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## Higeom.math.msu.su

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 which

*X *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 each

*x ∈ 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−*1

*h ∈ 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*] =

*hλ*(

*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 that

*GsX*(

*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 function

*λ *:

*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 surjection

*rK *:

*D*(

*λ*)

*→ D*(

*λK*); by (2.1),

*rK *is equivariant with respect to the actions of

*G *and

*G/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

*Q*correspond 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 group

*G *there exist various functorial models for a contractible space

*EG *on which

*G*acts freely, and with closed orbits; any such

*EG *is ﬁnal amongst well-behavedfree

*G*-spaces and

*G*-equivariant homotopy classes of maps. The quotient map

*EG → EG/G *is then a universal principal

*G*-bundle, where

*EG/G *=

*BG *is a

*classifying 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 *: C

*n × *C

*n → *C

*n*, deﬁned by

*µ*(

*y*1

*, . . , yn, z*1

*, . . , zn*) = (

*y*1

*z*1

*, . . , ynzn*)
for any

*n > *0. We abbreviate the set

*{*1

*, . . , n} *to [

*n*], and write the subspace

*{z zi *= 0 for

*i /∈ ω} *as C

*ω ≤ *C

*n *for any subset

*ω ⊆ *[

*n*]. Similarly, we denote thecompact torus (

*S*1)

*ω *by

*T ω ⊂ *(C

*×*)

*ω*, as a subspace of

*{z zi ̸*= 0

*} ⊂ *C

*ω*. Finally,we write C

*δ *and

*Tδ < *C

*n *for the diagonal line and its unit subcircle respectively.

Example 2.1. Let

*X *be C

*n*, and

*a *:

*T n × *C

*n → *C

*n *the restriction of

*µ*. For
any

*z ∈ *C

*n*, 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 *: C

*n/T n → *R

*n *to the non-negative coordinate cone, induced by

*h*(

*z*1

*, . . , zn*) = (

* z*1

* *2

*, . . , zn *2)

*,*
and the standard inclusion of R

*n *in C

*n *speciﬁes a canonical section

*s *: R

*n → *C

*n*.

The isotropy blocks are the subspaces (C

*×*)

*ω ⊂ *C

*n*, whose closures C

*ω *are the ﬁxedpoint sets Fix(

*T *[

*n*]

* ω*). The blocks project to the interiors of the correspondingfaces R

*ω *of the polyhedron R

*n*, whose closures are the faces themselves. The
characteristic map

*λs *: R

*n → S*(

*T n*) assigns the subtorus

*T *[

*n*]

* ω *to the interior of
the face R

*ω*, and the derived form associated to

*s *is the canonical homeomorphism

*fs *: (

*T n × *R

*n*)

*/ ∼ −→ *C

*n.*
Example 2.2. For any ordered pair (

*p, q*) of positive integers, let

*Z*(

*p, q*) be
the product of unit spheres

*S*2

*p*+1

*× S*2

*q*+1

*⊂ *C

*p*+

*q*+2 and

*a *the product action of

*T p*+1

*× T q*+1. The orbit space is homeomorphic to the product

*∆p × ∆q *of standardsimplices in R

*p*+1

*× *R

*q*+1
, by (2.3). Let

*K *be the 2–torus

*T *(

*p, q*)

*< T p*+1

*× T q*+1 of
( (

*t*1

*, . . , t*1)

*, *(

*t*2

*, . . , t*2

*, t−*1

*t*
where

*t*1,

*t*2

*∈ S*1. Then

*M *(

*p, q*) =

*Z*(

*p, q*)

*/T *(

*p, q*) is a 2

*nd 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 (

*x*1

*, . . , xn*;

*u*1

*, . . , 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

*S*2

*p*+1

*× S*2

*q*+1.

VICTOR M BUCHSTABER AND NIGEL RAY
By construction,

*M *(

*p, q*) is the projectivisation of the complex (

*q *+ 1)–plane
bundle C

*q ⊕ η *over C

*P p*, where

*η *is the canonical line bundle. So

*M *(1

*, *1) is a 2

*nd*

stage Bott tower [

**53**], and is diﬀeomorphic to a Hirzebruch surface; it is also the

*bounded flag manifold B*2 [

**24**], consisting of ﬂags 0

*< L*1

*< L*2

*< *C3 for which

*L*2 contains the ﬁrst coordinate line; then (

*u, v*)

*∈ T *2 acts on

*L*1 by (1

*, v, uv*), and

on

*L*2 by (1

*, *1

*, u*). The case

*q *= 0 reduces to the quotient space

*S*2

*p*+1

*/Tδ*, and is

diﬀeomorphic to C

*P p*.

In fact

*M *(

*p, q*) may be extended to a

*k*th stage tower, by iterating the projec-
tivisation procedure, and replacing C

*q ⊕ 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 C

*n*+1

* *0, and

*a *: C

*× × *(C

*n*+1

* *0)

*→ *(C

*n*+1

* *0) the
restriction of

*µ*. The quotient space is the algebraic variety C

*P n*, on which thealgebraic torus (C

*×*)

*n *acts with a single dense principal orbit. In this context, thetoric coordinates identify C

*P 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

*S*represents 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 > n*vectors in the integral lattice Z

*n < *R

*n*, which determine

*m *positive half-lines knownas

*rays*; the rays intersect the unit sphere

*Sn−*1

*⊂ *R

*n *in

*m *points. Initially, weinsist that these be distributed so as to form the vertices of a simplicial subdivision

*KΣ*, none of whose faces contains antipodal points. The set of convex polyhedraobtained by taking the inﬁnite cone on each face of

*KΣ*, 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 R

*n *as a union of closed cones.

For any fan

*Σ *in R

*n*, we may construct a compact toric variety

*XΣ*. It is
covered by aﬃne varieties

*Xσ *as

*σ *ranges over the cones of

*Σ*, where

*Xσ *and

*Xσ′*are glued along

*Xτ *whenever

*σ *and

*σ′ *have a common subcone

*τ *. The algebraictorus (C

*×*)

*n *acts compatibly on the

*Xσ*, and therefore on

*XΣ*; 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

*XΣ *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

*XΣ *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

*XΣ *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

*XΣ *are homeomorphic to ﬁnite quotients of
R2

*n*, and the variety is an orbifold. If every cone is deﬁned by rays that extend toa basis of Z

*n*, then

*Σ *is

*regular*, and

*XΣ *is smooth; and if

*KΣ *is the boundary ofa convex simplicial polytope, then

*Σ *is

*polytopal*, and

*XΣ *is projective.

One of Demazure's motivations for introducing toric varieties was his interest in
their algebraic isomorphisms, and he proved that Aut(

*XΣ*) 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

*QΣ *of the maximal torus may be identiﬁed with a polyhedral
ball in R

*n*, whose bounding sphere is subdivided by the dual of

*Σ*. Every subspace

*Xσ *projects onto the dual of the corresponding face of

*KΣ*; 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

*XΣ *may then be expressed in derived form by

means of a homeomorphism

*fΣ *: (

*T n × QΣ*)

*/ ∼ −→ XΣ.*
The image of the associated section is known to algebraic geometers as a canonical

submanifold with corners, possibly singular, of

*XΣ *[

**82**,

*§*1.3]. If

*Σ *is polytopal,

then

*QΣ *is the corresponding simple polytope.

Example 3.1. The simplest example of a toric manifold is, of course, given by
C

*P n*, whose fan may be taken to be the

*n *+ 1 vectors

*e*
1,. . ,

*en*,

*−*
*j *in R

*n*;
it is regular and polytopal, being normal to the

*n*-simplex on 0,

*e*1, . . ,

*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 C

*P n*(

*w, *1

*, . . , *1), known as a weighted projective space,whereas the orbit space

*QΣ *is an

*n*–simplex for all values of

*w*.

We describe generalisations of C

*P 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

*v*1, . . ,

*vm*, and the vertex set
as

*V *; we also assume that the faces

*σ ⊆ V *include the empty face ∅. So

*σ ∈ K*and

*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

*S*Z(

*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

*fj*denote the number of faces of dimension

*j*, for 0

*≤ j ≤ n − *1, and write

*f *(

*K*) forthe vector (

*f*0

*, . . , 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 (

*h*0

*, . . , hn*). Finally, let the

*g-vector g*(

*K*) be given

by (

*g*0

*, . . , g*[

*n/*2]), where

*g*0 = 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 *=

*h*0 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

*XΣ*
corresponding to the fan deﬁned by

*KP *. By construction,

*Σ *is both simplicial andpolytopal, so

*XΣ *is a projective toric orbifold. The Danilov-Jurkiewicz theoremtherefore describes its rational cohomology ring by means of an isomorphism
Q[

*KP *]

*/J −→ H∗*(

*XΣ*; 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 ring

*H∗*(

*XΣ*; Z). Stanley applied the

*hard Lefschetz theorem *to

*H∗*(

*XΣ*; 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 2

*j*-dimensional components of the quotientring

*H∗*(

*XΣ*; Q)

*/*(

*w*), and therefore that

*g *is an

*M *-vector. In this context, theDehn-Sommerville equations are equivalent to rational Poincar´
e duality for

*XΣ*.

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

*P*itself. 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 R

*n *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 *2

*n*; 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 → *R

*k*. Here R

*k *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

*z*1, . . ,

*zn *may be written as

*zα *for
some integral vector (

*α*1

*, . . , αn*). So any Laurent polynomial

*f *(

*z*) =
deﬁnes a ﬁnite set

*A ⊂ *R

*n *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

*f*1(

*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

*XΣ *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

*XΣ*(

*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

*XΣ*(

*P *) with the symplectic reduction

associated to the moment map

*Φ *: C

*m → *R

*m−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 *=

*{L*1

*, . . , Lr} *of subspaces

*Lj < *C

*m*, 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 *C

*m *
*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

*ACS*forms 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 2

*n*–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 n*is 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

*Tλ*(

*Fj*)

*< T n*to 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

*Tλ*(

*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−*1

*h ∈ 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

*Xλ*(

*G*) = (

*T n/Tλ*(

*G*)

*× G*)

*/ ∼ *is a 2(

*n − d*)–dimensional

submanifold for any face

*G*; it is ﬁxed by the

*d*-torus

*Tλ*(

*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 of

*KP *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 ≥ *2

*n*; 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*]. Whenever

*P *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 for

*P 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 subtorus

*K*(

*λ*)

*< T m *of dimension

*m − n *that acts freely on

*ZK*. The quotient

*n*-torus

*T 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 ⊂ *C

*P j × *C

*P k *is the complex hypersurface dual to the tensor product

*η*1

*⊗ η*2of the canonical line bundles over the respective factors. It has dimension 2(

*j*+

*k−*1),and may be realised by projectivising the stabilisation

*η⊥ ⊕ *C

*k−j *of the orthogonal
complement of

*η*1 over C

*P j*. So its integral cohomology ring takes the form
Z[

*x, y*] (

*xj*+1

*, yk *+

*yk−*1

*x *+

*· · · *+

*yk−jxj*)

*,*
where

*x *is the pullback of the generator of

*H*2(C

*P j*; Z) and

*y ∈ H*2(

*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−*1

*x *+

*· · · *+

*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 (

*S*3)

*j × S*2

*k−*1 by the (

*j *+ 1)–

dimensional subtorus

*T *(

*j, k*)

*< T *2

*j*+

*k *of points

*{*(

*t*1

*, . . , tj, tj*+1

*, . . , tj*+1) :

*ti ∈ T *for 1

*≤ i ≤ j *+ 1

*}*;
thus

*T *(

*j, k*) acts freely. Projection onto (

*S*3)

*j *shows that

*Bj,k *is the projectivisationof the complex

*k*-plane bundle

*γ⊥ ⊕ *C

*k−j *over the bounded ﬂag manifold

*B*
Example 2.2. So its integral cohomology ring takes the form
Z[

*x*1

*, . . , xj, y*] (

*xi*(

*xi − xi−*1)

*, yk *+

*yk−*1

*xj *+

*· · · *+

*yk−jxj *: 1

*≤ i ≤ j*)

*,*
where

*xi *is the pullback of the eponymous element in

*H*2(

*Bj*); Z), and

*x*0 = 0. Ashort calculation reveals that

*yk−j *(

*yj *+

*yj−*1

*xj *+

*· · · *+

*xj*) now factorises as

*yk−j *(

*y *+

*x*1)(

*y − x*1 +

*x*2)

*. . *(

*y − xj−*1 +

*xj*)
in

*H*2

*j *(

*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 2

*m*+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 *)

*⊕ *C

*m−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

*ℓ *obeys the analogue of condition (

*∗*). There are

2

*m *such homomorphisms

*ℓ *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,

*ℓ *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 *(

*ℓ*).

Example 4.1. In the case of C

*P *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*
*∗ *(

*S*0), 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 2

*n*. 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 C

*P 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 *2

*n, 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*
*∗ *(C

*P ∞*), 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

*ui*1

*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 2

*n*–dimensional

*T k*-manifold (

*M, a*) that carries a

*T k*-equivariant structure

*cν*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−*2

*n*(

*BT k*), and therefore takes the form

*ui*1

*· · · uik ,*
1

*···ik*
So

*g*0

*.*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

*cν *is the complement of an equivariant structure

*cτ *on its stable

*tangent *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 sign

*σ*(

*x*) (in the alternative form above) and the weight vectors

*w*1(

*x*), . . ,

*wn*(

*x*) ofthe representation of

*T k *in

*Tx*(

*M *). The resulting localisation formula is

*Φ*(

*M, α, c*
( [

*wj*(

*x*)](

*u*1

*, . . , un*) )
where

*x *ranges over the ﬁxed point set of

*a *and [

*wj*(

*x*)](

*u*1

*, . . , 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

*S*1 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

*Xσ*, 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(

*D*2)

*σ × T V σ *of the product of 2-disks (

*D*2)

*V *, and

*DV *(

*σ ⊂ τ *) is the correspondinginclusion (

*D*2)

*σ×T V σ → *(

*D*2)

*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 a

*cofibrant 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 write

*DJ *(

*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 F

*p*[

*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

*k*th 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 *2

*n *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

*ℓ *:

*T F → T n*, the cohomology ring of the derived quasitoric manifold is given by anisomorphism

*E∗*[

*KP *]

*JE −→ E∗*(

*M *(

*ℓ*)+)

*,*
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 ∗*(

*S*2

*i*) for 0

*≤ i ≤ n*, and

*KO∗*(

*Σ*2

*j*C

*P *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∗ *(C

*P 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 *2

*n*)) by the

sum of the fundamental classes

*X*(

*G*) in

*H*
2

*n−*2

*k*(

*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 the

*hj − 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

*≥ *2

*m−n − *1

*,*
where

*ω *ranges over the subsets of [

*m*], and the

*Kω *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 *= (2

*π*)

*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*

{Lσ :

*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

*KΣ *corresponds to a combi-

natorial operation on the fan

*Σ*, and thence to a blow-up and blow-down on

*X*Σ.

It was veriﬁed in 1997 by Morelli and W lodarczyk [

**98**], who proved that any two

regular fans in R

*n *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 space

*H *of

*n × n *Hermitian matrices. The orbit

*Hλ *of the diagonal matrix

*d*(

*λ*1

*, . . , λn*)is diﬀeomorphic to the (

*n*2

*− n*)–dimensional ﬂag manifold of complete ﬂags in C

*n*,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

*Hλ *of

*Φ *:

*H → *R

*n*, which associates to any Hermitian matrix its

vector of diagonal entries. A classical theorem of Schur and Horn [

**6**] asserts that

the image

*Φ*(

*Hλ*) is the convex hull of the

*n*! points obtained by permuting the

coordinates of (

*λ*1

*, . . , λn*) in R

*n*. 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

*λ*1, . . ,

*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

*Hλ *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 R

*n *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

*S*2

*n*+1

*⊂ *C

*n*+1, and

*a *the
coordinatewise action of

*T n*+1. For any vector

*χ *of positive integers (

*χ*1

*, . . , χn*+1),let

*H *be the subcircle

*T *(

*χ*)

*< T n*+1 of points

*{*(

*tχ*1

*, . . , tχn*+1) :

*t ∈ T },*
and

*M *the quotient space

*S*2

*n*+1

*/T *(

*χ*); so

*M *is acted on by the quotient

*n*–torus

*Tχ *=

*T n*+1

*/T *(

*χ*). In this case

*M *is the

*weighted projective space *C

*P n*(

*χ*), andreduces to the standard projective space C

*P 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∗ *(C

*P 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 C

*P 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*

complexity 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

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street,
Moscow 119991, Russia

*E-mail address*: buchstab@mi.ras.ru
School of Mathematics, The University of Manchester, Oxford Road, Manchester
School of Mathematics, The University of Manchester, Oxford Road, Manchester

*E-mail address*: nigel.ray@manchester.ac.uk

Source: http://higeom.math.msu.su/research/files/buchstaber-ray-tt.pdf

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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