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On toric generators in the unitary and sp ecial unitary b ordism rings

based on joint work with Zhi Lu (Fudan University) Taras Panov
Moscow State University

International Conference Toric Topology, Number Theory and Applications, Khabarovsk, 612 September 2015

Taras Panov (Moscow University)

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1. Unitary b ordism

Elements of the unitary bordism ring U are the complex bordism classes of stably complex manifolds. A stably complex manifold is a pair (M , cT ) consisting of a smooth manifold M and a stably complex structure cT , where the latter is determined by a choice of an isomorphism

cT : T M RN -
between the stable tangent bundle of M and a complex vector bundle .

=

Theorem (MilnorNovikov)

Two stably complex manifold M and N represent the same bordism classes in U i their sets of Chern characteristic numbers coincide. U is a polynomial ring on generators in every even degree:
U Z[ai , i > 0], =
Taras Panov (Moscow University)

deg ai = 2i .
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Toric generators in bordism rings


Polynomial generators of U can be detected using a special characteristic class sn . It is the polynomial in the universal Chern classes c1 , . . . , cn n n obtained by expressing the symmetric polynomial x1 + § § § + xn via the elementary symmetric functions i (x1 , . . . , xn ) and replacing each i by ci .

sn [M ] = sn (T M ) M : the corresponding characteristic number.

Theorem

The bordism class of a stably complex manifold M 2i may be taken to be U the polynomial generator ai 2i i
si [M 2i ] = ‘1 ‘p

if i + 1 = p if i + 1 = p

s s

for any prime p , for some prime p and integer s > 0.

Problem

Find nice geometric representatives in (unitary) bordism classes; e.g., smooth algebraic varieties and/or manifolds with large symmetry groups.
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 3 / 26


2. Toric manifolds

A toric variety is a normal complex algebraic variety V containing an algebraic torus (C½ )n as a Zariski open subset in such a way that the natural action of (C½ )n on itself extends to an action on V . We only consider nonsingular complete (compact in the usual topology) toric varieties, also known as toric manifolds. Projective toric manifolds V are determined by simple n-dimensional lattice polytopes P . Irreducible torus-invariant divisors on V (or connected torus-invariant submanifolds of codimension 2) are the toric subvarieties of complex codimension 1 corresponding to the facets of P . We assume that there are m facets of P , denote the corresponding inward-pointing normal primitive vectors by a1 , . . . , am , and denote the corresponding codimension-1 subvarieties by V1 , . . . , Vm . Each ai denes a one-dimensional subgroup of the torus, which xes pointwise the corresponding subvariety Vi .
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 4 / 26


Theorem (DanilovJurkiewicz)

Let V be a toric manifold of complex dimension n. The cohomology ring H (V ; Z) is generated by the degree-two classes vi dual to the invariant submanifolds Vi , and is given by
H (V ; Z) Z[v1 , . . . , vm ]/I , =
deg vi = 2,

where I is the ideal generated by elements of the following two types: vi1 § § § vik such that the facets i1 , . . . , ik do not intersect in P ;
m

ai , x vi , for any vector x Zn .
i =1

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It is convenient to consider the integer a11 § . . = . .

n ½ m-matrix § § a1m . .. . .

an1 § § § anm whose columns are the vectors ai written in the standard basis of Zn . Then the n linear forms aj 1 v1 + § § § + ajm vm corresponding to the rows of vanish in H (V ; Z).

Theorem

There is the following isomorphism of complex vector bundles:
T V Cm
-n

1 § § § m , =

where T V is the tangent bundle, Cm-n is the trivial (m - n)-plane bundle, and i is the line bundle corresponding to Vi , with c1 (i ) = vi . In particular, the total Chern class of V is given by
c (V ) = (1 + v1 ) § § § (1 + vm ).
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 6 / 26


Example
The complex projective space CP n is the toric manifold, whose corresponding polytope is an n-simplex P = n , and 1 0 0 -1 . = 0 . . . 0 . . 0 0 1 -1 The cohomology ring H (CP n ) is given by

Z[v1 , . . . , vn

+1

]/(v1 § § § vn

+1

, v1 - v

n +1

, . . . , vn - vn

+1

) Z[v ]/(v =

n +1

),

where v is any of the vi , and

T CP n C § § § ï =ï

(n + 1 summands),

where is the tautological (Hopf ) line bundle over CP n , and is its ï conjugate, or the line bundle corresponding to a hyperplane CP n-1 CP n .
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 7 / 26


Example
Given n1 , n2 > 0 and a sequence of integers (i1 , . . . , in2 ), dene

V = CP (
i

i1

§§§

in2

C),

where denotes the i th tensor power of over CP n1 when i 0 and the i th tensor power of otherwise. Then V is the total space of a bundle over ï CP n1 with bre CP n2 . It is a projective toric manifold with n1

1 0 0 -1 . 0 . . 0 . . . 0 0 1 -1 = i1 1 0 . . 0 ... . in2 0 0

0
0 0 1

0

-1 . . . -1

n2

The polytope P here is combinatorially equivalent to a product n1 ½ n2 .
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 8 / 26


Example (continued)
The cohomology of V is

H (V ) Z[v1 , . . . , vn =

1 +1

, vn

1+

2

,...,v

n1 +n2 +2

]/I ,

where I is generated by the elements

v1 § § § v i1 v
n1 +1

n1 +1

, vn

1 +2

§ § § vn

1 +n2 +

2

, v1 - vn
1 +1

1 +1

, . . . , vn1 - vn
1 +n2 +

1+

1

,
2

+v

n1 +2

-v

n1 +n2 +2

, . . . , in2 vn

+ vn

1

- vn

1 +n2 +

.

In other words,

H (V ) Z[u , v ] u n1 +1 , v (v - i1 u ) § § § (v - in2 u ) , = where u = v1 = § § § = vn1 +1 and v = vn1 +n2 +2 .
The total Chern class is

c (V ) = (1 + u )n

1 +1

(1 + v - i1 u ) § § § (1 + v - in2 u )(1 + v ).

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Toric generators in bordism rings

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3. Toric representatives in unitary b ordism classes

The classical family of generators for U is formed by the Milnor hypersufaces H (n1 , n2 ). Each H (n1 , n2 ) is a hyperplane section of the Segre embedding CP n1 ½ CP n2 CP (n1 +1)(n2 +1)-1 and may be given explicitly by the equation

z0 w0 + § § § + zn1 w

n1

=0
n1

in the homogeneous coordinates [z0 : § § § : zn1 ] CP [w0 : § § § : wn2 ] CP n2 , assuming that n1 n2 .

and

Also, H (n1 , n2 ) can be identied with the projectivisation CP ( ) of a certain n2 -plane bundle over CP n1 . The bundle is not a sum of line bundles when n1 > 1, so H (n1 , n2 ) is not a toric manifold in this case.

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Buchstaber and Ray introduced a family B (n1 , n2 ) of toric generators of U . Each B (n1 , n2 ) is the projectivisation of a sum of n2 line bundles over the bounded ag manifold BF n1 . Then B (n1 , n2 ) is a toric manifold, because BF n1 is toric and the projectivisation of a sum of line bundles over a toric manifold is toric. We have H (0, n2 ) = B (0, n2 ) = CP n2 -1 , so sn2 -1 [H (0, n2 )] = sn2 -1 [B (0, n2 )] = n2 . Furthermore,

s

n1 +n2 -1

[H (n1 , n2 )] = s

n1 +n2 -1

[B (n1 , n2 )] = -

n1 + n2 n1

for n1 > 1.

The fact that each of the families {[H (n1 , n2 )]} and {[B (n1 , n2 )]} generates the unitary bordism ring U follows from the well-known identity gcd

n , 0
=

1

if

p

n=p if n = p

s s

for any prime p , for a prime p and s > 0.
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Taras Panov (Moscow University)

Toric generators in bordism rings


We proceed to describing another family of toric generators for U . Given two positive integers n1 , n2 , dene

L(n1 , n2 ) = CP ( Cn2 ),
where is the tautological line bundle over CP n1 . It is a projective toric manifold with n1

-1 . . . -1

1 0 0 -1 . 0 . . 0 . . . 0 0 1 -1 = 110 . 0 0 .. 000

0
0 0 1

0

n2
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 12 / 26


The cohomology ring is given by

H L(n1 , n2 ) Z[u , v ] =
with u n1 v
n2

u

n1 +1

,v

n2 +1

- uv

n2

L(n1 , n2 ) = 1.

There is an isomorphism of complex bundles

T L(n1 , n2 ) C2 p § § § p ( p ) § § § , ï ï ïï =ï
where is the tautological line bundle over L(n1 , n2 ) = CP ( Cn2 ). The total Chern class is
n1 +1 n2

c L(n1 , n2 ) = (1 + u )n
with u = c1 (p ) and v = c1 ( ). ï ï We also set L(n1 , 0) = CP
Taras Panov (Moscow University)

1 +1

(1 + v - u )(1 + v )n

2

n1

and L(0, n2 ) = CP n2 .
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Lemma
sn

For n2 > 0, we have
1 +n2

L(n1 , n2 ) =

n1 +n2

0

-

n1 +n2

1

+ § § § + (-1)n

1 n1 +n2 n1

+ n2 .

Theorem (Lu-P.)

The bordism classes [L(n1 , n2 )] Proof. s
= (-1)
n1 +n2

2(n1 +n2 )

U

generate the ring U .

L(n1 , n2 ) - 2L(n1 - 1, n2 + 1) + L(n1 - 2, n2 + 2) + (-1)n
1 n1 +n2 n1

n1 -1 n1 +n2 n1 -1

- 2(-1)n

1-

1 n1 +n2 n1 -1

= (-1)

n1 n1 +n2 +1 n1

It follows that any unitary bordism class can be represented by a disjoint union of products of projective toric manifolds. Products of toric manifolds are toric, but disjoint unions are not, as toric manifolds are connected. A disjoint union may be replaced by a connected sum, representing the same bordism class. However, connected sum is not an algebraic operation, and a connected sum of two algebraic varieties is rarely algebraic.
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Connected representatives in all bordism classes can be constructed by appealing to quasitoric manifolds [Davis and Januszkiewicz], which provide a topological generalisation of projective toric manifolds. A quasitoric manifold is a smooth 2n-dimensional closed manifold M with a locally standard action of a (compact) torus T n whose quotient M /T n is a simple polytope. An omniorientation of a quasitoric manifold provides it with an intrinsic stably complex structure. One can form equivariant connected sum of quasitoric manifolds, but the resulting invariant stably complex structure does not represent the cobordism sum of the two original manifolds. A more intricate connected sum construction is needed, as described in [Buchstaber, P. and Ray]. The conclusion, which can be derived from the above construction and any of the toric generating sets {B (n1 , n2 )} or {L(n1 , n2 )} for U , is as follows:

Theorem (Buchstaber-P.-Ray)

In dimensions > 2, every unitary bordism class contains a quasitoric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the torus action.
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4. Sp ecial unitary b ordism

A stably complex manifold (M , cT ) is special unitary (an SU -manifold) if c1 (M ) = 0. Bordism classes of SU -manifolds form the special unitary bordism ring SU . The ring structure of SU is more subtle than that of U . Novikov 1 described SU Z[ 2 ] (it is a polynomial ring). The 2-torsion was described by Conner and Floyd. We shall need the following facts.

Theorem

The kernel of the forgetful map SU U consists of torsion. Every torsion element in SU has order 2. SU Z[ 1 ] is a polynomial algebra on generators in every even 2 degree > 2:
1 SU Z[ 1 ] Z[ 2 ][yi : i > 1], 2=

deg yi = 2i .

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U U Let : 2n 2n-2 be the homomorphism sending a bordism class [M 2n ] to the bordism class [V 2n-2 ] of a submanifold V 2n-2 M dual to c1 (M ). It satises (a § b ) = a § b + a § b - [CP 1 ] § a § b .

U Let W2n be the subgroup of 2n consisting of bordism classes [M 2n ] such 2n of which c 2 is a factor vanishes. The that every Chern number of M 1 restriction of the boundary homomorphism : W2n W2n-2 is dened.

The direct sum W = i 0 W2i is not a subring of U : one has 2 [CP 1 ] W2 , but c1 [CP 1 ½ CP 1 ] = 8 = 0, so [CP 1 ] ½ [CP 1 ] W4 . /

W is a commutative ring with respect to the twisted product a b = a § b + 2[V 4 ] § a § b ,
where § denotes the product in U and V 4 is a stably complex manifold 2 with c1 [V 4 ] = -1, e.g. V 4 = CP 1 ½ CP 1 - CP 2 .
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Set

mi =

1

if

p

i +1=p if i + 1 = p

s s

for any prime p , for some prime p and integer s > 0,

U so that [M 2i ] 2i represents a polynomial generator i si [M 2i ] = ‘mi .

W is a polynomial ring on generators in every even degree except 4: W Z[x1 , xi : i > 2], x1 = [CP 1 ], deg xi = 2i , = with si [xi ] = mi mi -1 and the boundary operator : W W , 2 = 0, given by x1 = 2, x2i = x2i -1 , and satisfying the identity (a b ) = a b + a b - x1 a b .

Theorem

Theorem

There is an exact sequence of groups


SU SU SU SU 0 - 2n-1 - 2n - W2n - 2n-2 - 2n-1 - 0, SU = where is the multiplication by the generator 1 Z2 , is the forgetful homomorphism, and = - .

Toric generators in bordism rings 612 Sep 2015 18 / 26

Taras Panov (Moscow University)


We have

W Z[ 1 ] Z[ 1 ][x1 , x2k 2= 2

-1

, 2x

2k

- x1 x2k
-1

-1

: k > 1],
2k -1

2 where x1 = x1 x1 is a -cycle, and each x2k

, 2x2k - x1 x

is a -cycle.

Theorem
SU There exist elements yi 2i , i > 1, such that s2 (y2 ) = -48 and

if i is odd, if i is even and i > 2. These elements are mapped as follows under the forgetful homomorphism : SU W :
si (yi ) = mi mi -1 2mi mi -1
2 y2 2x1 ,

y2k

-1

x2k

-1

,

y2k 2x2k - x1 x

2k -1

,

k > 1.

1 In particular, SU Z[ 1 ] embeds into W Z[ 2 ] as the polynomial subring 2 2, x generated by x1 2k -1 and 2x2k - x1 x2k -1 .
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5. Toric generators of the

SU

-b ordism ring

Proposition

An omnioriented quasitoric manifold M has c1 (M ) = 0 if and only if there exists a linear function : Zn Z such that (i ) = 1 for i = 1, . . . , m. Here the i are the columns of characteristic matrix. In particular, if some n vectors of 1 , . . . , m form the standard basis e1 , . . . , en , then M is SU i the column sums of are all equal to 1.
Corollary

A toric manifold V cannot be SU . Proof. If (i ) = 1 for all i , then the vectors i lie in the positive halfspace of , so they cannot span a complete fan.
Theorem (Buchstaber-P.-Ray)

A quasitoric SU -manifold M
Taras Panov (Moscow University)

2n

U represents 0 in 2n whenever n < 5.
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Toric generators in bordism rings


Example
Assume that n1 = 2k1 is positive even and n2 = 2k2 + 1 is positive odd, and consider the manifold L(n1 , n2 ) = CP ( Cn2 ). We change its stably complex structure to the following:

T L(n1 , n2 ) R4 p p § § § p p p ( p ) § § § ï ïï ï ï =ï
2k1 2k2

and denote the resulting stably complex manifold by L(n1 , n2 ). It has

c L(n1 , n2 ) = (1 - u 2 )k1 (1 + u )(1 + v - u )(1 - v 2 )k2 (1 - v ),
so L(n1 , n2 ) is an SU -manifold of dimension 2(n1 + n2 ) = 4(k1 + k2 ) + 2.

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Example (continued)
L(n1 , n2 ) is an omnioriented quasitoric manifold over corresponding to the matrix
n1 =2k1 n1

½

n2

=

10 01 . .. . . . 00 00

0 §§§ 0 §§§ .. .. . . 0 1 0 0

0

01 0 -1 . . . . . . 01 1 -1 1 1 0 §§§ 0 1 §§§ . .. .. . . . . 00 0



0

01 0 -1 . 0. . 1 1

n2 =2k2 +1

The columns sum of this matrix are 1 by inspection.
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 22 / 26


Example
Given positive even n1 = 2k1 and odd omnioriented quasitoric manifold N (n1 11 10 0 §§§ 01 0 §§§ . .. .. .. . . . . . 00 0 1 00 0 0 = -1 n1 =2k1 1 0

n2 = 2k2 + 1, consider the , n2 ) over 1 ½ n1 ½ n2 with
01 0 -1 . . . . . . 01 1 -1 0 1 0 0 §§§ 0 0 1 0 §§§ 1 0 0 1 §§§ . . . .. ... . ... 000 0

0

0

0

01 0 -1 01 . . . . . . 11



The column sums are 1 by inspection, so N (n1 , n2 ) is a quasitoric SU -manifold of dimension 2(1 + n1 + n2 ) = 4(k1 + k2 ) + 4.
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 23 / 26

n2 =2k2 +1


Lemma

For k > 1, there is a linear combination y2k +1 of SU -bordism classes [L(n1 , n2 )] with n1 + n2 = 2k + 1 such that s2k +1 (y2k +1 ) = m2k +1 m2k . For k > 2, there is a linear combination y2k of SU -bordism classes [N (n1 , n2 )] with n1 + n2 + 1 = 2k such that s2k (y2k ) = 2m2k m2k -1 .

Proof. Calculating the characteristic numbers, we get
s s
n1 +n2

L(n1 , n2 ) = -

n1 +n2

1

+
2k 1

n1 +n2

2

+ §§§ - + §§§ -

n1 +n2 +1

N (n1 , n2 ) = 2 -n1 -

+

2k 2

n1 +n2 n1 -1 2k n1 -1

+ +

n1 +n2 n1 2k n1 .

.

It follows that

s

n1 +n2

L(n1 , n2 ) - L(n1 - 2, n2 + 2) =
2k +1 2i

n1 +n2 n1

-

n1 +n2 n1 -1

.

Now the rst statement of the lemma follows from gcd

-

2k +1 2i -1

, 0
k = m2

k +1

m2k .

The second statement is proved using similar (but more involved) identities.
Taras Panov (Moscow University) Toric generators in bordism rings 612 Sep 2015 24 / 26


The main result is as follows:

Theorem (Lu-P.)

There exist quasitoric SU -manifolds M 2i , i 5, with si (M 2i ) = mi mi -1 if i is odd and si (M 2i ) = 2mi mi -1 if i is even. These quasitoric manifolds 1 represent polynomial generators of SU Z[ 2 ].

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References

Victor Buchstaber and Taras Panov. Toric Topology. Mathematical Surveys and Monographs, vol. 204, American Mathematical Society, Providence, RI, 2015, 516 pages. Victor Buchstaber, Taras Panov and Nigel Ray. Toric genera. Internat. Math. Research Notices 16 (2010), 32073262. Zhi Lu and Taras Panov. On toric generators in the unitary and special unitary bordism rings. Preprint (2014); arXiv:1412.5084.

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