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Spaces of p olytop es, and cob ordisms of toric manifolds
Taras Panov Moscow State University Osaka City University

joint work with Victor Buchstab er and Nigel Ray
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1. Simple p olytop es. An arrangement of half-spaces is a collection H of subsets Si = {x V : (ai, x) - bi 0}, 1 i m, where ai V = Rn and bi R. When the intersection of the Si is b ounded, it forms a convex p olytop e P ; otherwise, it is a p olyhedron. We may sp ecify P by the matrix inequality AP x b.

Assume dim P = n, and no redundant half-spaces. So P has m facets Fi, defined by its intersection with the b ounding hyp erplanes Hi = {x V : (ai, x) = bi}.
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When the b ounding hyp erplanes are in general p osition, every vertex is the intersection of precisely n facets, and P is simple. The p ositive cone Rm = {h Rm : hi 0 for i = 1, . . . , m}.


2. Spaces of p olytop es. Following Khovanskii, we fix AP and identify the m-dimensional vector b with the arrangement H, and hence with the p olytop e P . The co ordinates bi describ e the signed distances of the hyp erplanes Hi from the origin 0 in V , so long as the normal vectors ai have length 1; otherwise, the distances have to b e scaled accordingly. The sign is p ositive or negative as 0 lies in the interior or exterior of Si resp ectively. Every vector in Rm may then b e identified with an analogous arrangement of halfspaces, obtained from H by parallel displacement of the Si. Some such arrangements define p olytop es, and others, dubb ed virtual p olytop es by Khovanskii, do not; in either case, we also describ e the corresp onding intersections as analogous. The zero vector
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is therefore identified with the virtual p olytop e {0}. An n-parameter family of examples is given by the translations of V , for which the corresp onding p olytop es are congruent to P .


In this context, we denote the m-dimensional vector space of p olytop es analogous to P by R(P ), and interpret the identification as an isomorphism k : Rm R(P ). We may interpret the matrix AP as a linear transformation V Rm. Since the p oints of P are sp ecified by the constraint AP x b, it follows that the intersection of the affine subspace AP (V ) - b with the p ositive cone Rm is a copy of P in Rm. Denote iP : V Rm; V AP (v ) - b.

So = k iP restricts to an affine emb edding P R(P ), for which (x) is the p olytop e congruent to P with origin at x, for all x P . In particular, (P ) is a submanifold of the p ositive cone R(P ) .
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3. (Quasi)toric manifolds. A (quasi)toric manifold (cf. Davis­Januszkiewicz) M = M 2n over a simple p olytop e P = P n has · an action of an n-dimensional torus T that lo cally lo oks like the standard T-action on Cn; · the orbit map : M P sending every set of orbits with the same isotropy group onto the interior of a face of P . In comparison with the smo oth compact toric varieties from algebraic geometry, (quasi)toric manifolds enjoy much larger flexibility for b oth
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top ological and combinatorial applications, while retaining most of the imp ortant prop erties of their algebraic counterpart.


Co dimension-two characteristic submanifolds Mi M , i = 1, . . . , m: · Mi = -1(Fi), or, equivalently, · Mi is a connected submanifold fixed p ointwise by a circle subgroup of T. We denote by i the canonical orientable 2-dimensional real bundle over M determined by Mi; it restricts to the normal bundle (Mi M ).
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where Fi is a facet of P ;


An omniorientation of M consists of a choice of orientation for M and each i.


A choice of orientation of i identifies it as a complex line one-dimensional isotropy subgroup TMi of Mi acts in the normal bundle (Mi M ). We orient the circle TMi that this action preserves the orientation determined by structure in i. Thereby we obtain a map : T m T, T Fi TMi ,

bundle. The fibres of the in such way the complex

called the characteristic map of M . Due to a non-singularity condition the kernel K () of is isomorphic to a (m - n)-dimensional torus.

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Thm 1 (Davis­Januszkiewicz). There is an isomorphism of real 2mplane bundles: (M ) R2(m-n) = 1 . . . m. Idea of pro of. Consider the pullback diagram
i ZP = T m â P / -Z T m â Rm/ = Cm





,

P

-P

i

Rm .

where (z1, . . . , zm) = (|z1|2, . . . , |zm|2). It determines a canonically framed manifold ZP .
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The map : T m T induces a principal K ()-fibration p : ZP = T m â P / - T â P / = M . The tangent bundle to ZP decomp oses as (ZP ) = p (M ) (p) where (p) is the tangent bundle along the fibres of p. The required bundle isomorphism comes from Szczarba's identification (M ) ( (p)/K ()) ( (iZ )/K ()) = 1 . . . m by noticing that b oth (p)/K () and (iZ )/K () are trivial real (m-n)plane bundles over M .


4. Stably almost complex structures. Thm 2. A choice of omniorientation of M , ordering of facets, and initial vertex of P gives rise to a canonical framing of the real 2(m - n)bundle (iZ )/K () (p)/K () over M , thereby determining a canonical T-invariant stably complex structure for M . Prop 3. The equivalence class of the stably complex structure on M defined in Thm 2, and therefore the corresp onding complex cob ordism class, dep ends on only on a choice of orientations for M and for each normal bundle (Mi M ).

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5. Complex cob ordisms. Thm 4 (Buchstab er­Ray'01, corrected by Buchstab er­Ray­P.). In dimensions > 2, every complex cob ordism class contains a toric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the action of the torus. Idea of pro of. Start with the additive basis in the complex cob ordism ring consisting of toric manifolds, constructed by Buchstab er and Ray in 1999. Then one needs to replace the disjoint union (representing the cob ordism sum) by something connected. Given two cob ordism classes represented by 2n-dimensional omnioriented toric manifolds M1 and M2, with quotient p olytop es P1 and
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P2 resp ectively, we need to construct a third such manifold M , with quotient p olytop e P , representing the sum of the cob ordism classes of M1 and M2. This is done using the connected sum construction.


Ex 5. Connected sum of CP 2 and CP 2. The resulting stably complex structure on the manifold CP 2 # CP 2 b ounds, that is, represents the zero cob ordism class.
(0,1) ' $ 4 (-1,-1) 4 4 &% 4 4 4 4 4 (1,0) 4 4 4

#

4

4 4' 4 4 4

(0,1)44

4 4 4

(0,1) (-1,-1)

$

'

$

&% (1,0)

=
&%

(-1,-1)

(-1,-1) (1,0)

However, it is not p ossible to take the connected sum of two copies of CP 2 with the standard omniorientation by a pro cedure like this. A mo dification is needed here.
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Ex 6. Connected sum CP 2 # S # CP 2.
d (0,1) d (1,0) (0,1) d ' ' d (-1,-1) d d 4#d 4 d 4 d 4 4 ,1) 4 (1,0) (1,0)d (0 4 d 4 d 4 4 4 4 4 (-1, 4 4' (1,0) ' ( -1,-1)
(0,1)

(1,0)

-1)

4 4 #4

=
(-1,-1)

(0,1)


(1,

0) (0,1)


[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] . . , . . . (in Russian), , , 2004. (Extended version of [1]).

[3] Victor M Buchstab er and Nigel Ray. Tangential structures on toric manifolds, and connected sums of p olytop es. Internat. Math. Res. Notices, 4:193­219, 2001.
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[4] Michael W Davis and Tadeusz Januszkiewicz. Convex p olytop es, Coxeter orbifolds and torus actions. Duke Math. J., 62(2):417­ 451, 1991.

[5] Alexandr V Pukhlikov and Askold G Khovanskii. Finitely additive measures of virtual p olyhedra (Russian). Algebra i Analiz, 4(2): 161­185, 1992.