Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2008manchester-talk.pdf Äàòà èçìåíåíèÿ: Sun Apr 8 16:18:16 2012 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:12:44 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m-a 17

Moment-angle manifolds: recent developments and p ersp ectives
Taras Panov Moscow State University

International Conference "New Horizons in Toric Top ology" 7­12 June 2008, Manchester


Moment-angle manifolds from simple p olytop es. Rn Euclidean vector space. Consider a convex p olyhedron P = {x Rn : (a i, x ) + bi Assume: a) dim P = n; b) no redundant inequalities (cannot remove any inequality without changing P ); c) P is b ounded; d) b ounding hyp erplanes Hi = {(a i, x ) + bi = 0}, 1 i m, intersect in general p osition at every vertex, i.e. there are exactly n facets of P meeting at each vertex.
2

0 for 1

i

m} ,

a i Rn, bi R.


Then P is an n-dim convex simple p olytop e with m facets Fi = {x P : (a i, x ) + bi = 0} = P Hi and normal vectors a i, for 1 i m.

The faces of P form a p oset with resp ect to the inclusion. Two p olytop es are said to b e combinatorially equivalent if their face p osets are isomorphic. The corresp onding equivalence classes are called combinatorial p olytop es.

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We may sp ecify P by a matrix inequality P = {x : AP x + b P 0}, where AP = (aij ) is the m â n matrix of row vectors a i, and b P is the column vector of scalars bi. The affine injection iP : Rn - Rm, emb eds P into Rm = {y Rm : yi x AP x + b P 0}.

Now define the space ZP by a pullback diagram ZP -Z Cm

i

(z1, . . . , zm)


P Here iZ is a T m-equivariant emb edding.

- Rm

iP

(|z1|2, . . . , |zm|2)
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Prop 1. ZP is a smo oth T m-manifold with canonically trivialised normal bundle of iZ : ZP Cm. Idea of pro of. 1) Write the image iP (Rn) Rm as the set of common solutions of m - n linear equations m cj k (yk - bk ) = 0, 1 j m - n; k=1 2) replace every yk by |zk |2 to get a representation of ZP as an intersection of m - n real quadratic hyp ersurfaces:
m k=1

cj k |zk |2 - bk

(

)

= 0, for 1 j m - n.

3) check that 2) is a non-degenerate intersection, i.e. the gradient vectors are linearly indep endent at each p oint of ZP . ZP is called the moment-angle manifold corresp onding to P .
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Original Davis­Januszkiewicz construction. Given y = (y1, . . . , ym) Rm, set T (y ) = {t = (t1, . . . , tm) T m : ti = 1 if yi = 0} T m. Regard Cm as the identification space Rm â T m/ , where (y , t ) = (y , t ) iff y = y and t -1t T (y ). Then iZ : ZP Cm emb eds ZP as a subspace P â T m/ in Rm â T m/. Cor 1. The top ological typ e of ZP is determined by the combinatorial typ e of P . In fact, the T m-equivariant smo oth structure on ZP is also unique [Bosio­Meersseman].
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Simplicial complexes. K : an (abstract) simplicial complex on the set [m] = {1, . . . , m}. = {i1, . . . , ik } K a simplex; always assume K . Ex 1. Given P as ab ove, set KP = = {i1, . . . , ik } : Fi1 . . . Fik = in P , the b oundary complex of the dual (or p olar) p olytop e of P . It is a sphere triangulation, i.e. |KP | = S n-1.
{ }

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Moment-angle complexes. D2 C unit disk. Given {1, . . . , m}, set B := {(z1, . . . , zm) (D2)m : |zi| = 1 if i } / The moment-angle complex ZK :=

K

= (D2)| | â (S 1)m-| |.

B (D2)m.

Prop 2. ZK has a T m-action with quotient cone K : ZK

2 - (D)m



,

cone K - Im where K is the barycentric sub division of K ; = {i1, . . . , ik } e = (1, . . . , m), where i = 0 if i and i = 1 if i . /
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If K = KP for a simple p olytop e P , then cone K can b e identified with P , and ZKP with ZP ! Moreover, Prop 3. a) Assume |K | = S n-1 (a sphere triangulation with m vertices). Then ZK is an (m + n)-manifold; ) Assume K is a triangulated manifold. Then ZK \ Z is an op en manifold, where Z = T m. Ex 2. Z n = S 2n+1. For n = 1, S 3 = D2 â S 1 S 1 â D2 D2 â D2.

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First summary. So · · · far we had real quadratic complete intersection determined by P ; identification spaces P â T m/ and | cone K | â T m/; p olydisk subspace K B (D2)m.

The three spaces agree when K = KP , but there is no quadratic description of ZK for non-p olytopal K . Question 1. Is there something similar to the real quadratic description of ZP in the case of non-p olytopal sphere triangulations K ? In fact, complex this way those of despite ZP is defined as a real complete intersection, it is a manifold (we need to multiply it by S 1 if dim ZP is o dd). In we get a family of non-K¨ ahler complex manifolds, generalising Hopf and Calabi­Eckmann [Bosio­Meersseman].
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Partial pro duct space. As was noticed by N. Strickland, by replacing (D2, S 1) in the definition of ZK by an arbitrary pair of spaces (X, W ), we get a generalised m.-a. complex, or partial pro duct space ZK (X, W ). In more detail, given {1, . . . , m}, set B (X, W ) := {(x1, . . . , xm) X m : xi W if i }, / and ZK (X, W ) :=

K

B (X, W ) X m.

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Co ordinate subspace arrangement complements. A co ordinate subspace in Cm may b e written as L = {(z1, . . . , zm) Cm : zi1 = . . . = zik = 0}, where = {i1, . . . , ik }. Co ordinate subspace arrangements in Cm are parameterised by simplicial complexes K on m vertices. Their complements are then given by U (K ) = Cm \

K /

L .

Prop 4. There is a T m-equivariant deformation retraction U (K ) - ZK . Pro of. U (K ) = ZK (C, C), ZK = ZK (D2, S 1), and (D2, S 1) (C, C). This gives a homotopy equivalence.
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Homotopy fibre realisation of ZK . The original Davis­Januszkiewicz space is the Borel construction DJ (K ) := E T m âT
m

ZK .

Prop 5. There is a canonical homotopy equivalence DJ (K ) - ZK (CP , ), , ) = m = (CP )m . where ZK (CP K B T B T Cor 2. (a) ZK hofibre
(
K

BT BT m ;

)

(b) H (DJ (K )) = HT m (ZK ) = Z[K ], where
/(

Z[K ] = Z[v1, . . . , vm] vi1 · · · vik : {i1, . . . , ik } K / is the face ring (or the Stanley­Reisner ring) of K , deg vi = 2.
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)


Cohomology calculation. Thm 1. [Baskakov-Buchstab er-P, Franz] There is an isomorphism of (bi)graded algebras
, H (ZK ; Z) = TorZ[v ,...,v ](Z[K ], Z) m 1 [ ] H [u , . . . , u ] Z[K ]; d , = m 1

where dui = vi, dvi = 0 for 1 H p(ZK ) = Cor 3. [Ho chster'1975]


i

m. In particular, TorZ[v ,...,v ](Z[K ], Z). m 1
-i,2j

-i+2j =p

-i,2j H j -i-1(K ), TorZ[v ,...,v ](Z[K ], Z) = m 1 | | =j

where K is the full sub complex (the restriction of K to the subset {1, . . . , m}).
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You can rewrite the ab ove in terms of P instead of K : Cor 4.
-i,2j (Z ) = Tor-i,2j H j -i-1(P ), H (Z[P ], Z) = P Z[v1 ,...,vm ] | | =j where P = i Fi, the union of facets of P b elonging to .

Cor 5. [Goresky­MacPherson] Hi U (K ) =
( )
K

H 2m-2||-i-2(linkK ),

where K = { : [m] \ K } is the Alexander dual complex of K . /

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The ab ove cohomology ring calculation for H (ZK ) translates into explicit pro duct formula in terms of Ho chster's full sub complexes [Baskakov]. Also, de Longueville's description of the pro duct in the cohomology of co ordinate subspace arrangement complements in terms of links follows from Baskakov's result by applying the Alexander duality. Thm 2. [Bahri­Bendersky­Cohen­Gitler] ZK

K /

||+2|K |.

This generalises to ZK (X, W ).
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Back in 2001 we were able to make the following calculations. Ex 3. Let K = m disjoint p oints. Then U (K ) = Cm
\

{zi = zj = 0},

1 i
the complement of the union of all co dim 2 co ordinate planes, and H (U (K )) = H
( m

(S k+1)(k-

1)(m k

).

)

k=2

Ex 4. Let P b e an m-gon, so KP is the b oundary of m-gon. Then U (KP ) = Cm
\
i-j =0,1

{zi = zj = 0};
mo d m

ZP is an (m + 2)-dim manifold, and H (ZP ) = H (U (KP )) = H
(m-2
-) k+1 â S m-k+1 )#(k-1)(mk 2) . # (S k=2 17


Thm 3. [Grbi´ c­Theriault] If K is a shifted complex (e.g., a k-skeleton of m-1, or m disjoint p oints), then ZK (and U (K )) is homotopy equivalent to a wedge of spheres. The pro of uses the homotopy fibre realisation of ZK and elab orated unstable homotopy techniques. The numb er of spheres in the wedge and their dimensions are also given. Thm 4. [Bosio­Meersseman] If P is obtained by applying a "vertex cut" op eration to n several times (e.g., P is an m-gon), then ZP is diffeomorphic to a connected sum of spaces of the form S i â S j . The pro of uses real quadratic realisation of ZP and equivariant surgery techniques. The numb er of spheres is also given. Polytop es P describ ed in the ab ove theorem achieve the lower b ound for the numb er of faces in a p olytop e with the given numb er of facets. The dual complexes KP are known to combinatorialists as stacked.
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Ex 5. Let K = 4 p oints. Then ZK (S 3)6 (S 4)8 (S 5)3. Ex 6. Let P b e a p olytop e obtained by applying a vertex cut to 3 trice. Then ZP = (S 3 â S 7)#6 # (S 4 â S 6)#8 # (S 5 â S 5)#3. There should b e some general principle underlying b oth calculations!

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Warning. In general, top ology of ZP is much more complicated than that of the previous examples. E.g., if P is obtained from a 3-cub e by cutting two non-adjacent edges, then ZP has non-trivial triple Massey pro ducts in cohomology [Baskakov].

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Quasitoric manifolds. Assume given P as ab ove, and an integral n â m matrix
=

1 0 . . . 0

0 1 . . . 0

. . .

. . . ..

. 0 1,n+1 . 0 2,n+1 . . . .. . . . 1 n,n+1

. . .

. . . ..

. 1,m . 2,m . . . . . n,m



satisfying the condition the columns of j1 , . . . , jn corresp onding to any vertex v = Fj1 · · · Fjn form a basis of Zn. We refer to (P, ) as a combinatorial quasitoric pair.
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Define K = K () := ker( : T m T n) = T m-n. Prop 6. K () acts freely on ZP . The quotient M = M (P, ) := ZP /K () is the quasitoric manifold corresp onding to (P, ). It has a residual T n-action (T m/K () = T n) satisfying the two Davis­Januszkiewicz conditions: a) the T n-action is lo cally standard; b) there is a projection : M P whose fibres are orbits of the T n-action.
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Algebraic and Hamiltonian toric manifolds. Algebraic and symplectic geometers would recognise in the ab ove construction of a quasitoric manifold M from ZP a generalisation of the symplectic reduction construction of a Hamiltonian toric manifold. In the latter case we take = At ; then M is a toric manifold P corresp onding to the Delzant p olytop e P = {x Rn : (a i, x ) + bi 0 for 1 i m}, a i Zn, bi R. Here we additionally assume the normal vectors a i to b e integer, and the Delzant condition: for every vertex v = Fi1 . . . Fin of P , the corresp onding normal vectors a i1 , . . . , a in form a basis of Zn to b e satisfied. Then ZP is the level set for the moment map µ : Cm Rm-n corresp onding to the Hamiltonian action of K = Ker = Ker At on Cm.
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Cohomological rigidity phenomenon. Problem 1. Do es a graded isomorphism H (M ) = H (M ) imply a homeomorphism of quasitoric manifolds M and M ? Rem 1. Equivariant cohomology (as an algebra over H (B T n)) do es determine the top ological typ e of a quasitoric manifold [Masuda]. A q-t manifold M is cohomologically rigid if its homeomorphism typ e is determined by the cohomology ring. A non-rigid q-t manifold would provide a counterexample to Problem 1. A simple p olytop e P is cohomologically rigid if its combinatorial typ e is determined by the cohomology ring of any q-t manifold over P . In other words, P is rigid if for any q-t M P and M P isomorphism H (M ) = H (M ) implies P P . There are examples of non-rigid p olytop es [Masuda­Suh]. These are obtained by applying a "vertex cut" to a 3-simplex trice. The corresp onding m-a manifolds ZP are also diffeomorphic!
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In p ositive direction, the following is known: · pro duct CP 1 â . . . â CP 1 is rigid [Masuda-P]; · n-dim cub e I n is rigid [Masuda-P]; · pro duct CP i1 â . . . â CP ik is rigid [Masuda-Suh]; · pro duct of simplices i1 â . . . â ik is rigid [Masuda-Suh]. The pro ofs use a result of Dobrinskaya on decomp osability of a quasitoric manifold over a pro duct of simplices into a tower of fibrations. Also, most 3-dim simple p olytop es with few facets are rigid [ChoiSuh]; the only known non-rigid p olytop es are obtained as multiple vertex-cuts.
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How to establish rigidity of p olytop es? Face vector of P is easily recovered from H (M ); so if there is only one combinatorial typ e P with the given face vector, then P is rigid. But this is a rare situation; usually more subtle combinatorial invariants are required.
-i,2j Set -i,2j (P ) := -i,2j (ZP ) = dim TorQ[v ,...,v ](Q[P ], Q). m 1

Prop 7 P and P -i,2j (P

([Choi-P-Suh]). Assume M and M are q-t manifolds over resp ectively. Then H (M ) = H (M ) implies -i,2j (P ) = ) for all i, j .

It follows that if there is only one combinatorial typ e P with given bigraded Betti numb ers, then P is rigid. In this way the rigidity of many 3-dim p olytop es with few facets (e.g. a do decahedron) is established.
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Application to complex cob ordism. Define complex line bundles i : ZP âK Ci M , 1 i m,

where Ci is the 1-dim complex T m-representation defined via the quotient projection Cm Ci onto the ith factor.

Thm 5. There is an isomorphism of real vector bundles
= m -n - · · · . M R 1 m

This endows M with the canonical equivariant stably complex structure. So we may consider its complex cob ordism class [M ] U .
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Thm 6. [Buchstab er­Ray­P] Every complex cob ordism class in dim > 2 contains a quasitoric manifold. The complex cob ordism ring U is multiplicatively generated by the cob ordism classes [Hij ], 0 i j , of Milnor hyp ersurfaces Hij = {(z0 : . . . : zi)â(w0 : . . . : wj ) CP i â CP j : z0w0 + . . . + ziwi = 0}. But Hij is not a quasitoric manifold if i > 1. Idea of pro of 1) Replace each Hij by a quasitoric (in fact, toric) manifold Bij so that {Bij } is still a multiplicative generator set for U . Therefore, every stably complex manifold is cob ordant to the disjoint union of pro ducts of Bij 's. Every such pro duct is a q-t manifold, but their disjoint union is not. 2) Replace disjoint unions by certain connected sums. This is tricky, b ecause you need to take account of b oth the torus action and the stably complex structure.
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[1] Victor M Buchstab er and Taras E Panov. Torus Actions and Their Applications in Top ology and Combinatorics. University Lecture Series, vol. 24, Amer. Math. So c., Providence, R.I., 2002. [2] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Spaces of p olytop es and cob ordism of quasitoric manifolds. Moscow Math. J. 7 (2007), no. 2; arXiv:math.AT/0609346. [3] Megumi Harada, Yael Karshon, Mikiya Masuda, Taras Panov, eds. Toric Top ology. Contemp. Math., vol. 460, Amer. Math. So c., Providence, R.I., 2008. [4] Mikiya Masuda and Taras Panov. Semifree circle actions, Bott towers, and quasitoric manifolds. Sb ornik Math., to app ear (2008); arXiv:math.AT/0607094. [5] Taras Panov. Cohomology of face rings, and torus actions, in "Surveys in Contemp orary Mathematics". London Math. So c. Lecture Note Series, vol. 347, Cambridge, U.K., 2008, pp. 165­ 201; arXiv:math.AT/0506526.
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