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Torus Actions and Complex Cob ordism
Taras Panov Moscow State University

joint with Victor Buchstab er and Nigel Ray

Bratislava Top ology Symp osium, 7­11 Septemb er 2009


Thm. Every complex cob ordism class in dim > 2 contains a quasitoric manifold.

In other words, every stably complex manifold is cob ordant to a manifold with a nicely b ehaving torus action. In cob ordism theory, all manifolds are smo oth and closed.
n n M1 M2 (co)b ordant if there is a manifold with b oundary W n+1 such that W n+1 = M1 M2.

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Complex b ordism: work with complex manifolds. complex mflds almost complex mflds stably complex mflds Stably complex structure on a 2n-dim manifold M is determined by a choice of isomorphism
= 2(l-n) - c : M R

where is an l-dim complex vector bundle. Complex b ordism classes [M , c ] form the complex b ordism ring U with resp ect to the disjoint union and pro duct. U = Z[a1, a2, ...], dim ai = 2i Novikov'60.

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Quasitoric manifolds: 2n-dimensional manifolds M with a "nice" action of the torus T n (after Davis­Januszkiewicz);

· the T n-action is lo cally standard (lo cally lo oks like the standard T n-representation in Cn); · the orbit space M /T n is an n-dim simple p olytop e P .

Examples include projective smo oth toric varieties and symplectic manifolds M with Hamiltonian actions of T n (also known as toric manifolds).

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Polytop es and moment-angle manifolds. Rn: Euclidean vector space. Consider a convex p olyhedron P = {x Rn : (a i, x ) + bi 0 for 1 i m} , a i Rn, bi R.

Assume dim P = n, no redundant inequalities, P is b ounded, and b ounding hyp erplanes Hi = {(a i, x ) + bi = 0}, 1 i m, intersect in general p osition at every vertex. 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 of facets. i m. At every vertex meets an n-tuple

Two p olytop es are said to b e combinatorially equivalent if their face p osets are isomorphic.
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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

- Rm

iP

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

Here iZ is a T m-equivariant emb edding.
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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 m - n linear equations k=1 cj k (yk - bk ) = 0, 1 j m - n; 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 . In fact, the top ological typ e of ZP dep ends only on the combinatorial typ e of P (the original construction of Davis­Januszkiewicz).
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Quasitoric manifolds from combinatorial data. Assume given P as ab ove, and an integral n â m matrix 1 0 . . . 0 1,n+1 0 1 . . . 0 2,n+1 = . . .. . . . . . .. . . . . 0 0 . . . 1 n,n+1 satisfying the condition the columns of j1 , . . . , jn corresp onding to any vertex p = Fj1 · · · Fjn form a basis of Zn. We refer to (P, ) as the combinatorial quasitoric pair.


. . .

. . . ..

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



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Define K = K () := ker( : T m T n) = T m-n. Prop 2. 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 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 Z n , 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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Quasitoric representatives in cob ordism classes. 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 3 (Davis­Januszkiewicz). There is an isomorphism of real vector bundles
= 2(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 4 (Buchstab er­P­Ray). 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}. However, Hij is not a quasitoric manifold if i > 1. Idea of pro of of Thm 4. 1) Replace each Hij by a quasitoric 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 the disjoint unions by certain connected sums. This is tricky, b ecause we need to take account of b oth the torus action and the stably complex structure.

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Equivariant cob ordism and the universal toric genus. X a T k -space. There are 3 equivariant complex cob ordism theories: ·

stably tangentially complex T k -bundles over X . · M U k (X ) = lim[S V X+, M UT k (W )]T k : homotopic T k -cob ordisms; T here M UT k (W ) is the Thom T k -space of the universal |W |dimensional complex T k -vector bundle |W |, and S V is the unit sphere in a T k -representation space V .
· U (E T k âT k X ): Borel T k -cob ordisms.

U :T

k

(X ): geometric T k -cob ordisms: set of cob ordism classes of

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There are natural transformations of cohomology theories
U :T k (X ) - M UT k (X ) - U (E T k âT k X ). Restricting to X = pt we get a map := · : U :T k - U (B T k ) = U [[u1, . . . , uk ]], which we refer to as the universal toric genus. It assigns to the cob ordism class [M , c ] -2nk of a 2n-dimensional T k -manifold M

the "cob ordism class" of the map E T k âT k M B T k . We may write (M , c ) =


U :T

g (M )u ,

U where = (1, . . . , k ) Nk , u = u11 · . . . · uk k , g (M ) 2(||+n). U We have g0(M ) = [M ] 2n. How to express the other co efficients g (M )?
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U Ray's basis in (B T k ).

Consider the pro duct of unit 3-spheres (S 3)j =
{

(z1, . . . , z2j ) C2j : |zi|2 + |zi+j |2 = 1 for 1

i

j

}

with the free T j -action by (t1, . . . , tj )·(z1, . . . , z2j ) = (t-1z1, t-1t-1z2, . . . , t-11t-1zj , t1zj +1, . . . , tj z2j ) 1 12 j- j The quotient Bj := (S 3)j /T j is the b ounded flag manifold. It is a " Bott tower", i.e. a j -fold iterated 2-sphere bundle over B0 = . For 1 i j there are complex line bundles

i : (S 3)j âT j C - Bj via the action (t1, . . . , tj ) · z = tiz for z C. For any j > 0 have an explicit isomorphism ¯ (Bj ) Cj = 1 12 · · · j -1j 1 · · · which defines a stably cplx structure c on Bj with [Bj , c ] j j

¯ j ,
U = 0 in 2j .
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U Prop 5. The basis element b 2||(B T k ) dual to u U (B T k ) is represented geometrically by the classifying map

: B - B T k for the pro duct 1â · · · â k of line bundles over B = B1 â . . . â Bk . Let T = T 1 â . . . â T k ,and (S 3) = (S 3)1 â . . . â (S 3)k , on which T acts co ordinatewise. Define G (M ) := (S 3) âT M , where T acts on M via the representation
1 (t1,1, . . . , t1,1 ; . . . ; tk,1, . . . , tk,k ) - (t-, , . . . , t-1 ). 11 k,k

The stably complex structure c on G (M ) is induced by the structures c and c on the base and fibre of the bundle M G (M ) B .
U Thm 6. The manifold G (M ) represents g (M ) in 2(||+n).
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Hirzebruch genera and equivariant extentions. R a (graded) commutative ring with unit.
U : R a Hirzebruch genus.

Every genus has a T k -equivariant extension
k Uk T := · : :T - R[[u1, . . . , uk ]].

We have
T k (M , c ) = (M ) + (g (M )) u . | |>0

In particular, the T k -equivariant extension of the universal genus ug = U U id : is ; hence the name "universal toric genus".
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Rigidity and fibre multiplicativity. A genus is multiplicative with resp ect to N when (E ) = (N )(B ) holds for every bundle E B of stably complex manifolds with compact connected structure group and fibre N . If is multiplicative with resp ect to every N , then it is fibre multiplicative.
k Uk The genus is T k -rigid on M when T : :T - R[[u1, . . . , uk ]] is k constant, i.e. satisfies T (M ) = (M ). k If T is rigid on every M , then is T k -rigid.

In fact, T 1-rigidity suffices to imply G-rigidity for any compact Lie group G. We therefore refer simply to rigidity in case k = 1. It follows that is rigid whenever (G (M )) = 0 for | | > 0. Prop 7. If a genus is multiplicative with resp ect to M , then it is T k -rigid on M . Pro of. The B b ound for | | > 0, so apply to the bundle M G (M ) B . Ex 8. The signature is fibre multiplicative over any simply connected base, so it is a rigid genus.
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Isolated fixed p oints. For any x Fix(M ), have the representation rx : T k GL(l, C) associated to the T k -invariant structure c : M R2(l-n) . The fibre x decomp oses as Cn Cl-n, where rx has no trivial summands on Cn, and is trivial on Cl-n. Also, c ,x induces an orientation of x(M ). For any x Fix(M ), the sign (x) is +1 if the isomorphism c,x p i x(M ) - x(M ) R2(l-n) - x = Cn Cl-n - Cn resp ects the canonical orientations, and -1 if it do es not. So (x) compares the orientations induced by rx and c ,x on x(M ), and if M is almost complex then (x) = 1 for every x Fix(M ). The non-trivial summand of rx decomp oses into 1-dimensional representations as rx,1 . . . rx,n, and we write the integral weight vector of rx,j as wj (x) := (wj,1(x), . . . , wj,k (x)), for 1 j n. We refer to the collection of signs (x) and weight vectors wj (x) as the fixed p oint data for (M , c ).
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Each weight vector determines a line bundle
wj (x) := wj,1(x) · · · wj,k (x) 1 k

over B T k , whose first Chern class is a formal p ower series [wj (x)](u) :=
2 in U (B T k ).




a [wj,1(x)](u1) 1 · · · [wj,k (x)](uk ) k
m cMU(j ) in 1 k ).

Here [m](uj ) denotes the p ower series 2 U (CP ), and the a are the co efficients of cMU(1 · · · 1 Mo dulo decomp osables we have that [wj (x)](u1, . . . , uk ) wj,1u1 + · · · + wj,k uk .

Thm 9 (Lo calisation formula). For any stably tangentially complex M 2n with isolated fixed p oints, the equation 1 (M ) = (x) j =1 [wj (x)](u) Fix(M )
- is satisfied in U 2n(B T k ).
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Quasitoric manifolds revisited. Quasitoric manifolds provide a vast source of examples of stably complex T n-manifolds with isolated fixed p oints, for which calculations with the fixed p oint data and Hirzebruch genera can b e made explicit. Thm 10. For any quasitoric manifold M with combinatorial data (P, ) and fixed p oint x = Fj1 . . . Fjn , let N (P )x b e a matrix of column vectors normal to Fj1 , . . . , Fjn , let x b e square submatrix of of column vectors j1, . . . , jn, and Wx b e the matrix determined by t Wxx = In (unit n-matrix). Then 1. the sign (x) is given by sign det(x N (P )x)
( )

2. the weight vectors w1(x), . . . wn(x) are the columns of Wx.

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Elliptic genera. Buchstab er intro duced the formal group law d(u1) - d(u2) Fb(u1, u2) = u1c(u2) + u2c(u1) - au1u2 - u2u2 u1c(u2) - u2c(u1) 1 2 over the graded ring R = Z[a, cj , dk : j 2, k 1]/J , where deg a = 2, deg cj = 2j and deg dk = 2(k + 2); also, J is the ideal of asso ciativity relations, and c(u) := 1 +

j 2

cj uj ,

d(u) :=


k 1

dk uk .

Thm 11. The exp onential series fb(x) of Fb may b e written analytically as exp(ax)/(x, z ), where (x, z ) = (z - x) exp( (z ) x), (z ) (x)

(z ) is the Weierstrass sigma function, and (z ) = (ln (z )). Moreover, R Q is isomorphic to Q[a, c2, c3, c4] as graded algebras.
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The function (x, z ) is known as the Baker­Akhiezer function asso ciated to the elliptic curve y 2 = 4x3 - g2x - g3. It satisfies the Lam´ equation, and is imp ortant in the theory of nonlinear integrable e equations. Krichever studies the genus kv corresp onding to the exp onential series fb, which therefore classifies the formal group law Fb. Analytically, it dep ends on the four complex variables z , a, g2 and g3.
U Cor 12. The genus kv : R induces an isomorphism of graded ab elian groups in dimensions < 10.

Thm 13. Let M 2n b e an S U quasitoric manifold; then (1) the Krichever genus kv vanishes on M 2n U (2) M 2n represents 0 in 2n whenever n < 5. Conjecture 14. Theorem 13(2) holds for all n.

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Further applications to rigidity. Prop 15. For any series f over a Q-algebra A, the corresp onding Hirzebruch genus f is T k -rigid on M only if the functional equation 1 (x) =c f (wj (x) · u) j =1 Fix(M ) is satisfied in A[[u1, . . . , uk ]], for some constant c A. The quasitoric examples CP 1, CP 2, and the T 2-manifold S 6 are all instructive. Ex 16. A genus f is T -rigid on CP 1 only if the equation 1 1 + = c, f (u) f (-u) holds in A[[u]]. The general analytic solution is u f (u) = , where q (0) = 1. 2 ) + cu/2 q (u An example is provided by the To dd genus, ftd(u) = (ez u - 1)/z . In fact td is multiplicative with resp ect to CP 1.
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2 Ex 17. A genus f is T 2-rigid on the stably complex manifold CP(1,-1) only if the equation

1 1 1 - + =c f (u1)f (u2) f (u1)f (u1 + u2) f (-u2)f (u1 + u2) holds in A[[u1, u2]]. The general analytic solution satisfies f (u1) + f (u2) - cf (u1)f (u2) f (u1 + u2) = . 1 - cf (u1)f (u2) So f is the exp onential series of 2-parameter To dd genus t2 (also known as the Tx,y -genus), with c = y + z and c = y z . Cor 18 (Musin). The 2-parameter To dd genus t2 is universal for rigid genera.

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Ex 19. A genus f is T 2-rigid on the almost complex manifold S 6 only if the equation 1 1 + =c f (u1)f (u2)f (-u1 - u2) f (-u1)f (-u2)f (u1 + u2) holds in A[[u1, u2]], for some constant c. The general analytic solution is of the form exp(ax)/(x, z ), and f coincides with Krichever's exp onential series fb. Thm 20. Krichever's generalised elliptic genus kv is universal for genera that are rigid on S U -manifolds.

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[1] Victor M. Buchstab er and Taras E. Panov. Torus Actions and Their Applications in Top ology and Combinatorics. Volume 24 of University Lecture Series, 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, 219­242; arXiv:math.AT/0609346.

[3] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Toric genera. Preprint (2009); arXiv:0908.3298.

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