Contemporary Mathematics An Invitation to Toric Topology:
Vertex Four of a Remarkable Tetrahedron
Victor M Buchstaber and Nigel Ray 1. An Invitation
Motivation. Sometime around the turn of the recent millennium, those of
us in Manchester and Moscow who had been collaborating since the mid-1990sbegan using the term toric topology to describe our widening interests in certainwell-behaved actions of the torus. Little did we realise that, within seven years, asignificant international conference would be planned with the subject as its theme,and delightful Japanese hospitality at its heart.
When first asked to prepare this article, we fantasised about an authorita- tive and comprehensive survey; one that would lead readers carefully through thefoothills above which the subject rises, and provide techniques for gaining sufficientheight to glimpse its extensive mathematical vistas. All this, and more, would beilluminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale avail-able. Simultaneously, we understood that at least as valuable a service could berendered to conference participants by an invitation to a wider mathematical au-dience - an invitation to savour the atmosphere and texture of the subject, toconsider its geology and history in terms of selected examples and representativeliterature, to glimpse its exciting future through ongoing projects; and perhaps tolocate favourite Osaka lectures within a novel conceptual framework. Thus wasborn the Toric Tetrahedron TT , which identifies aspects of algebraic, combinato-rial, and symplectic geometry as the precursors of toric topology, and symbolisesthe powerful mathematical bonds between all four areas.
The Tetrahedron is the convex hull of these vertex disciplines, and every point has barycentric coordinates that measure the extent of their respective contribu-tions. We introduce the vertices in chronological order (a mere two years separates 2000 Mathematics Subject Classification. Primary 57R19, 57S25; Secondary 14M25, 52B20, 53D20, 55P15.
Key words and phrases. Davis-Januszkiewicz space, Hamiltonian geometry, homotopy col- imit, moment-angle complex, quasitoric manifold, Stanley-Reisner algebra, subspace arrangement,toric geometry, toric variety.
0000 (copyright holder) VICTOR M BUCHSTABER AND NIGEL RAY second and third), not least because time has acted like a Morse function in defin-ing a flow down the 1–skeleton, as TT has emerged from the unknown. Such flowson convex polytopes play an important rˆ ole in toric topology! Similar geometrical analogies have suggested other useful insights as we continue to refine our under-standing of the Tetrahedron.
So our primary aim is to issue a concise invitation to the study of T T , in which we avoid many technical details and offer an abbreviated bibliography. By way
of compensation, we invite readers to sample influential publications for each of
the vertex disciplines, and several of the edges and facets. We propose one or two
which lie at their source, and a few important survey articles; the latter combine
expert overviews with comprehensive bibliographies. Throughout our discussion we
view the torus as a unifying force that maintains the integrity of the Tetrahedron,
and provides a bridge between its far-flung regions. The most common context
for its actions lies in the theory of manifolds, which arise repeatedly in singular,
smooth, and more highly structured forms.
Before accepting our invitation, readers might also like to consult web-based re- sources which chart the rise of toric topology. These include: archives of Transpen-nine Topology Triangle meetings 18, 43, 56 and 59; the 2004 Keldysh CentenaryConference in Moscow; the Osaka meeting itself; and the Osaka City UniversitySummer School. There is also the Manchester Toric Topology Page, and the con-ference page for New Horizons in Toric Topology that is scheduled to take place inManchester during 2008. These may be found at We assume that readers have a basic knowledge of algebraic topology. Never- theless, we emphasise our convention that homology and cohomology groups H∗(X)and H∗(X) of a topological space X are always reduced, and that their unreducedcounterparts H∗(X+) and H∗(X+) require the addition of a disjoint basepoint. Thesame convention also holds for generalised theories such as cobordism and K-theory.
During the last 18 months, many colleagues have helped us to prepare this work (sometimes unwittingly) and we thank them all. Those who deserve special mentioninclude Tony Bahri, Galina Buchstaber, Kostya Feldman, and Taras Panov. Weapologise in advance to any whose work we have omitted or misrepresented; nodoubt such sins will be brought to our attention with due speed! We are alsograteful to the editors of these Proceedings and their counterparts at the AMS forall their support and encouragement.
In order to present our invitation on a suitably decorative background, we offer some elementary observations on group actions. In spite of their simplicitythey have been prominent in toric topology throughout its development, and haverecently fed back into the original vertex disciplines to good effect. We thereforerefer to them in terms that anticipate their reappearance below.
AN INVITATION TO TORIC TOPOLOGY The idea of a group G acting on a set of elements X has existed since Galois ushered in the revolutionary era of abstract algebra in the 1830s. The case in whichX denotes a set of points and G a group of symmetries was studied forty yearslater by Klein, and proved to be an equally dramatic catalyst for the developmentof geometry. We draw inspiration from both points of view — in particular, weconsider the poset S(G) of subgroups of G, ordered by inclusion .
We write any G-set as a pair (X, a), where a : G × X → X is the function a(g, x) = g · x that describes the left action of symmetries g on points x. For eachx ∈ X, we denote its isotropy subgroup by Gx ≤ G and its orbit by Gx ⊆ X;so G/Gx and Gx correspond bijectively, and Gw and Gx are conjugate in G forany w ∈ Gx. The partition of X into disjoint orbits is the kernel of the quotientfunction q : X → X/G. A section for a is a right inverse s : X/G → X of q, and isspecified by choosing a preferred representative s(Gx) for each orbit; s determinesa characteristic function λs : X/G → S(G), by λs(Gx) = Gs(Gx).
Alternatively, suppose given a characteristic function λ : Q → S(G) on an ar- bitrary set Q. The derived set of λ is defined by D(λ) = (G × Q)/ ∼ , where the equivalence relation is generated by (g, q) (h, q) whenever g−1h ∈ λ(q).
So D(λ) is a G-set with respect to the canonical action g · [h, q] = [gh, q], whoseisotropy subgroups are given by G[h,q] = (q)h−1 for any h ∈ G and q ∈ Q. Theorbits are the subsets {[h, q] : h ∈ G}, and projection D(λ) → Q onto the secondfactor is the associated quotient map; a canonical section is given by s(q) = [1, q]for any q ∈ Q. By definition, D(λ) is initial amongst G-sets (X, aX ) equipped withfunctions X → Q that are constant on orbits, and sections sX : Q → X such thatGsX(q) = λ(q) for any q ∈ Q.
Simple calculation confirms that the constructions D(λ) and λs are mutually inverse, and therefore that they establish a fundamental correspondence betweencharacteristic functions and G-sets with sections. In particular, every choice ofsection s for (X, a) leads to a G-equivariant bijection fs : D(λs) −→ X , satisfying fs[g, Gx] = g · s(Gx). We refer to D(λs) as a derived form of (X, a).
For any (X, a) with section s, we partition X/G by the kernel of λs, and pull the partition back to X along the projection q. Similarly, for any characteristic functionλ : Q → S(G) we partition Q by the kernel of λ, and pull the partition back to D(λ)along projection onto the second factor. These partitions are interchanged in theobvious fashion by the fundamental correspondence, and we refer to the blocks ofall four as isotropy blocks.
Now suppose that K E G is a normal subgroup, and consider the surjection S(G) → S(G/K) of posets induced by taking quotients. Any characteristic func-tion λ : Q → S(G) projects to λK : Q → S(G/K), and gives rise to a surjectionrK : D(λ) → D(λK); by (2.1), rK is equivariant with respect to the actions of G andG/K, and the canonical section sK : Q → D(λK) is given by rK · s. We may theninterchange the rˆ oles of G and G/K, by starting with an epimorphism e : H → G that has kernel L, and lifting λ to λe : Q → S(H). We obtain compositions −→ D(λK) −→ Q D(λe) −→ D(λ) −→ Q , VICTOR M BUCHSTABER AND NIGEL RAY which factorise and extend the original quotient function q in turn. The correspond-ing compositions for G-sets are X → X/K → X/G and Y → Y /L → (Y /L)/G.
In order to impose geometric flesh on these set theoretic bones, we proceed by assuming that a is a continuous action of the topological group G on a topologicalspace X. Every isotropy subgroup is necessarily closed, so the characteristic mapλs : X/G → C(G) takes values in the poset of closed subgroups, and is continuouswith respect to the lower topology on C(G), whose subbasic closed sets are of theform H↑ = {J H ≤ J}. Thus λ−1(H↑) is closed in X/G, and its inverse image under q is a component of the fixed point set Fix(H) for any closed H ≤ G.
Alternatively, suppose given a characteristic map λ : Q → C(G) for some topo- logical space Q. The derived space D(λ) is obtained by topologising the derivedset (2.1) so that projection onto Q is a quotient map; the inherited G-action andcanonical section are then continuous, and D(λ) is initial in the topological context.
The constructions D(λ) and λs are mutually inverse, and define a fundamental topological correspondence between characteristic maps and G-spaces with sections.
In particular, every choice of continuous section for an arbitrary G-space (X, a)leads to a G-equivariant homeomorphism fs : D(λs) −→ X ; as before, we refer to D(λs) as a derived form of (X, a). The closures of the isotropyblocks form coverings of X and D(λ) by fixed point sets, and are identified by thefundamental correspondence. The closures of the quotient blocks in X/G and Qcorrespond similarly.
As we explore TT , we impose increasingly stringent geometrical conditions on a, whose justification and significance will become apparent. The common philosophyis to view the quotient map q : X → X/G as a singular G-bundle, and to interpretthe geometry of X in terms of G and X/G using an appropriate derived form. Thisapproach has a long and distinguished history in equivariant topology, and featuresin attempts such as those of J¨ anich [62] and Davis [34] to classify Lie group actions
on smooth manifolds.
One additional aspect of the topological situation is important, namely the interplay between homotopy theory and the action a. For any topological groupG there exist various functorial models for a contractible space EG on which Gacts freely, and with closed orbits; any such EG is final amongst well-behavedfree G-spaces and G-equivariant homotopy classes of maps. The quotient mapEG → EG/G is then a universal principal G-bundle, where EG/G = BG is aclassifying space for G, and unique up to homotopy equivalence.
Any G-space X may be replaced by the homotopy equivalent EG×X, on which G acts freely by g · (e, x) = (g · e, g · x). So projection onto the first factor representsthe homotopy class of maps to the final object, and classifies a principal G bundleover the quotient EG ×G X. The latter space is the Borel construction on (X, a),otherwise known as the homotopy quotient, and is equivalent to X/G when G actsfreely on X. In other cases, EG ×G X has superior topological properties to X/G,and toric topology deals as much with homotopy quotients as with orbit spaces.
These considerations illustrate the power of algebraic topology to generalise constructions that are purely algebraic. Whenever G is discrete, for example, the
cohomology ring H∗(BG; Z) is isomorphic to the group theoretic cohomology of G.
In consequence, Milnor's original version [74] of BG stimulated the development of
continuous cohomology theory for topological groups.
AN INVITATION TO TORIC TOPOLOGY Henceforth, we restrict G to a compact n–dimensional torus T n unless otherwise stated, on the grounds that toroidal symmetry is implicit in many physical systems
because the coordinates may be thought of as angles. We sometimes assume that
X is an object of a category that is particularly convenient for homotopy theory,
such as the k-spaces of [96].
Examples. We now invite readers to consider three families of examples that
illustrate our background principles in action. The first is a local model for manyothers, and the second and third are generalised and extended later. All three arebased on the coordinatewise multiplication map µ : Cn × Cn → Cn, defined by µ(y1, . . , yn, z1, . . , zn) = (y1z1, . . , ynzn) for any n > 0. We abbreviate the set {1, . . , n} to [n], and write the subspace {z zi = 0 for i /∈ ω} as Cω ≤ Cn for any subset ω ⊆ [n]. Similarly, we denote thecompact torus (S1)ω by T ω ⊂ (C×)ω, as a subspace of {z zi ̸= 0} ⊂ Cω. Finally,we write Cδ and Tδ < Cn for the diagonal line and its unit subcircle respectively.
Example 2.1. Let X be Cn, and a : T n × Cn → Cn the restriction of µ. For any z ∈ Cn, the isotropy subgroup T n is T [n] ω, where ω is the unique subset of [n] for which z ∈ (C×)ω; the orbit T nz is the ω –dimensional torus T ωz. There is ahomeomorphism h : Cn/T n → Rn to the non-negative coordinate cone, induced by h(z1, . . , zn) = ( z1 2, . . , zn 2), and the standard inclusion of Rn in Cn specifies a canonical section s : Rn → Cn.
The isotropy blocks are the subspaces (C×)ω ⊂ Cn, whose closures Cω are the fixedpoint sets Fix(T [n] ω). The blocks project to the interiors of the correspondingfaces Rω of the polyhedron Rn, whose closures are the faces themselves. The characteristic map λs : Rn → S(T n) assigns the subtorus T [n] ω to the interior of the face Rω, and the derived form associated to s is the canonical homeomorphism fs : (T n × Rn)/ ∼ −→ Cn. Example 2.2. For any ordered pair (p, q) of positive integers, let Z(p, q) be the product of unit spheres S2p+1 × S2q+1 Cp+q+2 and a the product action ofT p+1 × T q+1. The orbit space is homeomorphic to the product ∆p × ∆q of standardsimplices in Rp+1 × Rq+1 , by (2.3). Let K be the 2–torus T (p, q) < T p+1 × T q+1 of ( (t1, . . , t1), (t2, . . , t2, t−1t where t1, t2 ∈ S1. Then M (p, q) = Z(p, q)/T (p, q) is a 2nd stage Dobrinskaya tower
[40], and is a 2(p + q)–dimensional smooth manifold, equipped with an action of
the quotient (p + q)–torus T p+q+2/T (p, q). The associated factorisation (2.2) takes
the form Z(p, q) → M (p, q) → ∆p × ∆q; many homotopic sections exist for the
projection, because the product of simplices is contractible.
The isotropy blocks of ∆p ×∆q corresponding to subtori T p+q−k are the relative interiors of the k–dimensional faces, for 0 ≤ k ≤ p + q; their closures are the facesthemselves. The corresponding components of the fixed point sets in M (p, q) arelower-dimensional 1st or 2nd stage towers. The coordinates induced on M (p, q)take the form (x1, . . , xn; u1, . . , up), where x lies in ∆p × ∆q and u in T p+q; theyare singular over the boundary. All these structures on M (p, q) lift to naturally totheir counterparts in S2p+1 × S2q+1.
VICTOR M BUCHSTABER AND NIGEL RAY By construction, M (p, q) is the projectivisation of the complex (q + 1)–plane bundle Cq ⊕ η over CP p, where η is the canonical line bundle. So M (1, 1) is a 2nd
stage Bott tower
[53], and is diffeomorphic to a Hirzebruch surface; it is also the
bounded flag manifold B2 [24], consisting of flags 0 < L1 < L2 < C3 for which
L2 contains the first coordinate line; then (u, v) ∈ T 2 acts on L1 by (1, v, uv), and
on L2 by (1, 1, u). The case q = 0 reduces to the quotient space S2p+1/Tδ, and is
diffeomorphic to CP p.
In fact M (p, q) may be extended to a kth stage tower, by iterating the projec- tivisation procedure, and replacing Cq ⊕ η with a sequence of more general bundles.
The details are provided by Dobrinskaya [40], and require care. The correspond-
ing bounded flag manifolds Bk are significant contributors to complex cobordism
theory [23], [88], as we shall explain below.
Example 2.3. Let X be Cn+1 0, and a : C× × (Cn+1 0) (Cn+1 0) the restriction of µ. The quotient space is the algebraic variety CP n, on which thealgebraic torus (C×)n acts with a single dense principal orbit. In this context, thetoric coordinates identify CP n with an equivariant compactification of (C×)n.
3. The Toric Triangle
We now invite readers to focus on the oldest facet of the Toric Tetrahedron, which we label ACS and call the Toric Triangle. Each of the vertices A, C, and Srepresents appropriate aspects of a familiar discipline, whose key ideas and selectedliterature we introduce in this section. The edges of the triangle are populatedwith interdisciplinary work that involves two of the vertices in some proportion,and interior points represent activity that combines all three.
The first vertex A. This is the original vertex of the Triangle, and represents
algebraic geometry; or more specifically, the study of toric varieties. These appear
to have been introduced in 1970 by Demazure [37], whose work is therefore the
source of the entire Toric Tetrahedron. Demazure's constructions also became
known as torus embeddings, and an early survey was provided by Danilov [33] in
1978. Twenty years later, Cox [30] popularised the term toric geometry for the
expanding array of ideas that surrounded the vertex, and updated his survey in
2002 to take account of developments in more distant regions of TT [31].
The seminal books of Ewald [44], Fulton [50], and Oda [82] presented the
topic to readers with little background in algebraic geometry, and showed singularexamples to be as important as those that are smooth; indeed, the term toricmanifold is sometimes used to distinguish the latter. For the sake of brevity, werestrict attention to compact toric varieties, whether singular or smooth.
An essential ingredient of the toric geometer's worldview is the concept of a complete fan (which we abbreviate to fan henceforth). Every fan arises from m > nvectors in the integral lattice Zn < Rn, which determine m positive half-lines knownas rays; the rays intersect the unit sphere Sn−1 Rn in m points. Initially, weinsist that these be distributed so as to form the vertices of a simplicial subdivision, none of whose faces contains antipodal points. The set of convex polyhedraobtained by taking the infinite cone on each face of , with vertex the origin, isthen a simplicial fan Σ. By convention, {0} is included as the cone on the emptyface ∅. If we allow the cones to be spanned by linearly dependent rays, we obtain AN INVITATION TO TORIC TOPOLOGY a less restricted notion of fan that is of sufficient generality for our purposes below;in every case, a fan decomposes Rn as a union of closed cones.
For any fan Σ in Rn, we may construct a compact toric variety . It is covered by affine varieties as σ ranges over the cones of Σ, where and Xσ′are glued along whenever σ and σ′ have a common subcone τ . The algebraictorus (C×)n acts compatibly on the , and therefore on ; in particular X{0}is naturally isomorphic to (C×)n, and the action is by multiplication. Every conecontains the subcone {0}, so X{0} is dense in and forms the principal orbit ofthe global action. There is also a natural map of posets from the cones of Σ tothe subtori of (C×)n, ordered by inclusion. The construction of in this fashionis tantamount to using the fan as a combinatorial blueprint for compactifying theprincipal orbit (C×)n in such a way that coordinatewise multiplication extends toa (C×)n-action. This viewpoint has been extremely influential during the growthof toric geometry.
Associating to Σ actually establishes a fundamental varietal correspondence between general fans and compact toric varieties, and the algebraic properties ofthe fan are reflected in the geometry of the corresponding variety. If Σ is simplicial,for example, then the singularities of are homeomorphic to finite quotients of R2n, and the variety is an orbifold. If every cone is defined by rays that extend toa basis of Zn, then Σ is regular, and is smooth; and if is the boundary ofa convex simplicial polytope, then Σ is polytopal, and is projective.
One of Demazure's motivations for introducing toric varieties was his interest in their algebraic isomorphisms, and he proved that Aut() is a compact Lie group
in any nonsingular case, with maximal torus T n < (C×)n acting by restriction.
Subsequently, his results were extended to toric orbifolds by Cox [29], and to the
general case by B¨
The orbit space of the maximal torus may be identified with a polyhedral ball in Rn, whose bounding sphere is subdivided by the dual of Σ. Every subspace
projects onto the dual of the corresponding face of ; in particular, X{0}
projects onto the interior of the ball. Following Fulton [50, §4.1], for example, the
underlying topological T n-space of may then be expressed in derived form by
means of a homeomorphism
: (T n × QΣ)/ ∼ −→ XΣ. The image of the associated section is known to algebraic geometers as a canonical
submanifold with corners, possibly singular, of [82, §1.3]. If Σ is polytopal,
then is the corresponding simple polytope.
Example 3.1. The simplest example of a toric manifold is, of course, given by CP n, whose fan may be taken to be the n + 1 vectors e 1,. . , en, j in Rn; it is regular and polytopal, being normal to the n-simplex on 0, e1, . . , en. More generally, the vectors e 1,. . , en, −w j form a fan Σ for any nonnegative in- teger w; it is simplicial, but regular only when w = 1. For w > 1, the correspondingtoric variety is an orbifold CP n(w, 1, . . , 1), known as a weighted projective space,whereas the orbit space is an n–simplex for all values of w.
We describe generalisations of CP n(w, 1, . . , 1) in §5.
VICTOR M BUCHSTABER AND NIGEL RAY The second vertex C . The study of geometrical objects such as regular
polyhedra forms one of the oldest branches of mathematics, and reaches back thou-
sands of years. Nevertheless, we invite readers to consider combinatorial geometry
as the second vertex of TT , on the grounds that we shall focus on those aspects of
the subject that have come of age since connections with toric varieties were first
discovered. The edge AC was exposed in spectacular fashion by Stanley's solution
[91] to one half of McMullen's conjecture in 1980; however brief, his work is the
obvious source for the emergence and development of C as an independent vertex.
Two of the most influential surveys of polytope theory have been Gr¨
book [54], and Ziegler's more recent lectures of 1995 [99]. The latter is now quoted
by authors working in many areas of TT , and emphasised the principle that simple
polytopes are by far the most amenable to general discussion. More specific to
the vertex C itself is Stanley's key text [92], where he transformed traditional
invariants of polytope theory into powerful algebraic machinery for the study of
abstract simplicial complexes K. His pivotal construction was the face ring R[K],
otherwise known as the Stanley-Reisner algebra of K, over a commutative ring R.
Whenever possible we write the vertices of K as v1, . . , vm, and the vertex set as V ; we also assume that the faces σ ⊆ V include the empty face ∅. So σ ∈ Kand ρ ⊆ σ imply that ρ ∈ K. For algebraic purposes we often insist that thevertices are graded by real dimension 2, and rewrite subsets ω ⊂ V as squarefree j in the polynomial algebra SZ(V ) on the vj . The face ring of K is then defined as the quotient SR(V )/(ω / ∈ K), and the set of monomials divisible by faces of K forms an additive R-basis.
In order to place McMullen's conjecture in context, we recall that the dual, or polar polytope P ∗, of any simple polytope P is simplicial, and therefore boundedby a simplicial sphere KP . For example, the n-cube In is simple, and its polar isthe cross-polytope, whose boundary KIn triangulates Sn−1. This is also the dualityof (3.1), because KP defines the normal fan of P , whose rays may be taken to beintegral by small deformations of P if necessary.
Historically, integral vectors f (P ) and h(P ) were defined for simple polytopes in terms of KP , as follows. For any simplicial complex K of dimension n − 1, let fjdenote the number of faces of dimension j, for 0 ≤ j ≤ n − 1, and write f (K) forthe vector (f0, . . , fn−1); let f−1 = 1 count the empty face. Then define integers j for 0 ≤ j ≤ n by the polynomial equation j tn−j = j=1 j (t − 1)n−1−j , and write h(K) for the vector (h0, . . , hn). Finally, let the g-vector g(K) be given
by (g0, . . , g[n/2]), where g0 = 1 and gj = hj − hj−1. In case K = KP , there is a
beautiful Morse theoretic argument [17] involving flow along the 1-skeleton of P to
show that the Dehn-Sommerville equations
hj = hn−j 0 ≤ j ≤ n always hold. The case h n = h0 is Euler's equation j = 1 + (1)n−1.
Since 1959 [90] it has been possible to characterise integral vectors that realise
f (K) for some simplicial complex K. The corresponding problem for simplicial
polytopes has a longer history, and the correct formulation was only achieved when
McMullen stated his conjecture in 1971 [73]. Given an integral vector h, he pro-
posed two requirements: firstly, that the Dehn-Sommerville equations (3.2) hold;
and secondly, that the associated g-vector is of a combinatorial form referred to
AN INVITATION TO TORIC TOPOLOGY by Stanley as an M -vector. Stanley confirmed the necessity of McMullen's condi-
tions, by proving that g(P ) is an M -vector for any simple polytope P . Sufficiency
was established simultaneously by Billera and Lee [13], using completely different
methods. McMullen's characterisation is now known as the g-theorem.
The revolutionary aspect of Stanley's proof was his use of the toric variety corresponding to the fan defined by KP . By construction, Σ is both simplicial andpolytopal, so is a projective toric orbifold. The Danilov-Jurkiewicz theoremtherefore describes its rational cohomology ring by means of an isomorphism Q[KP ]/J −→ H∗(; Q) , where J denotes an ideal generated by linear terms associated to the rays of Σ.
If Σ is regular, the integral version of (3.3) also describes the cohomology ringH∗(; Z). Stanley applied the hard Lefschetz theorem to H∗(; Q), and de-duced the existence of a 2-dimensional cohomology class w with certain multi-plicative properties. His proof concluded by showing that the coordinates gj ofthe g-vector measure the ranks of the 2j-dimensional components of the quotientring H∗(; Q)/(w), and therefore that g is an M -vector. In this context, theDehn-Sommerville equations are equivalent to rational Poincar´ e duality for .
The Dehn-Sommerville equations actually remain true for a considerably wider class of simplicial complexes, whereas restrictions on g-vectors are less easy to
extend. For example, the conjecture that g(K) is an M -vector for every simplicial
sphere K appears still to be open. Similar mystery shrouds the f -vectors of convex
polytopes; an explicit classification was given by Steinitz [93] for dimension 3 in
1906, yet the situation for dimension 4 remains obscure [100]!
Since Stanley's initial applications, face rings with particular algebraic proper- ties such as Cohen-Macauley or Gorenstein have become important test-beds for
commutative algebra [18], and interest has grown in identifying simplicial com-
plexes K for which Z[K] is of a specific type. Relevant results include Reisner's
Theorem [89], which supplies homological criteria for K to be Cohen-Macaulay;
his criteria are satisfied by all simplicial spheres.
For any simplicial complex K, the face poset L(K) contains the faces of K, ordered by inclusion; the empty face is a minimal element, and the combinatorialtype of K is determined by the isomorphism class of (K). By analogy, the combi-natorial type of an arbitrary polytope P is defined to be the isomorphism class ofits face poset L(P ), which omits the empty face but includes the maximal face Pitself. So for any simple polytope P , the opposite poset Lop(P ) is isomorphic tothe face poset L(KP ). The f -, g-, and h-vectors of P are combinatorial invariants,because they depend solely upon L(P ), as do our applications to toric topologybelow. We therefore deal only with combinatorial polytopes henceforth, on theunderstanding that they may be represented by face posets or affine realisations asthe occasion demands. We emphasise that n–dimensional polytopes in Rn may becombinatorially equivalent, but not affinely isomorphic.
The third vertex S . The origins of symplectic structure may also be traced
back down the years, as far as the profound reformulation of Newtonian mechanicsthat was begun by Lagrange, and continued by Hamilton during the first halfof the 19th century. Hamilton's choice of coordinates q and p for phase space,representing position and momentum respectively, led eventually to the notion ofa symplectic manifold (M, ω). The phase space M is smooth, and ω is a closed VICTOR M BUCHSTABER AND NIGEL RAY non-degenerate 2-form, whose existence forces the dimension of M to be even.
Typical examples of symplectic manifolds include the total space of any cotangentbundle, which admits a canonical ω, and an arbitrary K¨ ahler manifold, equipped with the imaginary part of the defining metric. The former example illustrates theimportance of noncompact phase spaces.
Many mechanical systems admit groups of symmetries, which may be inter- preted as diffeomorphisms of M 2n; in the symplectic setting such symmetries pre-
serve the 2-form, and are known as symplectomorphisms. Experience with classical
systems such as pendulums and spinning tops suggests that effective Hamiltonian
actions of a compact torus T k are important special cases, which require k ≤ n
and determine a moment map Φ : M → Rk. Here Rk is realised as the dual Lie
algebra of T k, and Φ is constant on the T k-orbits. When the phase space is com-
pact, Atiyah [3] and Guillemin and Sternberg [56] proved in 1982 that the image
Φ(M ) is always a convex polytope. These works are the source of the third vertex
S, whose associated discipline is the symplectic geometry of toroidal symmetry,
moment maps, and convex polytopes.
The second edition [6] of Audin's book offers an inspirational survey of de-
velopments up to 2004. The overarching theme is Hamiltonian actions of tori,with special attention being paid to the half-dimensional case; in this situation,the real-valued coordinate functions of Φ form a completely integrable system, andthe associated torus action is called completely integrable. It follows that the fixedpoint set of any such action is nonempty, and that its elements are isolated.
An important technique in Hamiltonian geometry is that of symplectic reduc- tion, which compensates for the fact that the quotient of a symplectic manifold(M, ω) by a Hamiltonian action of T k need not be symplectic. Reduction involvesforming the level submanifold Z = Φ−1(x) of a regular value of the moment map,and imposing a symplectic structure on the quotient Z/T k, which is better behaved,and at worst an orbifold. For each factorisation of T k into a product of subtori,the reduction may be performed in corresponding stages. Symplectic reduction hasplayed an important rˆ ole in fully understanding the relationship between the ver- tices A and S of the Tetrahedron, but enters rarely into our discussions below. We
therefore refer interested readers to [6] for further details.
The edge AC . After the existence of a wider toric world was revealed in
[91], the edge AC was quickly colonised. One pioneer was Khovanskii, whose work
[64], [65] resonated with that of Stanley by introducing torus actions into problems
where none were initially apparent.
Every monomial in n complex variables z1, . . , zn may be written as for some integral vector (α1, . . , αn). So any Laurent polynomial f (z) = defines a finite set A ⊂ Rn containing those α for which the coefficient cα ∈ C is
nonzero, and the Newton polyhedron of f is the convex hull of A. Now suppose
that Wf ⊂ (C×)n is the complete intersection defined by a non-degenerate system
f1(z) = · · · = fk(z) = 0 of k ≤ n polynomial equations, with respective Newton
polyhedra 1, . . , ∆k. Khovanskii considered the problem of computing invariants
such as the arithmetic genus χ(Wf ) in terms of the number of integral points interior
to the ∆j; his results generalised Kushnirenko's theorem [6] for the case k, where
Wf is discrete. He proceeded by forming the normal fan Σ of the Minkowski sum of
the ∆j, interpreted the corresponding toric variety as the compactification of
(C×)n, and proved that the closure cl(Wf ) ⊂ XΣ is nonsingular and conveniently
AN INVITATION TO TORIC TOPOLOGY embedded. He completed his computation by applying Ehler's description [43]
of the Chern classes of a nonsingular toric variety (now settled on the edge AT ).
Additional details may be found in [50, §5], for example.
A more unconventional topic on this edge was furnished by work of Sturmfels and Sullivant[94], who considered the cut polytope Cut(G) associated to an arbi-
trary finite graph G = (V, E), which is well-known in combinatorial optimization.
It is rarely simple for larger graphs, so the projective toric variety XG determined
by its normal fan is usually singular. The authors carried out experimental calcu-
lations in the ring H∗(XG : C), and discussed applications to algebraic statistics.
The edge AS . The importance of the edge AS was demonstrated in 1988 by
Delzant's characterisation [36] of non-singular projective toric varieties in terms
of symplectic geometry. For any n-polytope P with regular normal fan Σ(P ), he
invested the toric variety (P ) with a symplectic form for which the action of T n
is completely integrable, and deduced that the image of the associated moment map
recovers P . His proof proceeded by identifying (P ) with the symplectic reduction
associated to the moment map Φ : Cm → Rm−n of a certain (m − n)–dimensional
subtorus K < T m that depends upon P . Delzant also established the uniqueness of
his construction, by showing that any two compact, connected symplectic manifolds
with completely integrable torus actions are Hamiltonian diffeomorphic whenever
their moment maps have the same image.
The edge CS . The existence of the edge CS emphasises the rˆ
ment map in linking classical examples of convex polytopes to the phase spacesof interesting mechanical systems. We shall discuss examples with a topologicalcomponent in Section 5.
We may also interpret Delzant's classification theorem as a fundamental sym- plectic correspondence between primitive polytopes and symplectic manifolds with
completely integrable actions. This has been extended to weighted polytopes and
symplectic toric orbifolds by Lerman and Tolman [69].
The facet ACS . A major occupant of this facet is the study of arrangements
A = {L1, . . , Lr} of subspaces Lj < Cm, which began before the advent of toric
geometry, and may now be interpreted as a limit of activity within the interior
of TT . The source is Arnold's calculation of the integral cohomology ring of the
coloured braid group [2] in 1969, which stimulated a rapidly expanding interest in
arrangements and their complements Cm L , denoted by U (A). Because they arise as configuration spaces of various classical mechanical systems, their properties
are central to many problems in algebraic, combinatorial, and symplectic geometry.
An extensive general survey was produced by Bj¨
orner [14] in 1994, and several
other overviews are available.
In 1988 Goresky and MacPherson [51] determined the additive structure of
H∗(U (A); Z), as a direct sum of homology groups associated to the intersection
poset I
(A). Subsequently, Cox [30, Theorem 4.1] showed how to interpret Delzant's
constructions in terms of symplectic structure on the complements of a particular
class of complex arrangements; these are now of major importance to toric topology,
and will also be discussed in Section 5.
VICTOR M BUCHSTABER AND NIGEL RAY 4. Vertex Four - Toric Topology
Overview. Many phenomena associated with the Toric Triangle refer to topo-
logical spaces that are equipped with additional geometric structure, and it is nat-ural to search for their analogues in a purely topological context. Such results arethe substance of toric topology, our fourth vertex T , whose convex hull with ACSforms the Toric Tetrahedron.
The pioneering work is due to Davis and Januszkiewicz in 1991, whose paper [35] is the source of toric topology. They introduced a class of well-behaved actions
a of T n on 2n–dimensional manifolds M that are more amenable to topological
analysis than those of toric geometry. Here we make the additional assumption that
M and a are smooth, although such considerations and their consequences were left
to readers in [35]. The pairs (M, a) are now known as quasitoric manifolds, and
are defined by two conditions. The first is a weakening of the varietal structure by
requiring only that a be locally standard, and so ensures that M/T n is a manifold
with corners; the second identifies the orbit space as a simple polytope P n. Since
P n is contractible, a admits many smooth sections.
The faces of the polytope are the closures of its isotropy blocks, amongst which the m facets Fj are distinguished by their dimension n − 1, and by the isotropysubcircles T (Fj) < T n associated to their interiors, for 1 ≤ j ≤ m. Because P nis simple, every face G of dimension n − d is a unique intersection ∩d facets; the isotropy subtorus T (G) < T n associated to its interior has rank d, andis the isomorphic image of the product ×d T (F ). The facets lift to closed facial submanifolds Xj ⊂ M of dimension 2(n − 1), which are fixed by the circles T (Fj)for 1 ≤ j ≤ m. The Xj intersect transversally, and G lifts to the closed submanifold ∩d X of dimension 2(n − d), which we abbreviate to X(G) ⊂ M; it is fixed by the d–torus T (G).
Alternatively, suppose given a simple polytope P n with m facets Fj, and a characteristic map λ : P n → T (T n) to the poset of subtori. Assume that the kernelof λ partitions P n by the interiors of its faces, and associates a circle (Fj) < T nto the interior of the facet Fj, for 1 ≤ j ≤ m. Finally, suppose that the productmap ×d ) → T n is actually an isomorphism onto its image T λ(G) for every face G = ∩d of P ; this is condition () of Davis and Januszkiewicz, which guarantees that λ is determined by the circles (Fj).
The derived manifold of λ is given by topologising (2.1) as M (λ) = (T n × P n)/ ∼ , where (g, p) (h, p) whenever g−1h ∈ λ(p), and imposing the natural smooth struc-
ture suggested by Davis [34]. The canonical T n-action and the section s(q) = [1, q]
are also smooth, and (G) = (T n/Tλ(G) × G)/ ∼ is a 2(n − d)–dimensional
submanifold for any face G; it is fixed by the d-torus (G). Conversely, any
quasitoric manifold (M, a) with smooth section s admits a characteristic map
λs : P n → T (T n), where λs(p) = T (G) for any p in the interior of G.
These constructions are mutually inverse, and determine a fundamental qua- sitoric correspondence between characteristic maps and quasitoric manifolds. Inparticular, every choice of smooth section s for an arbitrary quasitoric manifold(M, a) leads to a T n-equivariant homeomorphism fs : M (λs) −→ M , AN INVITATION TO TORIC TOPOLOGY which may be upgraded to a diffeomorphism by appeal to [34]. As in [24], we refer
to M (λs) as the derived form of (M, a).
The derived manifold (4.1) is a version of Vinberg's construction [95], as
adapted by Davis and Januszkiewicz. The characteristic map depends only on thecombinatorial type of P , and may be reformulated in terms of face posets as an orderreversing map λ : L(P n) → T (T n), or an order preserving map λ : L(KP ) → T (T n).
In the latter context, condition () specifies the values of λ on arbitrary faces ofKP in terms of those on its m vertices.
Whenever such λ exist, we deem P n and KP to be supportive, and describe the associated M (λ) as lying over P n. But there are many unsupportive simple
polytopes! An intriguing family of examples arises when KP is the n–dimensional
cyclic polytope Cn [99], for any number k of vertices satisfying k ≥ 2n; in other
words, no quasitoric manifolds can lie over the dual polytopes.
In pursuing their cohomological calculations, Davis and Januszkiewicz defined an auxiliary space DJ (K) for arbitrary simplicial complexes K, and proved the sem-inal fact that H∗(DJ (K)+; Z) realises the Stanley-Reisner algebra Z[K]. WheneverP supports a quasitoric manifold (M, a), they exhibited an isomorphism Z[KP ]/J −→ H∗(M+; Z) [35, Theorem 4.14], where J is the ideal generated by the first Chern classes of n
complex line bundles over DJ (KP ), associated to the isotropy subcircles of a. This
is the toric topologist's version of the Danilov-Jurkiewicz theorem (3.3) for toric
varieties. It arises from the fibration
M −→ DJ(KP ) −→ BT n, which is built into Davis and Januszkiewicz's model for DJ (KP ) as the homotopyquotient ET n ×T n M ; the homotopy type of the latter does not depend on thechoice of (M, a). Philosophically, DJ (KP ) is the homotopy theorists' substitute forP n, whose T n-bundle ET n × M → DJ(KP ) desingularises the quotient map.
The construction of the second auxiliary space ZK in [35] was equally signifi-
cant, and closely related to that of the derived manifold (4.1). It admits a canonicalaction a of T m, whose orbit space PK is the simple polyhedral complex dual to K;if K = KP then PK = P n for any simple polytope. When KP is supportive, everycharacteristic map λ : PK → T (T n) satisfying condition () determines a subtorusK(λ) < T m of dimension m − n that acts freely on ZK. The quotient n-torusT m/K(λ) acts on the orbit space ZK/T m−n, which is readily identifiable with thederived quasitoric manifold M (λ). By varying the choice of λ, we run through afamily of subtori K(λ) < T m in the poset T (T ), and the corresponding factorisa-tions ZK → M (λ) → P n display all possible quasitoric manifolds over P n in theform of (2.2). From this point of view, the homotopy quotient ET m ×T m ZK isa natural model for DJ (KP ), and may be identified with any ET n ×T n M (λ) byfactoring out the freely acting subtorus K(λ).
The next major impact on the subject was made in 2002 by the survey of
Buchstaber and Panov [20], which contains a wealth of fascinating extensions, gen-
eralisations, and applications of the ideas of [35]. Several of these were responsible
for establishing new regions of TT , and are introduced in the context of the appro-
priate edges and faces below. So far as the vertex T is concerned, Buchstaber and
Panov emphasised the importance of working with arbitrary simplicial complexes
K, and constructed both DJ (K) and the moment-angle complex ZK by more direct
VICTOR M BUCHSTABER AND NIGEL RAY and functorial methods, as unions of subspaces indexed by the faces of K. These
were developed in [86] as diagrams over the face category cat(K), whose objects
are the faces of K and morphisms their inclusions. In particular, ZK and DJ(K)
were identified as homotopy colimits of certain exponential diagrams, which now
underlie a homotopy theoretic industrial zone at the vertex T .
Highlights. We have reached the most seductive passage of our invitation! As
the Toric Tetrahedron has acquired form and substance, new areas of toric topologyhave been revealed, spawning examples, insights, and results that illuminate manytopological phenomena. We present a small selection of our favourites.
1) Non-quasitoric manifolds. There is considerable insight to be gained from understanding why certain families of likely-looking manifolds cannot be quasitoric.
We focus on the Milnor hypersurfaces Hj,k, for 1 ≤ j ≤ k. By definition, Hj,k ⊂ CP j × CP k is the complex hypersurface dual to the tensor product η1 ⊗ η2of the canonical line bundles over the respective factors. It has dimension 2(j+k−1),and may be realised by projectivising the stabilisation η⊥ ⊕ Ck−j of the orthogonal complement of η1 over CP j. So its integral cohomology ring takes the form Z[x, y] (xj+1, yk + yk−1x + · · · + yk−jxj), where x is the pullback of the generator of H2(CP j; Z) and y ∈ H2(Hj,k; Z) is thefirst Chern class of the canonical line bundle along the fibre. If Hj,k is quasitoricthen its cohomology ring takes the form (4.2), and yk−j (yj + yj−1x + · · · + xj)must lie in an ideal of H∗(Hj,k; Z) generated by squarefree monomials and linearfactors. A straightforward calculation shows that this is impossible; so Hj,k admitsno suitable action of T j+k−1.
By way of comparison, consider the 2(j + k − 1)–dimensional quasitoric man- ifold Bj,k constructed in [23]. It is the quotient of (S3)j × S2k−1 by the (j + 1)–
dimensional subtorus T (j, k) < T 2j+k of points
{(t1, . . , tj, tj+1, . . , tj+1) : ti ∈ T for 1 ≤ i ≤ j + 1}; thus T (j, k) acts freely. Projection onto (S3)j shows that Bj,k is the projectivisationof the complex k-plane bundle γ⊥ ⊕ Ck−j over the bounded flag manifold B Example 2.2. So its integral cohomology ring takes the form Z[x1, . . , xj, y] (xi(xi − xi−1), yk + yk−1xj + · · · + yk−jxj : 1 ≤ i ≤ j), where xi is the pullback of the eponymous element in H2(Bj); Z), and x0 = 0. Ashort calculation reveals that yk−j (yj + yj−1xj + · · · + xj) now factorises as yk−j (y + x1)(y − x1 + x2) . . (y − xj−1 + xj) in H2j (Bj,k; Z). Note that γj is the pullback of η along its classifying map, whichinduces a monomorphism from (4.3) to (4.4) by mapping x to xj and fixing y.
2) Stably complex structures. Davis and Januszkiewic [35, §6] showed that
every quasitoric manifold admits a stably complex structure, but did not remark
that many inequivalent choices are possible. On bounded flag manifolds Bn, for
example, some structures bound and others do not [27], with serious consequences
for complex cobordism. To be precise, we invest (M, a) with an omniorientation
[24], by choosing orientations for M itself, and for every facial submanifold Xj;
so there are 2m+1 omniorientations in total. The normal 2–plane bundle of every
embedding Xj ⊂ M is also oriented, and extends to a complex line bundle σj over
AN INVITATION TO TORIC TOPOLOGY M for 1 ≤ j ≤ m, by the Pontryagin-Thom construction. An omniorientation givesrise to a unique isomorphism τ (M ) Cm−n ∼ j=1 j , and hence to a canonical stably complex structure on M .
The additional structure is captured by a fundamental omnioriented correspon- dence, between omnioriented quasitoric manifolds and dicharacteristic homomor-
phisms ℓ
: T F → T n, where F denotes the set of m > n facets of an oriented
combinatorial polytope P n, and obeys the analogue of condition (). There are
2m such homomorphisms for every characteristic map λ : P n → T (T n), because
any isotropy circle λ(p) = T (F ) admits two isomorphisms TF → T (F ) from the
coordinate subcircle T F ; there is one for each orientation of T (F ), corresponding
to the orientations of the X(F ) ⊂ M (λ). Also, is represented on Lie algebras by
an integral n × m matrix, which may be refined to [In : Λ] for some n × (m − n)
matrix Λ, by careful choice of bases. We label the pair (P, Λ) as the quasitoric
combinatorial data
[22] underlying the omnioriented quasitoric manifold M ().
Example 4.1. In the case of CP 3, the polytope P is 3 and n is 3; so m−n = 1 and Λ is a 3–dimensional column vector. For the omniorientation induced by thestandard toric varietal structure, this vector is (1, −1, −1)t, whereas one of the 32alternatives yields (+1, −1, +1)t. All three Chern numbers are nonzero in the firstcase, but zero in the second.
For the Dobrinskaya tower M (3, 3) of Example 2.2, P is 3 × ∆3 and n is 6; so m − n = 2 and Λ is 6 × 2. The omniorientation given by 1 1 1 1 1 1 is induced by a toric varietal structure, and 3 of the 29 alternatives are given bynegating one or both of the columns.
3) Complex cobordism. The Thom spectrum M U is universal amongst complex oriented ring spectra, and is one of the most important objects in stable homotopytheory. Complex bordism ΩU ( · ) and cobordism Ω∗ ( · ) are the associated homology and cohomology theories, and may be defined in terms of stably complex manifolds
as explained by Conner and Floyd [28] and Quillen [87] respectively. The coefficient
ring π∗(MU ) is isomorphic to ΩU
(S0), and abbreviated to ΩU ; this is the celebrated complex cobordism ring, shown by Milnor [75] and Novikov [78] to be isomorphic
to a graded polynomial algebra Z[zn : n ≥ 1], where zn has dimension 2n. Sum and
product of cobordism classes are induced by disjoint union and cartesian product
of manifolds respectively. No truly canonical choice of polynomial generators is
known, although zp−1 may be represented by CP p−1 for any prime p; the remaining
zn are represented by linear combinations of the Milnor hypersurfaces Hj,k, where
j +k −1. Such results are obtained by calculating Chern numbers, which completely
determine complex cobordism classes.
The essential feature of [23] was to exhibit omniorientations of the Bj,k to act
as substitutes for the Milnor hypersurfaces. It follows that every complex cobor-
dism class may be represented by a disjoint union of quasitoric manifolds. The
connected analogue is considerably more subtle, and relies on the construction of
connected sums that are compatible with omniorientations; this was finally achieved
in [22], where every class of dimension greater than 2 is represented by an omniori-
ented quasitoric manifold. The analogous problem for connected algebraic varieties
remains unsolved, having been posed by Hirzebruch in 1958.
The value of these ideas lies in the principle that quasitoric representatives (M 2n, a) offer an alternative source of discrete data for any complex cobordismclass. The fundamental omnioriented correspondence, for example, gives rise to VICTOR M BUCHSTABER AND NIGEL RAY quasitoric combinatorial data (P, Λ), whereas the isolated fixed points of any T n-
action give rise to classical local invariants; theoretically, we may express the latter
in terms of the former. A crucial case is that of the sign σ(x) = ±1 of any fixed
point x. Its expression in terms of combinatorial data was obtained by Panov [83],
and compares the global orientation of M with the orientation of the tangent space
Tx(M ) induced by the n incident facial submanifolds. Signs are therefore indispens-
able for formulating the connected sum of quasitoric manifolds [22]. Alternatively,
σ(x) compares the orientation underlying the stably complex structure on M with
that induced on Tx(M ) by the T n-action; so σ(x) is always positive when M is
complex or almost complex.
The quasitoric manifolds Bn were introduced to complex cobordism theory in 1986 [88], as iterated sphere bundles with bounding omniorienations. They were
shown to represent the basis elements bn for ΩU
(CP ∞), dual to powers un of the first cobordism Chern class. So their cartesian products Bi × · · · × B the basis elements bi · · · b · · · uik.
(BT k), dual to monomials ui1 4) Hirzebruch genera. Classic genera associated to complex analytic manifolds include the Euler characteristic and Todd genus. These were extended to almost
complex manifolds, and thence to complex cobordism classes, as they became ex-
pressible in terms of Chern numbers [59]. In modern parlance, a Hirzebruch genus
is a homomorphism f : ΩU
∗ → R∗ of graded rings into any commutative R∗, and is equivalent to a formal group law over R∗ [79]. Historically, f was specified by
a formal power series t/ef (t) in R∗ ⊗ Q[[t]], and ef (t)) was identified with the ex-
ponential of the formal group law as soon as the concept was available [19], [80].
The construction and computation of genera and their equivariant analogues has a
long and illustrious history [4], [5], [39], [58], [60], [65], [66], [68], [81].
A prime example is the universal toric genus Φ, which may be evaluated on any 2n–dimensional T k-manifold (M, a) that carries a T k-equivariant structure on its stable normal bundle. Then Φ(M, a, cν) is given by the geometric cobordismclass [ET k ×T k M → BT k] in Ω−2n(BT k), and therefore takes the form ui1 · · · uik , 1 ···ik So g0.0 is precisely the cobordism class [M ] in ΩU , and we may represent g by a connected manifold that fibres over Bi × · · · × B with fibre M . Crucially, the base bounds, so we may relate the multiplicative properties of any Hirzebruchgenus to the rigidity properties of its T k-equivariant extension.
We may restrict attention to examples (M, a, cν) for which a has isolated fixed points only, and is the complement of an equivariant structure on its stabletangent bundle; these include all toric and quasitoric manifolds. In order to in-terpret (4.5) in terms of local invariants at each fixed point x, we require the signσ(x) (in the alternative form above) and the weight vectors w1(x), . . , wn(x) ofthe representation of T k in Tx(M ). The resulting localisation formula is Φ(M, α, c ( [wj(x)](u1, . . , un) ) where x ranges over the fixed point set of a and [wj(x)](u1, . . , uj) is the appro-
priate multivariable wj-series for the universal formal group law. The fact that the
irregular part of (4.6) must vanish imposes severe restrictions the possible signs and
weight vectors. The formula reduces to the S1 version of [25], and was obtained
AN INVITATION TO TORIC TOPOLOGY by Krichever [67] for almost complex manifolds (so σ(x) = 1). It was applied in
[26] to determine the complex bordism classes of classical homogeneous spaces. For
quasitoric manifolds we have k, and the signs and weights may be described explic-
itly in terms of the combinatorial data (P, Λ); the descriptions simplify significantly
for projective toric varieties.
Masuda [70] and Panov [83] also obtained formulae evaluating Hirzebruch's
original χy-genus in terms of combinatorial data. Both confirmed the well-knownfact that every smooth toric variety has Todd genus 1, in contrast to the arbitraryintegral values attainable on quasitoric manifolds.
5) cat(K)-diagrams. The appropriate framework for the study of toric homo- topy types is that of diagrams over the face category cat(K), meaning covariant
functors such as the exponential diagrams XK → top of [86]. Here top denotes
a category of pointed topological spaces such as k-spaces [96], of which X is an
object. Also, XK (σ) is the cartesian product , and XK (σ ⊂ τ ) is the inclusion
Xσ → Xτ induced by assigning the basepoint to the additional coordinates. In case
X is the classifying space BT of the circle, colim
BT K is precisely Buchstaber and Panov's construction BT σ for DJ (K) [20].
The corresponding descriptions of ZK and its quotients is subtler - and more revealing. Consider the diagram T V K : cat(K) top, for which T V K(σ) is thetorus T V σ and T V K (σ ⊂ τ ) is the projection T V σ → T V τ ; then colim is a point. There is a more complex diagram DK , for which DK (σ) is the subspace(D2)σ × T V σ of the product of 2-disks (D2)V , and DV (σ ⊂ τ ) is the correspondinginclusion (D2)σ×T V σ → (D2)τ ×T V τ . The projection DK(σ) → T V K(σ) is a ho-motopy equivalence for every σ, and induces a map colim DK → colim DK is Buchstaber and Panov's construction for Z K [20]. This exam-
ple shows how badly colimits behave under objectwise homotopy equivalences, butalso suggests a remedy. The homotopy colimit of T V K is defined by constructing acofibrant replacement diagram such as DK , and taking its ordinary colimit; in otherwords, the simpler diagram provides a weak equivalence hocolim T V K ≃ Z This procedure always works in favourable circumstances, although finding an appropriate cofibrant replacement may be difficult. For spaces X such as CW-complexes, the exponential diagram XK is already cofibrant, so we may also writeDJ (K) as hocolim BT K . On the other hand, any quotient of Z K by a subtorus T l < T V gives rise to a weak equivalence hocolim T V/l ≃ Z K /T l, where T V /l(σ) is T V σ/ Im(T l).
In particular, this expresses any quasitoric manifold M as a homotopy colimit, and extends the corresponding result [97, Proposition 5.3] for
toric varieties, where homotopy colimits and toric geometry were first associated.
The close relationship between derived forms and homotopy colimits actually hinges
on the fact that the nerve of cat(K) is the cone on the barycentric subdivision of
K, and may therefore be identified with the polyhedral complex PK .
Homotopy colimits over cat(K) are sometimes preserved by the standard func- tors of algebraic topology, so long as they can be defined in the target category.
The consequences of these ideas are studied in [85], along with many algebraic and
geometrical examples associated to the vertex T .
6) Homotopy types. Explicit calculations of homotopy types fall into the three standard categories of increasing complexity; rational, p-adic, and integral.
VICTOR M BUCHSTABER AND NIGEL RAY The rational homotopy type of any simply connected space X is most efficiently encoded by Sullivan's minimal model, which is a well-structured commutative differ-
ential graded algebra that is quasi-isomorphic to the commutative cochain algebra
A∗ (X) [47]. Spaces whose minimal models may be constructed directly from
their rational cohomology algebras are known as formal, and play a central rˆ rational homotopy theory. Important examples include spheres, compact connectedLie groups, and compact K¨ ahler manifolds.
The rationalisation of any DJ (K) or quasitoric manifold M retains significant homotopy theoretic information. A result of [76] confirms that DJ (K) is formal
for arbitrary K, and is applied in [85] to deduce the formality of M ; indeed, the
argument extends to cases such as the torus manifolds of [71]. By way of contrast,
moment-angle complexes cannot generally be formal, as follows from Baskakov's
discovery [10] of nontrivial Massey products in their rational cohomology for certain
K. His calculations employed the bigraded chain complex of ZK discussed in [20],
and have since been extended by Denham and Suciu [38]. Minimal models for
DJ (K) have been exhibited only for iterated joins of simplices and boundaries of
simplices [76], in which case the cohomology ring is a complete intersection and
uniquely determines the simply connected rational homotopy type.
So far as p-adic results are concerned, the p-completion of DJ (K) is uniquely determined by its mod-p cohomology ring Fp[K] whenever K is a join of skeleta of
simplices [77]. This holds for any prime p, and combines with the rational results
to show integral uniqueness for complete intersections.
The integral homotopy type of the ZK has long been of interest to combinato- rialists, as we shall explain in our discussion of the face ACT . Work of Grbi´ Theriault [52] showed that ZK is weakly equivalent to a wedge of spheres whenever
K is obtained by iterated pushouts of shifted complexes. We may then interpret
crucial elements of π∗(DJ(K)) in terms of the homotopy quotient fibration
T m −→ ZK −→ DJ(K) . In particular, π∗(DJ(K)) contains many Whitehead products, both iterated andgeneralised, that directly reflect the combinatorial structure of K.
The possibility of analysing stable homotopy types has also been raised by [8], which considers homotopy decompositions of the suspensions ΣZK and their
generalisations. These should be compared with earlier splittings of suspensions of
kth stage Bott towers into wedges of Thom complexes over lower stages [27], and
analogous splittings for Dobrinskaya towers.
7) Oriented cohomology theories. The integral cohomology of any DJ (K) or quasitoric manifold M 2n is free, so evaluating a complex oriented cohomology the-ory E∗( · ) on either space is relatively straightforward. Nevertheless, it is conve-nient to assume that the coefficient ring π∗E = E∗ is concentrated in even dimen-sions, and free of additive torsion. Then E∗(DJ (K)+) is isomorphic to the facering E∗[K], for example, because the corresponding Atiyah-Hirzebruch spectral se-quence collapses. Given any simple polytope P and dicharacteristic homomorphism: T F → T n, the cohomology ring of the derived quasitoric manifold is given by anisomorphism E∗[KP ] JE −→ E∗(M ()+), where J E is the ideal generated by the first E-theory Chern classes of the n complexline bundles of (4.2). These bundles are tensor products, so their Chern classes AN INVITATION TO TORIC TOPOLOGY embody the formal group law associated to E∗( · ). The resulting formulae areconceptually straightforward, and have been obtained independently by Stricklandand the authors. Complex cobordism and complex K-theory are good examples,involving the universal and multiplicative formal group laws respectively, althoughdetailed calculations with the former may require major technical expertise. Oneremarkable consequence is that the ΩU -module ΩU (M ) is generated by embedded quasitoric submanifolds for any quasitoric manifold M .
Computing the KO -theory of these spaces is much more difficult. A significant start was made by Bahri and Bendersky [7], who found a degree of order amidst
the algebraic chaos of their Adams spectral sequence approach. In particular, they
proved that KO ∗(M ) is additively isomorphic to a sum of KO ∗-modules of two
types; namely KO ∗(S2i) for 0 ≤ i ≤ n, and KO∗(Σ2jCP 2) for 0 ≤ j ≤ n − 2.
The splitting depends only on the cohomology ring H∗(M ; Z/2), and leads to the
BB numbers p and q, which enumerate the summands of the first and second type
respectively. More specific calculations were carried out in [27] for certain families
of Bott towers, whose KO ∗-algebra structures and BB numbers were determined.
The most general setting for these attempts is that of quaternionic oriented co- homology theory, for which quaternionic cobordism Ω∗ ( · ) is the universal exam- ple. Neither the quaternionic cobordism ring nor the module structure of Ω∗ (CP n) are known, so any applications to toric spaces are extremely interesting, but wellout of current reach.
5. New Faces - and Beyond
Having introduced the fourth vertex T , we conclude our invitation by adver- tising the seven new faces it defines, and speculating on future trends that shouldfurther increase the allure of the Tetrahedron.
The edge AT . Even before the vertex T was properly identified, several toric
geometers had strayed along the edge AT in their desire to investigate topological
invariants of toric varieties. For example, Ehler's formula [43] for the Poincar´
duals of the Chern classes of a nonsingular toric variety was utilised in Khovanskii's
work [65] on the edge AC. His formula represents the dual of ck(τ (M 2n)) by the
sum of the fundamental classes X(G) in H 2n−2k(M+; Z), taken over all the 2(n−k)–dimensional facial submanifolds of M . It extends immediately to quasitoricmanifolds, so long as M is omnioriented and the X(G) are invested with the inducedomniorientations. Indeed, the formula may then be interpreted in ΩU computes the the duals of the universal complex cobordism Chern classes.
Toric topologists have also visited AT , as exemplified by [22]. The authors
prove that any complex cobordism class may be represented by the quotient of a
free torus action on a real quadratic complete intersection. The importance of the
real quadratic viewpoint has recently been emphasised by Bosio and Meersseman
[16], in their construction of families of non-K¨
ahler complex manifolds using ZK.
The edge CT . Exploration of the edge CT motivated much of Buchstaber
and Panov's survey [20]. Their results include a notable generalisation of the Dehn-
Sommerville equations (3.2), for which they applied Poincar´
e duality to prove j − hn−j = (1)n−j χ(K) − χ(Sn−1) VICTOR M BUCHSTABER AND NIGEL RAY for any triangulated (n − 1)–manifold K, and 1 ≤ j ≤ n. It follows that thehj − hn−j are homotopy invariants, and independent of the triangulation.
This edge is also a natural habitat for Halperin's well-known toral rank conjec- ture, concerning the largest integer trk(X) for which a finite-dimensional topological
space X admits an almost free T trk(X)-action. Every isotropy subgroup is necessar-
ily finite, and the conjecture states that dim H∗(X+; Q) 2trk(X). If the conjecture
holds, then it follows from [21] that any (n − 1)–dimensional simplicial complex K
on vertices [m] satisfies the inequality
2m−n − 1 , where ω ranges over the subsets of [m], and the are induced subcomplexes.
When K is the boundary of ∆n, for example, the right hand side of (5.1) is 1, andequality holds. Computer calculations confirming the exponential growth requiredby the conjecture for cyclic polytopes are currently in progress by Gadjikurbanov.
The edge ST . The edge ST is becoming densely populated, as realisation
dawns that global properties of Hamiltonian actions are as much a part of equi-
variant topology as of the symplectic geometry that nurtured their growth. This
viewpoint is expounded by Guillemin, Ginzburg and Karshon [55], who provide a
beautiful survey of many exciting aspects of ST , and initiate several new ideas
that stray into adjacent faces of the Triangle.
For example, the famous Duistermaat-Heckman formula [42]
e⟨Φ,ξ⟩ωn = (2π)nn! is associated to the process of symplectic reduction, and is well known to math-
ematical physicists as the exact stationary phase formula. Atiyah and Bott and
Berligne and Vergne proved that it is also a special case of a localisation formula
of equivariant cohomology that arises in the presence of isolated fixed points; fur-
ther explanation is given in [55], for example. From this viewpoint, (5.2) is closely
related to the universal toric genus (4.6), as is evident from their formal similarity.
These considerations are relevant to the topology of symplectic circle-actions on a compact symplectic M . In dimension 4, McDuff proved that any such action is
Hamiltonian if and only if the fixed point set is nonempty, and gave a 6–dimensional
non-Hamiltonian example whose fixed point sets are 2-tori. Subsequently, Feldman
[46] studied the case of isolated fixed points, and deduced that the Todd genus of
M is necessarily 1 or 0; if 1, the action is Hamiltonian, and if 0, it is not.
The facet ACT . This facet is home to the development of a special class of
subspace arrangements, as introduced on ACS; here too, work represents the limitof activity within the interior of TT .
For any subset σ ⊆ [m] we write the coordinate subspace {z : zj = 0 if j ∈ σ} as Lσ ≤ Cm (otherwise known as C[m] σ in Example 2.1). So an arbitrary sim-
plicial complex K on vertices [m] determines the coordinate subspace arrangement
: σ /∈ K}, with complement U(K). So T m acts on U(K) coordinatewise, and
we may interpret U globally as a diagram of T m-spaces on the category of subcom-
plexes of the (m−1) simplex on vertices [m], for every m > 0. The relevance of U to
TT is epitomised by [20, Theorem 8.9], which establishes a natural T m-equivariant
AN INVITATION TO TORIC TOPOLOGY deformation retraction U (K) → ZK. In other words, we may study homotopy the-oretic properties of moment-angle complexes purely in terms of coordinate subspacearrangements. In particular, we obtain smooth structures on the ZK, Well before this link was established, Ziegler and ˇ c [101] brought the
theory of homotopy colimits to bear, and rederived Goresky and MacPherson's
algebraic decomposition of H∗(U (A); Z). For the case U (K), their ideas lie behind
our current formulation of ZK as a homotopy colimit over cat(K). For coordinate
subspace arrangements, the decomposition of H∗(U (K); Z) was also recovered in
[11] by applying a theorem of Hochster, and the link with ZK was exploited to
deduce the multiplicative structure; similar results were obtained independently by
Franz [48]. The algebraic decomposition has recently been realised topologically,
by splitting ΣZK [8].
Further techniques of unstable homotopy theory are now being focused on moment-angle complexes in this context. For example, iterated and generalisedSamelson products in the loopspaces ΩZK are under consideration, and their rela-tionship with the geometry of the arrangement and the combinatorics of K promisesto be particularly fruitful.
Other questions that were originally posed in the era of torus embeddings have also been formulated and solved in the context of the facet ACT . One such is the
weak Oda conjecture of 1978, concerning the factorisation of proper equivariant bi-
rational maps of nonsingular toric varieties into sequences of well-behaved blow-ups
and blow-downs. The conjecture may be approached using combinatorial topology,
because any bistellar move on the simplicial complex corresponds to a combi-
natorial operation on the fan Σ, and thence to a blow-up and blow-down on XΣ.
It was verified in 1997 by Morelli and W lodarczyk [98], who proved that any two
regular fans in Rn are connected by a sequence of bistellar operations through reg-
ular fans. The result was then extended to other nonsingular varieties by a process
of torification [1].
The facet AST . The facet AST may not yet have acquired a distinctive
literature, but two topics suggest themselves as candidates.
For any symplectic manifold (M, ω), let Diff(M ) denote the group of diffeo- morphisms of the underlying smooth manifold, and consider the subgroups Ham(M, ω) < Sym(M, ω) < Diff(M ) of Hamiltonian diffeomorphisms and symplectomorphisms respectively. Algebraic,
geometric and homotopy theoretic properties of these groups are discussed in Mc-
Duff's survey of 2003 [72], but many questions remain open. It is therefore natural
to consider examples in which M admits the action of a compact Lie group [63];
in particular, the situation for toric varieties and quasitoric manifolds has aroused
recent interest.
Harada and Landweber's study [57] of the equivariant K-theory of Hamiltonian
T n-spaces should also be located in this facet.
The facet CST . A beautiful circle of ideas that typify the facet CST arises
from the action by conjugation of the unitary group U (n) on the vector space spaceH of n × n Hermitian matrices. The orbit of the diagonal matrix d(λ1, . . , λn)is diffeomorphic to the (n2 − n)–dimensional flag manifold of complete flags in Cn,and admits a standard symplectic structure. The natural left action of T n < U (n)is Hamiltonian, and we may identify the corresponding moment map with the VICTOR M BUCHSTABER AND NIGEL RAY restriction to of Φ : H → Rn, which associates to any Hermitian matrix its
vector of diagonal entries. A classical theorem of Schur and Horn [6] asserts that
the image Φ() is the convex hull of the n! points obtained by permuting the
coordinates of (λ1, . . , λn) in Rn. The resulting simple convex polytope Πn−1 is
the famous permutahedron [99], and a section for Φ is given by lifting each of its
points p to the corresponding diagonal matrix d(p).
Now consider the manifold J 2(n−1) of n × n tridiagonal Hermitian matrices, with distinct eigenvalues λ1, . . , λn. According to [15], there is a non-standard
embedding J λ that serves as a symplectic reduction; the restriction of the symplectic form on is non-degenerate, and the restriction of the momentmap is a completely integrable system. In particular, J 2(n−1) is a toric and qua- sitoric manifold over Πn−1. Furthermore, the permutahedron is the polar of the
barycentric subdivision of ∆n−1, and is therefore (n − 1)-colourable. It follows
from [35, Corollary 6.10] that J 2(n−1) is stably parallelisable, and hence of serious
interest to stable homotopy theorists.
The interior. The interior of TT is relatively uncharted, but exploratory set-
tlements have been established and several further expeditions are under way.
Panov's work [84] is a recent example, which aims to exploit the relative ease
with which topologists construct the factorisation ZK → M → PK of (2.2) for any
quasitoric manifold M . Analogues for algebraic varieties and completely integrable
torus actions were obtained by several authors in the early 1990s, as described by
Cox [30]. The results may be interpreted as providing homogeneous coordinates
for the varieties in question, but depend on the notions of categorical quotient and
symplectic reduction respectively. Panov achieved a more satisfying unification by
adapting the concept of Kempf-Ness set from the theory of algebraic group actions
on affine varieties, and applying it to the coordinate subspace pair (U (K), ZK). His
procedure works for toric orbifolds, and is closest to the motivating example in the
case of projective toric varieties.
A more established exploration of the interior concerns the pursuit of mirror symmetry, first observed by theoretical physicists. Originally formulated as the
identification of partition functions on a pair of Calabi-Yau manifolds (V, V ∗), the
subject became intertwined with toric geometry under the influence of Batyrev,
as summarised in his seminal paper [12], for example. In 1999, Cox and Katz
published an entertaining survey [32] from the algebraic viewpoint.
A convex polytope P in Rn is reflexive if it is integral, contains the origin in its interior, and has integral polar P ∗; then P ∗ is also reflexive. Following the con-
struction of [64] in the case of a single generic hypersurface, Batyrev conjectured
that the corresponding pair (XP , XP ∗ ) of toric varieties is mirror symmetric. By
defining stringy Hodge numbers for reasonable singular varieties, he proposed an
extension of the simplest topological test for smooth pairs (V, V ∗) to be mirror sym-
metric, and proved that any (XP , XP ∗ ) satisfies his test. Work on his conjectures
is ongoing. An inspection of the bibliography of [49] confirms that the subject lies
close to the the facet ACS; nevertheless, it has a small but significant T -coordinate.
Many examples in mirror symmetry involve toric orbifolds, which arise alge- braically from a class of simplicial fans and topologically from characteristic func-tions that fail to satisfy Davis and Januszkiewicz's condition ().
AN INVITATION TO TORIC TOPOLOGY Example 5.1. Following Example 2.2, let Z be S2n+1 Cn+1, and a the coordinatewise action of T n+1. For any vector χ of positive integers (χ1, . . , χn+1),let H be the subcircle T (χ) < T n+1 of points {(1, . . , tχn+1) : t ∈ T }, and M the quotient space S2n+1/T (χ); so M is acted on by the quotient n–torus= T n+1/T (χ). In this case M is the weighted projective space CP n(χ), andreduces to the standard projective space CP n in case χ = (1, . . , 1).
Weighted projective spaces have inspired an impressive literature of their own, with links to each of the vertex disciplines; [41] is a well-known survey . Recent
work is increasingly topological, and includes the calculation [9] of the equivariant
cohomology ring H∗ (CP n(χ); Z); this doubles as a face ring for the weighted
quotient simplex of [69], and is described by generators and relations in terms of
piecewise polynomials on the associated fan. It is also isomorphic to the Chow ring
of CP n(χ).
Further research into the algebraic topology of quasitoric orbifolds is under active development, and includes applications of weighted lens spaces, homotopycolimits, weighted face rings, and the Bousfield Kan spectral sequence, togetherwith calculations in K-theory and cobordism. The topic certainly appears to meritinclusion deep within the interior of the Tetrahedron.
So far as future trends are concerned, applications to the motion planning prob- lem of robotics look especially intriguing. Moment-angle complexes ZK are known
to engineers as configuration spaces of planar linkages [61], and their topological
measures the instability of algorithms for determining paths between
initial and final states of the system [45]. Bounds on the topological complexity
depend on the homotopy type and cohomological structure of ZK, and are therefore
within reach for certain specific K.
Beyond TT . Experience shows that toric objects provide concrete realisa-
tions of abstract concepts in all four vertex disciplines, which not only stimulatethe interests of experts, but are also attractive and comprehensible to a generalmathematical audience. Such reasons suggest that the study of TT will continueto flourish. We therefore extend one final invitation to future authors, to engage inthe light-hearted but enlightening exercise of determining the toric coordinates oftheir own lectures and publications.
It seems likely that TT will eventually be subdivided as areas acquire sufficient theory, literature, and distinction to merit a separate identity.
imagine notation to accommodate this process! It is also fascinating to speculateon whether additional vertices lie buried in the future — although mathematicaland theoretical physicists' increasing involvement suggests that vertex P may soonemerge. We encourage readers to dream of a Toric Four-Simplex, and entertain uswith their fantasies at future toric conferences . .
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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street, Moscow 119991, Russia E-mail address: [email protected] School of Mathematics, The University of Manchester, Oxford Road, Manchester School of Mathematics, The University of Manchester, Oxford Road, Manchester E-mail address: [email protected]

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Geobios 37 (2004) 631–641 A new large barn owl (Aves, Strigiformes, Tytonidae) from the Middle Pleistocene of Sicily, Italy, and its taphonomical significance Une nouvelle espèce d'effraie géante (Aves, Strigiformes, Tytonidae) du Pléistocène moyen de Sicile, Italie, et son importance taphonomique Dipartimento di Scienze della Terra, University of Torino, Via Accademia delle Scienze 5, 10123 Torino, Italy