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Torus actions in topology and combinatorics
Victor Buchstaber, Alexander Gaifullin, Taras Panov
Lomonosov Moscow State University

Moscow State University, 15 March 2012

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Moscow, 15.03.2012

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?Toric Tetrahedron?
ori topology
t d d d d d d dt d d d d d d d t elgeri d geometry

gomintoris

t d



ympleti geometry
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 2 / 29


1. Combinatorics.
K simpliil omplex on the set [m] = {I, . . . , m} @e olletion of susets in [m] losed under inlusionAF
s = {i1 , . . . , ik } K simplex @or feA of dimension k - IF elwys ssume KF fi = fi (K) the numer of fes @simpliesA of dimension i F vet dim K = n - IF
f (K) = (f0 , f1 , . . . , fn-1 ) the f EvetorF f0 = m F he hEvetor h (K) = (h0 , h1 . . . , hn ) is de(ned from the identity

h0 t n + h1 t

n

-1

+ ћ ћ ћ + hn = (t - I)n + f0 (t - I)n

-1

+ ћ ћ ћ + fn

-

1

.

K is tringulted sphere if |K| n-1 F = ixmpleX the oundry of onvex nEdimensionl simpliil polytopeF
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 3 / 29


Restrictions on the number of faces
uestionX how to hrterise the f Evetors @or hEvetorsA for interesting lsses of simpliil omplexes @eFgFD polytopesD tringulted spheres or tringulted mnifoldsAc ixmples of restritions on f (K)X f0 - f1 + . . . + (-I)n Pfn f1 f0 f1
-
2

-

1

fn

-

1

= (K) hn - h0 = (K) - (
@hehn!ommerville reltionsAY if |K| = h2
n

n

-1

)Y

= nfn
-
i f0 2

-1

if K is tringulted sphere or mnifoldY if |K| =
n

hi = hn

-

1

Y

n+I nf0 -
n

h0
+1
2

h1

-

1

Y
n

h1

if |K| =

-

1

F

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Yaroslavl, 15.03.2012

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Theorem (BilleraLee, Stanley, 1980)
he following onditions re neessry nd su0ient for olletion (f0 , f1 , . . . , fn-1 ) to e the f Evetor of simpliil polytopeX @A hi = hn-i for i = H, . . . , nY @A h0 h1 h2 . . . h[n/2] Y @A . . . @ restrition on the growth of hi AF sn the proofD projetive tori vriety P is ssigned to polytope F he ohomology of P stis(es dim r 2i (P , Q) = hi ( ). hen @A follows from oinr? dulityD while @A nd @A follow from the e rrd vefshetz heorem for projetive vrietiesF

Problem (McMullen's conjecture)
ss it true tht the sme onditions @A!@A hrterise the f Evetors of tringulted spheresc
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 5 / 29


row to otin tringultions whih do not rise from polytopesc here re lssil exmplesX ? frnette sphere nd frukner sphere in dimension QY houle suspension on the onr? sphere in dimension SF e wore exmples n e onstruted using the opertions of suspensionD joinD nd istellr moves @or )ipsAF

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Yaroslavl, 15.03.2012

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Bistellar moves in dimension 2 and 3
p p p 'E p p p p p p d d d dp ' E p p d d d d d d p p d dp

p p ? ? ? ? ? 'E ? ? p hh phhhp h ?h h h p h h d ? ? h$p d? $ d?? $$ dp?$$$ p$

p p ? ? ? ? ? ? p hh p h? hh ? h h ? h h e d ? h$ p ' E d hh$ p e $ $ ?p edp?$$ ed$$ eg eg eg eg eg eg g g ep ep

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Yaroslavl, 15.03.2012

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Theorem (Pachner)
wo tringulted mnifolds re pieewise linerly equivlent if nd only if one n e tken into nother y sequene of istellr movesF st follows tht if we strt with nonEpieewiseEliner sphere tringultion @eFgFD from the doule suspension of the oinr? sphereA nd pply e istellr moves to itD then we never end up in polytopl tringultionF he ehviour of the hEvetor under istellr moves is esily ontrolledF here re softwre pkges @fistellrA exeuting this proedureF roweverD no ounterexmples to wwullen9s onjeture hve een found yet on this wyFFF

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Yaroslavl, 15.03.2012

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2. Topology.
gonsider unit polydis in Cm X

Dm = (z1 , . . . , zm ) Cm : |zi |
por eh s = {i1 , . . . , ik } [m]D set

I.

fI := (z1 , . . . , zm ) Dm : |zj | = I for j s . / he(ne the momentEngle omplex

ZK =
I

fI Dm
K

st is n invrint suset with respet to the oordintewise tion of the stndrd torus

Tm = (z1 , . . . , zm ) Cm : |zi | = I,
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology

i = I, . . . , m .
Yaroslavl, 15.03.2012 9 / 29


qiven pir of susets nd s [m]D set

( , )I = (x1 , . . . , xm ) m : xj for j s / =
i

Ч

I i

,
/
I

nd de(ne the polyhedrl produt of the pir ( , ) y

( , )K =
I

( , )I m .
K

hen ZK = (D, S)K D where S is the unit irleF enother exmpleX the omplement of oordinte suspe rrngement (K) = Cm \
{i1 ,...,ik }K /

{z Cm : zi1 = . . . = zik = H}
ЧK

= (C, C ) =
I

CЧ
K
i

CЧ ,
i

I

/

I

where C = C \ {H}F glerlyD ZK (K)F
Ч

Theorem (BuchstaberPanov)
@A here is deformtion retrtion (K) ZK Y @A vet |K| n-1 F hen ZK is mnifoldF =
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 10 / 29


3. Combinatorial commutative algebra.
K simpliil omplex on [m] = {I, P, . . . , m}F he fe ring @or the tnley!eisner ringA of K is Z[K] = Z[v1 , . . . , vm ]
vi1 ћ ћ ћ vik : {i1 , . . . , ik } K , / deg vi = P.

Theorem (BuchstaberPanov)
here is n isomorphism of @iAgrded lgers r (ZK ) =

Tor

Z[v1 ,...,vm ]

Z[K], Z .

Corollary
sf K is tringultion of n (n - I)Edimensionl mnifoldsD then oinr? e dulity for ZK implies the reltions hn
-
i

- hi = (-I)i (K) - (

n

-1

)

n i

@the generlised hehn!ommerville reltionsAF
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 11 / 29


he dimensions of the igrded omponents of the

Tor

EgroupsD



-i ,

2j

(Z[K]) := dim

Tor

-i ,2j Z[v1 ,...,vm ]

Z[K], Z

re sutle omintoril invrints of KF he fe numers of K re expressed in terms of he igrded fetti numers simpliil ohomology of KX
-i ,
2j

-i ,

2j

(Z[K])F

(Z[K]) n e lso omputed vi the

Theorem (Hochster)

Tor

-i ,2j Z[v1 ,...,vm ]

(Z[u ], Z) =
J

r
[m], |J |=
j

j

-i -

1

(u |J ),

where u |J is the full suomplex @the restrition of K to t {I, . . . , m}AF he numers -i ,2j (Z[K]) n e lso omputed e'etively using the ommuttive lger softwre pkge wuly PF
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 12 / 29


4. Bistellar moves in topology and geometry
p p p 'E p p p p

p p d d d dp ' E p p d d d d d dp p

d dp

p p ? ? ? 'E ? ? p hh phhh ? p h hh hh h h d ? $$p d ?? $ p $ $ dp$ dp?? $ $ ?

p p ? ? ? ? p h ? p h ? h h h h d e ?hhh p ' E d ?hhh p e $ $ dp$$$ dp$$$ ? ? e e eg eg eg eg g g egp egp

Theorem (Pachner, 1987)
por ny two v homeomorphi tringultions of the sme mnifoldD the (rst one n e trnsformed to the seond one y (nite sequene of istellr moves nd simpliil isomorphismsF
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 13 / 29


Applications of bistellar moves, stellar subdivisions, and related operations
1

gonstrution of lol omintoril formul for the (rst rtionl ontrygin lss of tringulted mnifold @qifullinD PHHRAF he wellEknown ql onjeture on the properties of the fe numers of )g simple polytopes is proved for the se of nestohedr @olodinD PHIHAF reise lower nd upper ounds for the )g numers re proved for mny importnt fmilies of )g simple polytopesD whih pper in di'erent res of mthemtisF @fuhster!olodinD PHIIAF hese estimtes re sed on the ft tht ll polytopes in these fmilies n e otined from ue y onseсutive truntions of odimension P fesF he formule for the numers of multiplitive genertors in given dimension of the rings of )g vetors of onvex polytopes @fuhsterEirokhovetsD PHIIAF
Toric Top ology Yaroslavl, 15.03.2012 14 / 29

2

3

Buchstab er, Gaifullin, Panov (MSU)


Applications of bistellar moves, stellar subdivisions, and related operations

4

ixistene of formul for the volume of simpliil REdimensionl polyhedron from its omintoril struture nd the set of its edge lengthsF he volume of n ritrry )exile REdimensionl polyhedron is onstntF @qifullinD PHIIA

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

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Combinatorial computation of the Pontryagin classes

ontrygin lsses re lssil invrints of mnifoldsF heir de(nition uses smooth struture on the mnifoldF sn IWSUGSV okhlinEhvrts ndD independentlyD hom proved the invrine of the rtionl ontrygin lsses under v homeomorphismsF his result leds nturlly to the prolem of omintoril omputtion of the rtionl ontrygin lsses of mnifold from tringultion of itF smportnt results on this prolem hve een otined y qrielov!qelfnd!vosik @IWUSAD wherson @IWUUAD vevittEourke @IWUVAD gheeger @IWVQAD qelfnd!wherson @IWWPAF roweverD omplete solution hs not een hievedF

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Toric Top ology

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Theorem (Gaifullin, 2004)
1

here exists n expliit lgorythm tht for ny @orientedA mEdimensionl omintoril mnifold u D omputes n (m - R)Edimensionl simpliil yle gm-4 (u , Q) representing the oinr? dul of the (rst ontrygin lss of u F e he yle is omputed from the tringultion u lollyF his mens tht = ,
dim

2

=m -

4

where the oe0ient depends only on the omintoril struture of the tringultion u in neighourhood of simplex F o e more preiseD depends only on the omintoril struture of the link v of in u F @v is tringultion of QEsphereFA
3

o ompute the oe0ient one needs to trnsform v to the oundry of REsimplex y mens of istellr movesX v = v1 v2 ћћћ vk = 4 nd then to tke the sum of ontriutions of ll these istellr movesF

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Toric Top ology

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Heron's formula
r d d d

a
r

db d d dr d

S 2 = p(p - a)(p - b)(p - c ) p= a+b+c
2

c
r r

r ? ? E ? ? ? ? r ? ? r ? ? ? ?

r

r

r

No formula for the area from edge lengths
18 / 29

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

Yaroslavl, 15.03.2012


CayleyMenger formula
vet Rn e n nEdimensionl simplex with verties p0 , p1 , . . . , pn nd let ij e the length of the edge pi pj F he gyley!wenger determinnt is given y H I I gw (p0 , . . . , pn ) = I F F F I hen 2 () = I H
2 01 2 02

I H F F F
2 01

I H F F F
2 02 2 12

F F F

2 12

2 0n

2 1n

2 2n

F ћћћ

ћ ћ ћ ћ F

ћ ћ ћ ћ F

ћ ћ ћ ћ

I
2 0n 2 1n 2 2n

F F F H

(-I)n+1 gw (p0 , . . . , pn ). Pn (n!)2
Toric Top ology Yaroslavl, 15.03.2012 19 / 29

Buchstab er, Gaifullin, Panov (MSU)


Computation of the volume of a simplicial polyhedron
Problem
uppose n QF por given simpliil nEdimensionl polyhedron in Rn D n we (nd formul for its volume in terms of its edge lengthsc

Equivalent problem
uppose n QF por given nEdimensionl polyhedron in Rn D n we (nd formul for its volume in terms of the intrinsi metris of the fesD tht isD in terms of the lengths of edges nd digonls of fesc here is formul mens tht the volume is root of polynomil
N

+ 1 ( )

N

-1

+ ћ ћ ћ + N ( ) = H,

where j re polynomils in the edge lengths of the polyhedronF n = QX iD itovD IWWTF n = RX iD qifullinD PHIIF n SX xuxyx
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 20 / 29


What is a simplicial polyhedron in R4 ?
xive nswerX e region ounded y losed QEdimensionl tringulted polyhedrl surfeF wore generl nswerX he oundry of polyhedron is QEdimensionl yle in R4 D iF eFD forml liner omintion @with integrl oe0ientsA of oriented onvex QEsimplies in R4 suh tht its lgeri oundry is zeroF por ny suh mnifold the generlised volume n e de(nedF
d 1 dd1

d

-1 2 c +1

s d d

d s 1d d d d

d 1

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Toric Top ology

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Main ideas of the proof (n = 4)
Theorem (Gaifullin, 2011)
por eh omintoril type of polyhedr there exists polynomil reltion ( , ) =
N

+ 1 ( )

N

-1

+ 2 ( )

N

-

2

+ . . . + N ( )

etween the volume of polyhedron nd the set of edge lengths of polyhedronF rere j ( ) re polynomils in edge lengths with rtionl oe0ientsF he proof of this theorem y indution on the numer of vertiesD then on the smllest vertex degreeD etF o simplify polyhedron the following moves re usedX e n dd @sutrtA the oundry of REdimensionl onvex simplex in R4 to the oundry of the polyhedronF hese moves re nturl nlogues of istellr movesF
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Flexible polyhedra
Denition
e )ex of polyhedron is ontinuous fmily of polyhedr t D H t ID of the sme omintoril type suh tht 0 = D the edge lengths of the polyhedr t re onstntD nd the polyhedr t1 nd t2 re not ongruent unless t1 = t2 F he guhy heoremX xo onvex polyhedron is )exileF frirdD IVWUX plexile selfEinterseted othedrF gonnellyD IWUUX pirst exmple of )exile emedded polyhedronF te'enD IWUVX he simplest known )exile emedded polyhedronF pogelsngerD IWVVX olyhedr in generl position re not )exileF lzD IWWVD thelD PHHHX plexile REdimensionl rossEpolytopesF st is unknown if )exile polyhedr exist in Rn D n
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology

SF
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Bricard's exible octahedron of the rst type

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Bricard's exible octahedron of the second type

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Toric Top ology

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Connelly's exible polyhedron. The rst example of an embedded exible polyhedron

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Steen's exible polyhedron. The simplest known embedded exible polyhedron

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The Bellows Conjecture

Conjecture (1978)
he generlized volume (t ) of )exile polyhedron is onstntF

Theorem
1 2

@itovD IWWTA he fellows gonjeture is vlid in dimension QF @qifullinD PHIIA he fellows gonjeture is vlid in dimension RF

Buchstab er, Gaifullin, Panov (MSU)

Toric Top ology

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Bibliography
I F wF fuhsterD xF uF irokhovetsD olytopesD pioni numersD ropf lgersD nd qusiEsymmetri funtionsD ussF wthF urvF 66XP @PHIIAD PUI!QTUF P itor fuhster nd rs novF orus etions nd heir epplitions in opology nd gomintorisF niversity veture eriesD volF 24D emerF wthF oFD rovideneD FsFD PHHPF Q ВF МF БухштаберD ТF ЕF ПановF Торические действия в топологии и комбинаторикеF МоскваD МЦНМОD PHHRF R itor fuhsterD rs novF ori opologyF rivD PHII!PHIPF S F wF fuhsterD F hF olodinD hrp upper nd lower ounds for nestohedrD szvestiyXwthemtis 75XT @PHIIAD IIHU!IIQQF T eF eF qifullinD vol formule for omintoril ontrygin lssesD szvestiyX wthemtisD 68XS @PHHRAD VTI!WIHF U eF eF qifullinD itov polynomils for polyhedr in four dimensions D rivXIIHVFTHIRF
Buchstab er, Gaifullin, Panov (MSU) Toric Top ology Yaroslavl, 15.03.2012 29 / 29