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TORIC TOPOLOGY AND COMPLEX COBORDISM
TARAS PANOV

Abstract. We plan to discuss how the ideas and methodology of Toric Topology can be applied to one of the classical sub jects of algebraic topology: finding nice representatives in complex cobordism classes. Toric and quasitoric manifolds are the key players in the emerging field of Toric Topology, and they constitute a sufficiently wide class of stably complex manifolds to additively generate the whole complex cobordism ring. In other words, every stably complex manifold is cobordant to a manifold with a nicely behaving torus action. An informative setting for applications of toric top ology to complex cobordism is provided by the combinatorial and convex-geometrical study of analogous polytop es. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. The latter is a yet another disguise of the moment-angle manifold, another familiar ob ject of toric topology. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular pro jective toric varieties (otherwise known as toric manifolds).

Contents 1. 2. 3. 4. 4.1. 4.2. 4.3. 4.4. 4.5. 5. 5.1. 5.2. 5.3. 5.4. Polytopes Moment angle manifolds Quasitoric manifolds Cobordism theories General notion of cobordism Oriented cobordism Complex cobordism Generalised (co)homology theories Main results on cobordism (Quasi)toric representatives in complex cobordism classes Equivariant stably complex structure on quasitoric manifolds Hi,j are not quasitoric Toric multiplicative generator set for U Constructing connected representatives: replacing the disjoint union by the connected sum References 3 4 8 11 11 11 12 13 13 16 16 17 17 19 21

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TARAS PANOV

Main Theorem . Every complex cobordism class in dim > 2 contains quasitoric representative.

In cobordism theory, al l manifolds are smooth and closed. Complex cob ordism. complex manifolds almost complex stably (almost) complex manifolds M n RN - - - - - M ----- Quasitoric manifolds. manifold M 2n with "nice" T n -action Ç locally standard action Ç The orbit space M 2n /T n is a simple polytope. Examples include pro jective smooth toric varieties and symplectic manifolds M with Hamiltonian action of T n .
complex bundle

2n


TORIC TOPOLOGY AND COMPLEX COBORDISM

3

1. Polytopes Rn : Euclidean vector space. P = {x Rn : a i , x + bi 0 for 1 i m}, a i Rn , bi R. Hi = { a i , x + bi = 0}, the ith bounding hyperplane. Assume: (1) dim P = n; (2) P is bounded. Then P is called a (convex) n-dimensional polytope. A supporting hyperplane H is characterised by the condition that P lies within one of the halfspaces determined by H . A proper face of P is the intersection with a supporting hyperplane. 0-dim faces are vertices. 1-dim faces are edges. (n - 1)-dim faces are facets. n-dim face is P . Also assume: (3) there are no redundant inequalities (cannot remove any inequality without changing P ); then P has exactly m facets; (4) bounding hyperplanes of P intersect in general position at every vertex; then there are exactly n facets of P meeting at each vertex. Then P is a simple n-dim polytope with m facets. The faces form a poset L(P ) with respect to the inclusion. Two polytopes are said to be combinatorial ly equivalent if their face posets are isomorphic. The corresponding equivalence classes are called combinatorial polytopes. Assume |a i | = 1. Then a i , x + bi is the distance from x Rn to the ith hyperplane Hi .


4

TARAS PANOV

2. Moment angle manifolds P a simple polytope given as above, a i = (ai1 , . . a1 a2 Set AP = . = (aij ) (m ç n-matrix), b P = . . a
m P

. , ain b1 . . . b

), 1 i m. . Then can write P as

m

P = {x : AP x + b iP : Rn P

0}.

Define iP (x ) = AP x + b P , iP : Rn Rm , so we have Rm Rm

= {(y1 , . . . , ym ) : yi 0}

iP (P ) is the intersection of an n-dim affine plane in Rm with Rm . Consider the m-torus T Then R
m m

= {(t1 , . . . , tm ) = (e

2 i

1

,...,e

2 i

m

) Cm ; i R}.

is the orbit space of the standard T m -action on Cm :

(t1 , . . . , tm ) Ç (z1 , . . . , zm ) = (t1 z1 , . . . , tm zm ). The orbit pro jection is Cm Rm , 2 (z1 , . . . , zm ) (|z1 | , . . . , |zm |2 ). Now define the space ZP from the pullback diagram ZP P So Z
P
Z Cm . iP Rm

i

is a T m -space and iZ : ZP Cm is a T m -equivariant embedding. {x1 0, x2 0, -x1 - x2 + 1 0} a triangle, 0 1 , -1

Example 2.1. P 2 = 1 AP = 0 -1

iP (R2 ) = {AP x + b P } = {y1 + y2 + y3 = 1} R3 , ZP P2 C3 , R3 ZP = {|z1 |2 + |z2 |2 + |z3 |2 = 1} S 5 . =

Prop osition 2.2. ZP is a smooth T m -manifold with the canonical trivialisation of the normal bund le of iZ : ZP Cm . Idea of proof. (1) Write the image iP (Rn ) Rm as the set of common solutions of (m - n) linear equations in yi , 1 i m. (2) Replace yi 's by |zi |2 's to get a representation of ZP as an intersection of (m - n) real quadratic hypersurfaces. (3) Check that (2) is a "complete" intersection, i.e. the gradients are linearly independent at each point of ZP .


TORIC TOPOLOGY AND COMPLEX COBORDISM

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In the presentation of P , let us fix a i , 1 i m, but allow for bi 's to change. Let us consider "virtual polytopes" analogous to P ("analogous" here means "keep a i 's, change bi 's"), so virtual polytope = arrangement of half-spaces. Let R(P ) be the space of virtual polytopes analogous P . : Rm bP + h R(P ) an isomorphism, P (h ) := {x : AP x + b P + h 0}
m

Remark 2.3. Sum in R Now define

corresponds to Minkowski sum of polytopes in R(P ).

P = iP : Rn R(P ). So P (y ) is the polytope congruent to P obtained by translating the origin to y Rn . Indeed, iP (y ) = AP y + b P and P (y ) = P (AP y ) = {x : AP x + b P + AP y 0} = P - y . Assume that the first n facets of P meet at a vertex v1 , called the initial vertex. So H1 Ç Ç Ç Hn = v1 in P , and therefore (H1 - h ) Ç Ç Ç (Hn - h ) = v1 (h ) is the initial vertex of P (h ). Denote di (h ) = distance between v1 (h ) and Hi + h , so di (h ) = 0 for 1 i n. Define C : Rm Rm C (b P + h ) = (dn In other words, C: R(P ) P (h ) (dn
+1 +1 -n

by

(h ), . . . , dm (h )). Rm-n , (h ), . . . , dm (h ))
A C -n

P Claim 1. The sequence 0 Rn - Rm - Rm

0 is exact.

Proof. Use the fact that di are metric invariants, so they take the same values on congruent polytopes. In what follows assume 1 0 . . . . . . . AP = 0 an+1,1 . . . . . . . am,1 a i = e i for 1 i n; so we have 0 ... 0 1 ... 0 . . . . . . . . . . . . . . . . 0 ... 1 = (aij ). . . . . . . an+1,n . . . . . . . . . . . . . . . . . . . . . . am,n

Example 2.4. : Rm R(P ) maps the basis vector e j to the virtual polytope P (-b P + e j ) =: Pj ; then di (Pj ) = -a ij
i,j

if 1 j n, if n + 1 j m, ç m matrix . . ... . . . . . . -an+1,n . -an+2,n ............ . -am,n

for n + 1 i m, .0 . 0 . . . . . . .1

and C is given by the (m - n) -an+1,1 -an+2,1 C = (cij ) = . . . . . . . . -am,1

1 0 ... 0

0 1 .... 0

. . .. .

. . . .


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TARAS PANOV

Proof of Proposition 2.2. Step (1). We can write iP (Rn ) = {y Rm : y = AP x + b = {y : C y - C b Step (2). Then
m P P

for some x Rn }

= 0}

(m - n linear equations in y Rm ).

ZP = {z Cm :
k=1

cj k (|zk |2 - bk ) = 0,

1 j m - n}

Step (3). Now we want to check that the gradients in the presentation of ZP in Step (2) are linearly independent at each point. Write zk = qk + -1rk ; then the gradients are given by 2(cj 1 q1 , cj 1 r1 , . . . , c
j m qm

, cj m rm ),

1 j m - n.

So the gradients form the rows q1 r 1 0 . . . . 0 0 q2 r2 R= . . . . . . . . . . . . . . . . 0 ..........

of the (m - n) ç 2m matrix 2C R, where ....... 0 0 ... 0 m ç 2m matrix . . . . . . . . . . . . 0 qm rm

Assume that qj1 = rj1 = Ç Ç Ç = qjk = rjk = 0 at z ZP so that (zj1 = Ç Ç Ç = zjk = 0). Then the corresponding facets Fj1 , . . . , Fjk of P intersect nontrivially. The condition C AP = 0 guarantees that the submatrix obtained form C by deleting the columns c j1 , . . . , c jk has rank m - n. Then rank of 2C R is also m - n. Z
P

is called the moment angle manifold corresponding to P .
P

Remark 2.5. It can be proved that the equivariant smooth structure on Z pends only on the combinatorial type of P . Summary (reminder). Given a simple polytope P = {x Rn : a i , x + bi 0 for 1 i m}, a i Rn , bi R with m facets Fi = {x Rn : a i , x + bi = 0} P, The facets are finely ordered, i.e. F1 Ç Ç Ç Fn = v
1

de-

1 i m.

the initial vertex
P

May specify P by the matrix inequality AP x + b AP : m ç n matrix of row vectors a i , b
P

0, where

Rm : column vector of scalar bi
P

The intersection of the affine subspace AP (Rn ) + b a copy of P in Rm : iP : Rn Rm , iP (x ) = AP x + b moment angle manifold Z
i
Z

with the positive cone Rm is

P

affine, injective

P

P

Cm
P

i

Rm

((z1 , . . . , zm )) = (|z1 |2 , . . . , |zm |2 ).


TORIC TOPOLOGY AND COMPLEX COBORDISM

7

We want to describe the isotropy subgroups of points of Z T m -action. We may write
m

P

with respect to the

T

m

=
i=1

Ti ,

where Ti := {(1, . . . , 1, t, 1, . . . , 1)} T m is the i-th coordinate subcircle. Given a multiindex I = {i1 , . . . , ik } [m] = {1, 2, . . . , m}, define the corresponding coordinate subgroup of T m as TI :=
iI

Ti T m .

Now take z Cm . Its isotropy subgroup with respect to the coordinatewise T m action is
m Tz = {t T m

: t Ç z = z } T m.

It is easy to see that
m Tz = T (z )

where (z ) = {i [m] : zi = 0} [m]. Obviously, every coordinate subgroup of T m arises as T(z ) for some z Cm . However not every coordinate subgroup of T m arises as the isotropy subgroup for some z ZP . The isotropy subgroups of the T m -action on ZP are described as follows. Given p P , set F (p) :=
pF
i

Fi .

It is the unique face of P containing p in its relative interior. Note Ç if p is a vertex, then F (p) = p; Ç if p intP , then F (p) = P . Now set T (p) =
pF
i

Ti T m .

Note that 0 dim T (p) n ( P n is simple). Now if z ZP , then (z ) P , and
m Tz = T ((z )).


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TARAS PANOV

3. Quasitoric manifolds Assume given P 1 0 = . . . . 0 as above, and an n ç m matrix 0 ... 1 ... .......... .......... ... 0 0 1,n+1 . 0 2,n+1 . ............... ............... 1 n,m+1 . . . . . . . 1,m . 2,m ........ ........ . n,m = (In , ),

In : n ç n unit matrix, : n ç (m - n) matrix, satisfying () the columns j1 , . . . , form a basis for Zn .
jn

corresponding to any vertex F

j

1

Ç Ç Ç Fjn of P

Definition 3.1. A combinatorial quasitoric pair is (P, ) as above. We may view as a homomorphism T K () = ker(T
m - T n ) T = m

T n . Now set .

m-n

Prop osition 3.2. K () acts freely on ZP . Proof. The map : T m T n is injective when restricted to T (p), for all p P . Therefore, K () meets every isotropy subgroup of the T m -action on ZP trivially. Definition 3.3. The quotient M (P, ) := ZP /K () is the quasitoric manifold corresponding to (P, ). The 2n-dimensional manifold M = M (P, ) has a T n T m /K ()-action which satisfies the two Davis- = Januszkiewicz conditions : (a) the T n -action : T n ç M 2n M 2n is local ly standard, or locally isomorphic to the standard coordinatewise representation of T n in Cn . More precisely, every x M is contained in a T n -invariant neighborhood U (x ) M for which there is a T n -invariant subset W Cn , an automorphism : T n T n , and a homeomorphism f : U (x ) W satisfying f (t y ) = (t )f (y ) for all t T n , y U (x ). (b) there is a pro jection : M P whose fibres are orbits of . It follows from the construction that M is canonically smooth. Question 3.4 (open). Unlike ZP , we don't know whether the equivariant smooth structure on M is unique. Example 3.5. Assume that the initial vertex v1 is the origin, and the first n normal vectors a 1 , . . . , a n form the standard basis of Rn . (We can always achieve this by applying an affine transformation). Then 1 0 . . . 0 an+1,1 . . . am,1 0 1 . . . 0 an+1,2 . . . am,2 At = P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 ... 0 1 an+1,n . . . am,n has the same form as , although with real (rather than integer) matrix elements. We can always achieve that P has integral coordinates of vertices without changing its combinatorial type. So we may assume aij Z. However, condition () on


TORIC TOPOLOGY AND COMPLEX COBORDISM

9

the minors of is more severe: there are combinatorial polytopes with no integral realisation satisfying (). But if you can realise P so that At satisfies () , then P M (P ) = ZP /K (At ) P is the projective toric variety corresponding to P . Example 3.6.

1. = Then

1 0 -1 0 1 -1

= At , and K () = (t, t, t) T 3 , the diagonal subcircle. P

M (P ) = ZP /K () = S 5 /S 1 CP 2 . = The T 2 -action is given by (t1 , t2 ) Ç (z0 : z1 : z2 ) = (z0 : t1 z1 : t2 z2 ) 10 01 The action is 2. = 1 , and M (P ) = CP -1
2

(the standard orientation is reversed).
-1 2 z2

(t1 , t2 ) Ç (z0 : z1 : z2 ) = (z0 : t1 z1 : t 3. = 10101 01010

)

0 , M (S 2 ç S 2 )#(S 2 ç S 2 ). = 1

The T n -action on M is free over the interior intP = P . p P ,
-1

(p ) = (p , t ),

: M P.

We orient M using the decomposition (p ,t ) M = p P t T n by insisting that (1 , 1 , . . . , n , n ) is a positive basis of (1 , . . . , n ) > 0 in p P = R
n (p ,t )

M whenever

and (1 , . . . , n ) > 0 in t T n .

This is similar to orienting Cn by the basis (e 1 , ie 1 , . . . , e n , ie n ). Corollary 3.7. M is canonical ly oriented by the orientations of P and T n . The facial (or characteristic ) submanifolds of M are defined as Mi :=
-1

(Fi ) = ZFi /K

for 1 i m.

ZFi is the fixed point set of ZP with respect to the action of Ti T m . So Mi M is fixed by the circle subgroup (Ti ) T n determined by the ith column of : T m T n. Let Ci denote the 1-dim complex T m -representation defined via the quotient projection Cm Ci onto the ith factor. Define ZP çK Ci = {(z , w) : z ZP , w Ci } /, (z , w) (z t i : ZP çK Ci M over M whose restriction to Mi is the normal bundle of the inclusion Mi M .
-1

, tw) for every t K.

Then we have a complex line bundle


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TARAS PANOV

Definition 3.8. The ominiorientation of M is a choice of orientation for M and for every Mi , 1 i m. By the above considerations, (P, ) determines a canonical omniorientation for M (P, ).


TORIC TOPOLOGY AND COMPLEX COBORDISM

11

4. Cobordism theories 4.1. General notion of cob ordism. All manifolds are closed, smooth.
n n n n Definition 4.1. M1 and M2 are (co)bordant (notation: M1 M2 ) if there exists n+1 n+1 a manifold W with boundary such that W = M1 M2 .

Prop osition 4.2. is an equivalence relation. Proof. (1) M M . Indeed, W = M ç [0, 1]; (2) M1 M2 M2 M1 obvious; (3) M1 M2 & M2 M3 = M1 M3 .

Denote by [M ] the cobordism equivalence class of M . O = {[M n ]} the set of cobordism classes of n-dimensional manifolds. n
n n n Prop osition 4.3. O an abelian group with respect to [M1 ] + [M2 ] = [M1 n n M2 ].

Proof. Zero is the cobordism class of an empty set, -[M ] = [M ]. In particular, O is a 2-torsion. n Set O := n0 O . n Prop osition 4.4. O is a ring with respect to [M1 ] ç [M2 ] = [M1 ç M2 ]. O is called the unoriented (co)bordism ring (in fact, it is a Z/2-algebra). 4.2. Oriented cob ordism. Now all manifolds are oriented. n n M1 M2 if there is an oriented W n+1 such that W = M1 M2 where M2 denotes M2 with orientation reversed. S O is defined in the same way as O except -[M ] = [M ]. So S O is no longer a 2-torsion! It is a Z-algebra. Remark 4.5. [M1 ] + [M2 ] = [M1 #M2 ]. In other words, M1 M2 M1 #M2 .


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TARAS PANOV

Example 4.6. 1. O Z/2 (with two cobordism classes and Ç = pt). 0= 2. O = 0 (every 1-manifold bounds). 1 3. O Z/2 with generator [RP 2 ]; 2= 2[RP 2 ] = [RP 2 #RP 2 ] = [K 2 ] = 0. Here K 2 is the Klein bottle (it bounds). 4. O 0 elementary, but hard. Established by Rohlin in 1951. 3= 5. O was completely calculated by Thom in 1954 using algebraic and homo topy methods. Example 1. S 0 2. S 1 3. S 2 4. S 3 5. S 4 6. S 4.7. Z. The generator is [pt]. = O = 0. O = 0 (every oriented 2-manifold bounds). O = 0 by Rohlin. O = Z with generator [CP 2 ]; hard. O was completely calculated by the efforts of several people by 1960.
O 2n+1

Exercise 4.8. RP

, CP

2n+1

bound.

4.3. Complex cob ordism. Idea: try to work with complex manifolds. This runs into a complication as W cannot be complex. The remedy is to consider complex structures on M up to "stabilisation", i.e. assume chosen a real bundle isomorphism c : (M ) Rk where (M ) denotes the tangent bundle, Rk a trivial real k -plane bundle over M , and a complex bundle over M . Definition 4.9. A (tangential ly) stably complex manifold is an equivalence class of pairs (M , c ) as above, where (M , c ) (M , c ) if there are some m, m and a complex bundle isomorphism Cm Cm such that the composition (M ) Rk C
m c id

-- C -- =
id

m

-- (M ) Rk Cm - - Cm is an isomorphism of real bundles. FACT 1. We can do cobordism with tangentially stably complex manifolds. The opposite element in the resulting cobordism group is given by -[M , c ] := [M , c ] where c : (M n ) Rk (the conjugate stably complex structure). If M is an (almost) complex manifold then it has the canonical tangentially stably complex structure id = c : (M ) (M ). Example 4.10. M = CP 1 . Then we have a complex bundle isomorphism : (CP 1 ) C = where is the Hopf line bundle. So [CP 1 , ] is the canonical stably complex structure. The opposite element -[CP 1 , ] is determined by the real bundle isomorphism (CP 1 ) R2 . Finally, the real bundle isomorphism : (CP 1 ) R2 = C2 gives rise to the trivial stably complex structure on CP 1 .

c


TORIC TOPOLOGY AND COMPLEX COBORDISM

13

FACT 2. U Z, genetated by [CP 1 ]. 2= 4.4. Generalised (co)homology theories. Definition 4.11. Let X be a "good" topological space. Define On (X ) as the set cobordism classes of maps M n X , where (M1 X ) (M2 X ) if there is W such that W = M1 M2 and the map M1 M2 X extends to W :

O (X ) satisfied 3 of 4 Steenrod axioms for homology theory. It is Ç homotopy invariant; Ç has exact sequences of pairs; Ç has the excision axiom. But O (pt) = O . The forth Steenrod axiom fails. So O (X ) gives rise to a generalised homology theory. We can also define the "cohomology theory" O (X ), with O (pt) = O
O -

(pt).

In other words, = O . - Other (co)bordism theories S O (X ), S O (X ), U (X ), U (X ) are treated similarly. Another common notation: use M O (X ), M S O (X ), etc. instead of O (X ), S O (X ), etc. 4.5. Main results on cob ordism. O : M, SO : M, w( M ) = 1 + w1 ( M ) + w2 ( M ) + . . . p( M ) = 1 + p1 ( M ) + p2 ( M ) + . . . c( ) = 1 + c1 ( ) + c2 ( ) + . . . total Stiefel-Whitney class total Pontrjagin class total Chern class of

U : (M , c , ),

Given a sequence = (i1 , i2 , . . . , ik ) such that i1 + 2i2 + Ç Ç Ç + k ik = n (a partition of n), define the corresponding characteristic numbers as
i i i w (M n ) = w11 w22 . . . wkk ( M ) M Z/2,

dim M = n, dim M = 4n, dim M = 2n,

p (M c (M
2n

4n

)=

, ) =

pi1 1 ci1 1

pi2 2 ci2 2

... ...

pik ( M ) k cik ( ) M k

M Z, Z,

where M denotes the fundamental homology class of M (with Z/2 or Z coefficients). Example 4.12. M 4 = CP 2 , = (M ) , (CP 2 ) C = . c( (M )) = (1 + u)3 = 1 + 3u + 3u2 , where u = c1 ( ) H 2 (CP 2 ),
c
1

c 2

2

c2 (CP ) = 3,

2

c (CP ) = 9,

2 1

2

u CP

2

= 1.

Theorem 4.13 (Thom, Milnor).


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TARAS PANOV

1. M1 M2 unoriented ly cobordant , w (M1 ) = w (M2 ). 2. [M1 ] - [M2 ] is a torsion element in S O , p (M1 ) = p (M2 ). 3. (M1 , 1 ) (M2 , 2 ) complex cobordant , cw (M1 , 1 ) = cw (M2 , 2 ). Theorem 4.14 (Thom'1954). O Z/2[{ai , i = 2k - 1}] with deg ai = i. So in = smal l dimensions, O Z/2[a2 , a4 , a5 , . . . ].. Moreover, we can take a2n = [RP 2n ]. = Theorem 4.15 (Novikov, Milnor, Averbuh, Wall, Rohlin, Thom). U Z[a1 , a2 , . . . ], deg ai = 2i; = SO deg bi = 4i. /T ors = Z[b1 , b2 , . . . ], Moreover,
SO

has only 2-torsion, which is completely described.

Remark 4.16. Over rationals, the cobordism rings look much simpler: U Z Q = Q[[CP 1 ], [CP 2 ], . . . ], S O Z Q = Q[[CP 2 ], [CP 4 ], . . . ]. In what follows we consider only complex cobordism. Write formally the total Chern class of (M 2n , ) as c( ) = 1 + c1 ( ) + Ç Ç Ç + cn ( ) = (1 + x1 ) . . . (1 + xn ), so ci ( ) = i (x1 , . . . , xn ) is the ith elementary symmetric function. Consider Pn (x1 , . . . xn ) = xn + Ç Ç Ç + xn and express it as a polynomial in elementary symn 1 metric functions, Pn (x1 , . . . , xn ) = sn (1 , . . . , n ). Definition 4.17. sn (M Theorem 4.18. [M 2n if and only if sn (M
2n 2n 2n

, ) = sn (c1 , . . . , cn ) M .

] can be taken as a multiplicative generator of U in degree p 1 if there is a prime p such that k = ps , else.
2n

, ) = ÁÅ(n + 1) where Å(k ) =
2n

in other words, sn (M

) = Á1 except for n = ps - 1 in which case sn (M

) = Áp.

Example 4.19. Can we take [CP n ] as a generator of Un ? 2 1. CP 1 : P1 (x1 ) = x1 , s1 (CP 1 ) = c1 CP 1 = 2. Since n = 1 = 21 - 1, [CP 1 ] is a generator or U . 2 2. CP 2 : P2 (x1 , x2 ) = x2 + x2 = (x1 + x2 )2 - 2x1 x2 = c2 - 2c2 , so s2 (CP 2 ) = 1 1 2 (c2 - 2c2 ) CP 2 = 3. Since n = 2 = 31 - 1, [CP 2 ] is a generator of U . 1 4 3. CP 3 : In general, sn (CP n ) = n + 1 (Exercise; use the fact (CP n ) C = Ç Ç Ç ). So for n = 3, s3 (CP 3 ) = 4. Since n = 3 = 22 - 1, one should have s3 (M ) = Á2 for a generator, and [CP 3 ] is not a generator! Example 4.20 (Milnor hypersurfaces). Given two integers 1 i j , consider the following hypersurface in CP i ç CP j : Hi,j = {(z0 : Ç Ç Ç : zi ) ç (w0 : Ç Ç Ç : wj ) CP i ç CP j : z0 w0 + Ç Ç Ç + zi wi = 0} Consider Ci
+1

C

j +1

embedded onto first i + 1 coordinates. },
+1

CP = {l C

i

i+1

E = {(l, ) : l a line in Ci So we have a fibration CP
j +1

, a hyperplane in C
i

j +1

containing l}.

E CP .


TORIC TOPOLOGY AND COMPLEX COBORDISM

15

Prop osition 4.21. E = Hi,j . Also, set H
0,j

= CP

j -1

. (Hi,j ) =
i+ j i+1

Exercise 4.22. si

+j -1

.

Corollary 4.23. U is multiplicatively generated by the set of cobordism classes {[Hi,j ], 0 i j }. Proof. Use the fact that gcd
1j k-1

k j

=

p if k = ps , 1 else.


16

TARAS PANOV

5. (Quasi)toric representatives in complex cobordism classes Theorem 5.1. In dim > 2, every complex cobordism class contains a quasitoric manifold, necessarily connected, whose stably complex structure is compatible with the action of the torus. Plan of pro of. 1. Identify equivariant stably complex structures on quasitoric manifolds. 2. Observe that Hi,j are not quasitoric manifolds. 3. Replace Hi,j by a toric manifold, denoted Bi,j , with the same characteristic number si+j -1 . This provides a set of toric multiplicative generators for U . 4. Replace disjoint unions by connected sums. This is tricky because we need to keep track of both the action and the stably complex structure. The above theorem provides a solution to a toric version of the following famous problem: Problem 5.2 (Hirzebruch). Describe cobordism classes in nected algebraic representatives.
U

which have con-

Example 5.3. We have U = [CP 1 ] . For k 1, the class k [CP 1 ] contains a 2 Riemanian surface of genus 1 - k . But k [CP 1 ] with k > 1 does not contain a connected algebraic representative. So the solution to the above problem in dim 2 is given by the inequality c1 (M ) 2. In dimension 4 (complex 2), some similar inequalities for c2 and c2 are known, but 1 the complete answer is open. 5.1. Equivariant stably complex structure on quasitoric manifolds. Recall: iZ : ZP Cm the framed T m -equivariant embedding of the moment-angle manifold, (P, ) a combinatorial quasitoric pair, 1 0 . . . 0 1,n+1 . . . 1,m 0 1 . . . 0 2,n+1 . . . 2,m = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 0 0 . . . 1 n,n+1 . . . n,m M (P, ) = ZP /K () the associated omnioriented quasitoric manifold, i : ZP çK Ci ZP /K = M a T = T m /K -equivariant C-line bundle over M . Theorem 5.4. There is a real bund le isomorphism (M ) R2(m-n) 1 Ç Ç Ç m . = Proof. There is a T m -invariant decomposition (ZP ) (iZ ) ZP ç Cm = obtained by restricting (Cm ) to ZP . Factoring out K = ker( : T (M ) ( /K ) ( (iZ )/K ) ZP çK Cm , =
m n

T n ) gives

where denotes the (m - n)-plane bundle of tangents along the fibres of ZP M . Both and (iZ ) are trivial real (m - n)-plane bundles. Moreover, the matrix AP provides a canonical framing (trivialisation) of Z , as described in Section 2. Similarly, the matrix provides a canonical choice of basis in K = ker , and therefore a canonical framing of . It remains to note that ZP çK Cm = 1 Ç Ç Ç m .


TORIC TOPOLOGY AND COMPLEX COBORDISM

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Remark 5.5. Everything is T m /K -invariant. Definition 5.6. Assume N is a G-manifold, : G ç N N the action. A stably complex structure c : (N ) Rk is said to be G-equivariant if - (N ) Rk - - - - (N ) Rk - - - - - is an isomorphism of complex bundles for every g G. Corollary 5.7. The quasitoric manifold M (P, ) admits a canonical T n -equivariant stably complex structure. Remark 5.8. Using the 1-1 correspondence combinatorial quasitoric pairs (P, ) omnioriented quaritoric manifolds
c
-1

d(g ,Ç)id

c

we see that the T n -equivariant stably complex structure is determined by the omniorientation. Changing the orientation of one Mi in the omniorientation data results in changing the corresponding i to its conjugate in the stably complex structure. This is equivalent to reversing the sign of the ith column in . 5.2. Hi,j are not quasitoric. Recall: H
i,j

= {(l, ) : l Ci

+1

a line, Cj

+1

a hyperplane containing l},

0 i j,
i

so Hi,j = CP ( ), where is the complex j -plane bundle whose fibre over l CP is the j -plane l in Cj +1 : CP
j -1

CP ( ) CP i . ui
+1

Theorem 5.9 (exercise). H (Hi,j ) Z[u, w] =

,v

j -i

(ui + ui

-1

w + Ç Ç Ç + uw

i-1

+ wi ) .

Theorem 5.10 (Davis-Januszkiewicz). H (M (P, )) = Z[u1 , . . . , um ]/I + J , where ui = c1 (i ) H 2 (M (P, )), I = {vi1 , . . . , vik : Fi1 Ç Ç Ç Fik = } the Stanley-Reisner ideal of P, J = {i,1 u1 + Ç Ç Ç + Corollary 5.11. H
i,j i,m um

,

1 i n }.

is not a quasitoric manifold for 2 i j . deg u = deg w = 2

Proof. Assume the converse. Comparing H 2 , we obtain 2 = m - n. Therefore, H (Hi,j ) = (Z[u1 , . . . , um ]/J )/I = Z[u, w]/I , where the ideal I has a basis consisting of elements of deg 4 decomposable into linear factors. This gives a contradiction. 5.3. Toric multiplicative generator set for U . Construction 5.12 (the bounded flag manifold Bn ). A bounded flag in Cn+1 is a complete flag U = {U1 Ç Ç Ç Un+1 = Cn+1 } such that Uk contains the coordinate subspace Ck-1 generated by the first k - 1 standard basis vectors, for 2 k n. Bn = {set of bounded flags in C There is a pro jection Bn Bn U =U /C1 = (U1 = U2 /C1 U2 = U3 /C1 Ç Ç Ç Un
-1 -1 n+1

}.

U =(U1 U2 Ç Ç Ç Un-1 Un Cn+1 ) = Un /C1 Cn )


18

TARAS PANOV

The fibre of Bn Bn-1 is CP 1 (to recover U1 we need to choose a line in U1 C). Get a tower of fibrations Bn B
n-1

Ç Ç Ç B2 B1 = CP 1 .

This is an example of a Bott tower of height n. Prop osition 5.13. The action Tn ç C
n+1

Cn+1 ,
n+1

(t, z) (t1 z1 , . . . , tn zn , z

)

induces a T n -action on Bn making it a quasitoric manifold over I n . Idea of proof. Bn = (P, ) where P = I n (an n-dimensional cube), and -1 0 . . . 0 1 -1 . . . 0 = In m = 2n, . . , .. .. . . . . . . 0 ... 1 -1 so K () T
2n

as
-1 1 t2

(t1 , . . . , tn ) (t1 , t

,t

-1 2 t3

,...,t
2

-1 n-1 tn

, t1 , t2 , . . . , tn ),

ZP = {(z1 , . . . z2n ) C : |zk | + |zn+k | = 1, 1 k n} = (S 3 )n . To identify ZP /K () with Bn , we do the following. Given (z1 , . . . , z2n ) ZP , define v 1 , . . . , v n+1 Cn+1
2n 2

v

n+1

=e

n+1

,

v k = zk e k + zk

+n v k+1

,

k = n, . . . , 1.

Then we get a pro jection ZP Bn , z U = (U1 U2 Ç Ç Ç Un C Uk = e 1 , . . . , e Now, define fi : Bi CP i , U = {U1 U2 . . . } U1 Ci Bi,j Bi So B
i,j +1 k-1 n+1

),

, vk .

,

and define Bi,j from the pullback diagram
f
i

H

i,j i

= CP ( )

- CP

= {(U, ) : U a bounded flag in Ci
j -1 i+j -1

+1

, a hyperplane in C

j +1

containing U1 }

and there is a fibration CP

Bi,j Bi . -action turning it into a quasitoric manifold

Prop osition 5.14. Bi,j has a T over I i ç j -1 .

Idea of proof. Like always with "flag" manifolds, pulling back along fi splits it into a sum of line bundles. So Bi,j is a pro jectivisation of a sum of line bundles over a toric manifold Bi . Under these circumstances, the torus action can be extended from the base to the total space.


TORIC TOPOLOGY AND COMPLEX COBORDISM

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Remark 5.15. Both Bi and Bi,j are toric manifolds, or Bott and generalised Bott towers respectively.
2 2 Lemma 5.16. Assume f : N1 i N2 i is a degree 1 map of stably complex mani2i folds, and N2 a j -plane complex bund le. Then

si

+j -1

(CP (f ( ))) = s

i+j -1

(CP ( ))

Theorem 5.17 (Buchstaber-Ray '98). {Bi,j } is the set of multiplicative generators of U consisting of toric manifolds. Proof. Indeed, s
i+j -1

(Bi,j ) = s

i+ j - 1

(Hi,j ) by the above Lemma.

5.4. Constructing connected representatives: replacing the disjoint union by the connected sum. Remark 5.18. We cannot find a toric representative in every cobordism class because e.g. T d(M ) = 1 and cn (M ) = (M ) > 0 for every toric manifold M . Construction 5.19 (connected sum of polytopes). P , P simple polytopes, finely ordered, of dim n: v0 = F1 Ç Ç Ç Fn , v0 = F1 Ç Ç Ç Fn : initial vertices.

Construction 5.20 (equivariant connected manifolds). 1 0 . . . . . . 1,n+1 . . . 0 1 . . . . . . 2,n+1 . . . = . . . . . ... . . . . . . . . . . . . . . . . 0 . . . . . . . . 1 n,n+1 . . . 1 0 . . . . . . 1,n+1 . . . 0 1 . . . . . . 2,n+1 . . . = . . . . . ... . . . . . . . . . . . . . . . . 0 . . . . . . . . 1 n,n+1 . . . 1 0 . . . . . . 1,n+1 0 1 . . . . . . 2,n+1 # = . . . . . ... . . . . . . . . . . . . 0 . . . . . . . . 1 n,n+1 M = M (P , ),

sum of quasitoric pairs and quasitoric
1,m 2,m

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

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



1,n+1

... ...




1,m

M = M (P , ),


20

TARAS PANOV

M := M (P #P , # ). Prop osition 5.21. M is the equivariant connected sum of M and M at and -1 (v1 ).
-1

(v1 )

Difficulty: Both M and M are oriented. The only possible obstruction to get the omniorientation of M #M right involves the associated orientations of M and M : the orientations must be preserved under the collapse maps p : M #M M and p : M #M M . Definition 5.22. Let w P be a vertex, w = Fi1 Ç Ç Ç Fin . The sign (w) is Á1: it measures the difference between the orientations induced on Tw M by i1 Ç Ç Ç in and by the orientation of M . It can be calculated by (w) = ui1 , . . . , u
2 i
n

M

where ui = c1 (i ) H (M ), and M H2n (M ) the fundamental class. Prop osition 5.23. M #v1 ,v1 M admits an orientation compatible with those of M and M if and only if - (v1 ) = (v1 ). In this case, [M #M ] = [M ] + [M ] in U . Lemma 5.24. Let M be an omnioriented quasitoric manifold of dimension > 2 over P . Then there exists an ominioriented M over P such that [M ] = [M ] in U and P has at least two vertices of opposite signs. Corollary 5.25. The main theorem. Example 5.26. How to find a quasitoric representative in 2[CP 2 ] U ? We have 4 c2 ([CP 2 ]) = 3 = number of vertices in a triangle , and c2 (2[CP 2 ]) = 6. So there is no quasitoric manifold over # = representing 2[CP 2 ], because has only 4 vertices. But it is possible to do over a hexagon:


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References
[1] M. Davis T. Januszkiewcz, Convex polytopes, Coyeter orbifolds and torus actions Duke Math. J. 62(2), 1991 [2] V. M. Buchstaber and T. Panov, Torus acitons and their applications in topology and combinatorics University Lecture Ser., v24, AMS, 2002 [3] P. E. Conner and E. E. Floyd, On the relationships between the cobordism and K-theory ~1964 [4] P. E. Conner and E. E. Floyd, Differentiable periodec maps ~1964 [5] R. E. Strong, Notes on cobordism theory ~1968 [6] V. M. Buchstaber and N. Ray, Tangential structures on toric manifolds and connected sum of polytopes IMRN 4, 2001; arxiv:math AT/0010025 [7] V. M. Buchstaber, T. Panov and N. Ray, Spaces of polytopes and cobordism of quasitoric manifolds arxiv:math AT/0609346