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Bott Towers and Equivariant Cobordism
based on joint works with Victor Buchstaber and Nigel Ray Taras Panov
Lomonosov Moscow State University

Geometry Days in Novosibirsk2014 International Conference dedicated to the 85th Anniversary of Yuri Grigorievich Reshetnyak 2427 September 2014

Taras Panov (MSU)

Bott Towers and Equivariant Cobordism

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1. Bounded ag manifolds and Bott towers
A bounded ag in C
n+1

is
+1

U = {U1 U2 · · · Un
such that Uk C
k -1

= Cn

+1

,

dim Ui = i }

= e 1, . . . , e

k -1

, k = 2, . . . , n .
n+1

Denote by BF n the set of all bounded ags in C

.

Theorem

BF n is a smooth compact toric variety under the action of the torus (Câ )n (Câ )n â BF n BF n
(t1 , . . . , tn ) · (w1 , . . . , wn , w
n +1

) = (t1 w1 , . . . , tn wn , w

n +1

),

BF n bounded ag manifold.
Taras Panov (MSU) Bott Towers and Equivariant Cobordism 2427 Sep 2014 2 / 16


n tautological line bundle over BF n , whose bre over U is U1 C. =

Proposition

BF n = CP (C
Proof.
Consider

n -1

), where n

-1

is over BF

n -1

.

BF n BF

n-1

U U = U /C

1

in C

2,...,n+1

= Cn ,

where U = {U1 U2 · · · Un-1 }, Uk = Uk +1 /C1 . To recover U from U one has to choose a line U1 in U2 = C1 U1 . Get a sequence of bre bundles with bre CP
1

BF n BF

n-1

· · · BF 1 = CP 1 pt

a Bott tower structure on BF n .
Taras Panov (MSU) Bott Towers and Equivariant Cobordism 2427 Sep 2014 3 / 16


A Bott tower is a sequence of bre bundles

Bn Bn
in which Bk = CP (C k

-1

· · · B1 = CP 1 pt
k -1

-1

) for a line bundle

over Bk

-1

.

Theorem
H (Bn ) H (Bn =
-1

)[un ]

2 un = c1 (

n -1

)un ,

where un = c1 (n ) and n is the tautological line bundle over Bn .
Example
When k = k for each k , we get Bn = BF n , the bounded ag manifold with the `intrinsic' structure of a Bott tower. We have

H (BF n ) H (BF =
Taras Panov (MSU)

n-1

2 )[un ] (un = un

-1 un

).
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Bott Towers and Equivariant Cobordism


2. Representing complex bordism classes
As a complex manifold BF n , represents a 2n-dimensional class in the complex bordism ring

U = {stably complex manifolds}/complex bordism relation

Theorem (Milnor, Novikov'1960)
U Z[a1 , a2 , . . .], =
dim ai = 2i .

A stably complex manifold M

2n

can be taken as a representative of an i
±1, n = p k - 1, ±p , n = p k - 1.

sn [M 2n ] =

Here sn is the characteristic class corresponding to the symmetric n n polynomial x1 + · · · + xn , where cn (T M 2n ) = (1 + x1 ) · · · (1 + xn ).
E.g., sn [CP n ] = n + 1, so [CP 1 ] = a1 , CP 2 = [a2 ], but CP 3 = [a3 ].
Taras Panov (MSU) Bott Towers and Equivariant Cobordism 2427 Sep 2014 5 / 16


Given i

j , consider C

i +1

C

j +1

and dene the Milnor hypersurface

Hij = ( , W ) :

is a line in C

i +1

, W is a hyperplane in C

j +1

,

W .
ij +i +j

It is given by the equation z0 w0 + · · · + zi wi = 0 in CP i â CP j CP E.g., H
22

.

= Fl (C3 ), complete ags in C3 .

Proposition
s
i +j -1

[Hij ] =

i +j . i

Therefore, {[Hij ], 0

i

j } generate the complex bordism ring U .
2.
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However, Hij is not a toric manifold when i
Taras Panov (MSU)

Bott Towers and Equivariant Cobordism


Theorem (BuchstaberRay)

The complex bordism ring U has a generator set consisting of toric manifolds.
Proof.
Consider the manifolds

Bij = (U , W ) : U is a bounded ag in C

i +1

,
+1

W is a hyperplane in Cj Bij
i

, U1 W .



BF

Hij CP

i

(U , W ) (U1 , W ) U U1

Bij = CP (sum of line bundles), so it is toric.

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Bott Towers and Equivariant Cobordism

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Quasitoric manifolds generalise toric manifolds topologically. A quasitoric manifold M 2n has an action of a torus T n with quotient a simple polytope P . Quasitoric manifolds have canonical T n -invariant stably complex structures, but are not complex or almost complex in general.

Theorem (BuchstaberPRay)

In dimensions > 2, every complex bordism class contains a quasitoric manifold.
It remains open whether ring generators ai of the complex cobordism ring U can be represented by toric or quasitoric manifolds. A partial result on this problem has been recently obtained by A. Wilfong.

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Bott Towers and Equivariant Cobordism

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3. The universal toric genus
Given a T k -manifold, one has a universal transformations between the three version of equivariant cobordism:
UT k (X ) MUT k (X ) U (ET k âT k X ) geometric homotopic Borel

For X = pt one gets a homomorphism of U -modules

: U

:T

k

U (BT k ) = U [[u1 , . . . , uk ]]

called the universal toric genus. It assigns to the equivariant cobordism class of a T k -manifold M the `cobordism class' of the map ET k âT k M BT k .
Taras Panov (MSU) Bott Towers and Equivariant Cobordism 2427 Sep 2014 9 / 16


We have

(M ) = [M ] +
: | |> 0

g (M ) u ,

in U (BT k ) = U [[u1 , . . . , uk ]], where [M ] U , u = u1 1 · · · uk k .

What are the coecients g (M )?

Taras Panov (MSU)

Bott Towers and Equivariant Cobordism

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The bounded ag manifold BF n is the quotient of

(S 3 )n = {(z1 , . . . , z2n ) C2n : |zk |2 + |zk
by the T n -action given by

+n

|2 = 1 , 1

k

n}

- -1 (z1 , . . . , z2n ) (t1 z1 , t1 1 t2 z2 , . . . , tn-1 tn zn , t1 z

n +1

, t2 z

n+2

, . . . , tn z2n )

This gives the stable splitting

¯ T (BF n ) Cn 1 1 2 · · · =¯

n-1 n

¯ 1 2 · · · n ¯ ¯ ¯

where k is the tautological line bundle over BF k pulled back to BF n . E.g., for n = 1 we obtain the standard isomorphism T CP 1 C , =¯ ¯ where = 1 is the tautological line bundle.
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Now we twist the torus action on (S 3 )n as follows:
- -1 (z1 , . . . , z2n ) (t1 z1 , t1 1 t2 z2 , . . . , tn-1 tn zn , t

1

-1

zn

+1

- , t2 1 zn

+2

- , . . . , tn 1 z2n ).

This gives the splitting

¯ T (BF n ) R2n 1 1 2 · · · =¯

n - 1 n

¯ 1 2 · · · n ,

U and the corresponding complex bordism class is zero in 2n , as an iterated sphere bundle.

Taras Panov (MSU)

Bott Towers and Equivariant Cobordism

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We denote by n 2n (CP ) the bordism class of BF n - CP .

n

Theorem (Ray)

The bordism classes {n : n 0} form a basis of the free U -module U (CP ) which is dual to the basis {u k : k 0} of the U -module U (CP ) = U [[u ]]. BF = BF
Similarly, dene 2|| (BT k ) the bordism class of
1

â · · · â BF

n

BT k .

Given a T k -manifold M , dene the bundle

G (M ) = (S 3 ) âT M - BF = (S 3 ) /T .

Theorem (Buchstaber-P-Ray)

The manifold G (M ) represents the coecient g (M ) in the expansion of the universal toric genus.
Taras Panov (MSU) Bott Towers and Equivariant Cobordism 2427 Sep 2014 13 / 16


4. Rigidity and bre multiplicativity
A genus is a homomorphism : U R where R is a commutative ring with unit (usually Z). By the Hirzebruch correspondence, a genus is determined by a series

f (x ) = x + · · · R Q[[x ]].
Namely,

n

(M ) =
i =1

xi , [M ] , f (xi )

where c (T M ) = (1 + x1 ) · · · (1 + xn ).

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Given a genus : U R , dene its equivariant extension

T : U :T k - U [[u1 , . . . , uk ]] - R Q[[x1 , . . . , xk ]] mapping [M ] (M ) and ui f (xi ).



h



Denition
A genus is rigid on M if T = (a constant). A genus is bre multiplicative with respect to M if

(N ) = (M )(B ) for any bre bundle N M B with structure group G of positive rank.

Theorem (Buchstaber-P-Ray)

A genus is rigid on M i it is bre multiplicative with respect to M .
Proof.
Use the expansion (M ) = [M ] + · · · with coecients represented by G (M ), a bundle over null-bordant base BF .
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References

Victor M. Buchstaber and Taras E. Panov. Toric Topology. A book project; arXiv:1210.2368. Victor M. Buchstaber, Taras E. Panov and Nigel Ray. Toric genera. Internat. Math. Res. Notices 2010, no. 16, 32073262. Victor M. Buchstaber, Taras E. Panov and Nigel Ray. Spaces of polytopes and cobordism of quasitoric manifolds. Moscow Math. J. 7 (2007), no. 2, 219242.

Taras Panov (MSU)

Bott Towers and Equivariant Cobordism

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