP 1 ). Then M = CP 1 (homeomorphism). This theorem is prove in two steps: They prove the theorem for quasitoric manifold M over I n. Then they show that I n is cohomologically rigid. 21/ 28

49 Theorem (Choi-Masuda-S, 2008) Let M be a quasitoric manifold over n i such that H (M) = H ( CP n i ). Then M = CP n i (homeomorphism). 22/ 28

50 Theorem (Choi-Masuda-S, 2008) Let M be a quasitoric manifold over n i such that H (M) = H ( CP n i ). Then M = CP n i (homeomorphism). Since we have shown that n i is cohomologically rigid, we have the following corollary. Corollary We can drop the condition over n i in the previous theorem. 22/ 28

51 Sketch of proofs of the main results We show that n i is cohomologically rigid. M: 2n-dim quasitoric manifold over P = t i=1 n i. N: 2n-dim quasitoric manifold over Q such that H (M) = H (N). 23/ 28

52 Sketch of proofs of the main results We show that n i is cohomologically rigid. M: 2n-dim quasitoric manifold over P = t i=1 n i. N: 2n-dim quasitoric manifold over Q such that H (M) = H (N). Since cohomology of quasitoric manifold contains information on face numbers of the base polytope, f 0 (P) = f 0 (Q) = n + t where f 0 denotes the facet number. 23/ 28

53 Sketch of proofs of the main results We show that n i is cohomologically rigid. M: 2n-dim quasitoric manifold over P = t i=1 n i. N: 2n-dim quasitoric manifold over Q such that H (M) = H (N). Since cohomology of quasitoric manifold contains information on face numbers of the base polytope, f 0 (P) = f 0 (Q) = n + t where f 0 denotes the facet number. On the other hand by our previous result H (M) = H (N) H (Z P : k) = H (Z Q : k) β i,j (P) = β i,j (Q) for all i and j. 23/ 28

54 Define σ(p) = j 2 jβ 1,2j (P). 24/ 28

55 Define σ(p) = j 2 jβ 1,2j (P). Note that 2σ(P) = sum of degree of all element of a minimal basis of the Stanley-Reisner ideal of P. 24/ 28

56 Define σ(p) = j 2 jβ 1,2j (P). Note that 2σ(P) = sum of degree of all element of a minimal basis of the Stanley-Reisner ideal of P. When P n = t i=1 n i, σ(p) = n + t. 24/ 28

57 Define σ(p) = j 2 jβ 1,2j (P). Note that 2σ(P) = sum of degree of all element of a minimal basis of the Stanley-Reisner ideal of P. When P n = t i=1 n i, σ(p) = n + t. Proposition If Q is a simple polytope with σ(q) = f 0 (Q), then P is a product of simplices. 24/ 28

58 Define σ(p) = j 2 jβ 1,2j (P). Note that 2σ(P) = sum of degree of all element of a minimal basis of the Stanley-Reisner ideal of P. When P n = t i=1 n i, σ(p) = n + t. Proposition If Q is a simple polytope with σ(q) = f 0 (Q), then P is a product of simplices. Since β i,j (P) = β i,j (Q) we have σ(p) = σ(q) = n + t. Hence by the proposition Q = t i=1 n i. 24/ 28

59 For other results, general idea of the proof goes as follow. We calculate bigraded Betti numbers β i,2j and use some combinatorial arguments to show that for the polytopes we are considering, their bigraded Betti numbers are distinct. 25/ 28

60 For other results, general idea of the proof goes as follow. We calculate bigraded Betti numbers β i,2j and use some combinatorial arguments to show that for the polytopes we are considering, their bigraded Betti numbers are distinct. This shows that the cohomology of their moment angle complexes are all distinct. 25/ 28

61 For other results, general idea of the proof goes as follow. We calculate bigraded Betti numbers β i,2j and use some combinatorial arguments to show that for the polytopes we are considering, their bigraded Betti numbers are distinct. This shows that the cohomology of their moment angle complexes are all distinct. This implies that the cohomology rings of the quasitoric manifolds over different polytopse are not isomorphic. 25/ 28

62 For other results, general idea of the proof goes as follow. We calculate bigraded Betti numbers β i,2j and use some combinatorial arguments to show that for the polytopes we are considering, their bigraded Betti numbers are distinct. This shows that the cohomology of their moment angle complexes are all distinct. This implies that the cohomology rings of the quasitoric manifolds over different polytopse are not isomorphic. This proves the results. 25/ 28

63 Are bigraded Betti numbers sufficient? NO! There are two polytopes with 10 facets P = Q = with identical bigraded Betti numbers (β 1,4,β 2,6,...,β 6,14 ) = (21, 64, 78, 40, 8, 0). 26/ 28

64 Are bigraded Betti numbers sufficient? NO! There are two polytopes with 10 facets P = Q = with identical bigraded Betti numbers (β 1,4,β 2,6,...,β 6,14 ) = (21, 64, 78, 40, 8, 0). We know the rigidity of 3-polytopes with 10 facets except for the above P and Q. 26/ 28

65 Are bigraded Betti numbers sufficient? NO! There are two polytopes with 10 facets P = Q = with identical bigraded Betti numbers (β 1,4,β 2,6,...,β 6,14 ) = (21, 64, 78, 40, 8, 0). We know the rigidity of 3-polytopes with 10 facets except for the above P and Q. We do not know whether P and Q are rigid or not, yet. 26/ 28

66 Rigidity of Cohen-Macauley complex Let K be a simplicial complex on [m] := {1,..., m}. The face ring or Stanley-Reisner ring of K is k[k] := k[x 1,..., x m ]/ < x i1 x ik {i 1,...,i k } K > 27/ 28

67 Rigidity of Cohen-Macauley complex Let K be a simplicial complex on [m] := {1,..., m}. The face ring or Stanley-Reisner ring of K is k[k] := k[x 1,..., x m ]/ < x i1 x ik {i 1,...,i k } K > A simplicial complex K of dimension n 1 is Cohen-Macauley if there exists a length n regular sequence λ 1,...,λ n in k[x], i.e., they are homogeneous, algebraically independent and k[k] is a free module over k(λ 1,...,λn). 27/ 28

68 Rigidity of Cohen-Macauley complex Let K be a simplicial complex on [m] := {1,..., m}. The face ring or Stanley-Reisner ring of K is k[k] := k[x 1,..., x m ]/ < x i1 x ik {i 1,...,i k } K > A simplicial complex K of dimension n 1 is Cohen-Macauley if there exists a length n regular sequence λ 1,...,λ n in k[x], i.e., they are homogeneous, algebraically independent and k[k] is a free module over k(λ 1,...,λn). All spherical simplicial complexes are Cohen-Macauley. 27/ 28

69 Definition cohomological rigidity of Cohen-Macauley complex A Cohen-Macauley complex K n 1 is cohomologically rigid if there is another C-M complex L n 1 and ideals I k[k] and J k[l] generated by length n regular sequences such that k[k]/i = k[l]/j, then K and L are combinatorially equivalent. 28/ 28

70 Definition cohomological rigidity of Cohen-Macauley complex A Cohen-Macauley complex K n 1 is cohomologically rigid if there is another C-M complex L n 1 and ideals I k[k] and J k[l] generated by length n regular sequences such that k[k]/i = k[l]/j, then K and L are combinatorially equivalent. So far there is no example of rigid Cohen-Macauley complex K such that K (P ) for a simple convex polytope. 28/ 28