Selected Title s i n Thi s Serie s

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n topology an d combinatorics , 200 2

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory ,

2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Frobeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf f an d Die tma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2 2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0 1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/024

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Torus Actions an d Thei r Applications i n Topolog y

and Combinatoric s

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University

LECTURE Series

Volume 2 4

Torus Actions an d Thei r Applications i n Topolog y

and Combinatoric s Victor M . Buchstabe r

Tar as E . Pano v

American Mathematica l Societ y Providence, Rhod e Islan d

EDITORIAL COMMITTE E

Jerry L . Bon a (Chair ) Nige l J . Hitchi n Jean-Luc Brylinsk i Nicola i Reshetikhi n

2000 Mathematics Subject Classification. P r i m a r y 52B70 , 57Q15 , 57R19 , 14M25 , 52B05 , 13F55, 52C35 .

ABSTRACT. Th e ai m o f thi s boo k i s t o presen t toru s action s a s a connectin g bridg e betwee n com -binatorial an d conve x geometr y o n on e side , an d commutativ e an d homologica l algebra , algebrai c geometry an d topolog y o n th e other . Th e establishe d lin k help s t o understan d th e geometr y an d topology o f a tori c spac e b y studyin g th e combinatoric s o f it s orbi t quotient . Conversely , subtles t properties o f a combinatoria l objec t ca n b e recovere d b y realizin g i t a s th e orbi t structur e fo r a prope r manifol d o r comple x acte d o n b y th e torus . Th e latte r ca n b e a symplecti c manifol d with Hamiltonia n toru s action , a tori c variet y o r manifold , a subspac e arrangemen t complemen t etc., whil e th e combinatoria l object s involve d includ e simplicia l an d cubica l complexes , polytope s and arrangements . Suc h a n approac h als o provide s a natura l topologica l interpretatio n o f man y constructions fro m commutativ e an d homologica l algebr a use d i n th e combinatoric s i n term s o f torus actions .

The expositio n center s aroun d th e theor y o f moment-angl e complexes , whic h provide s a n effec -tive wa y t o stud y invariant s o f triangulation s b y th e method s o f equivarian t topology . Th e tex t i s furnished wit h a larg e lis t o f bot h ne w an d well-know n ope n problem s o f relevance t o th e subject . We hop e tha t th e boo k wil l b e usefu l fo r topologist s a s wel l a s combinatorialist s an d wil l hel p t o establish eve n tighte r connection s betwee n th e subject s involved .

Library o f Congres s Cataloging-in-Publicatio n D a t a

Buchstaber, V . M . Torus action s an d thei r application s i n topolog y an d combinatoric s / Victo r M . Buchstaber ,

Tar as E . Panov . p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 24 )

Includes bibliographi c reference s an d index . ISBN 0-8218-3186- 0 (alk . paper ) 1. Torus (Geometry ) 2 . Topologica l spaces . 3 . Combinatoria l analysis . I . Panov , Tara s E. ,

1975- II . Title . III . Universit y lectur e serie s (Providence , R.I. ) ; 24. QA613.2.B83 200 2 514 /.3-dc21 200201824 3

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

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© 200 2 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http:/ /www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 7 0 6 0 5 0 4 0 3 0 2

Contents

Introduction 1

Chapter 1 . Polytope s 7 1.1. Definition s an d mai n construction s 7 1.2. Fac e vector s an d Dehn-Sommervill e equation s 1 1 1.3. Th e ^-theore m 1 5 1.4. Uppe r Boun d an d Lowe r Boun d theorem s 1 8 1.5. Stanley-Reisne r fac e ring s o f simple polytope s 2 0

Chapter 2 . Topolog y an d combinatoric s o f simplicia l complexe s 2 1 2.1. Abstrac t simplicia l complexe s an d polyhedron s 2 1 2.2. Basi c PL topology , an d operation s wit h simplicia l complexe s 2 3 2.3. Simplicia l sphere s 2 8 2.4. Triangulate d manifold s 2 9 2.5. Bistella r move s 3 1

Chapter 3 . Commutativ e an d homologica l algebr a o f simplicia l complexe s 3 5 3.1. Stanley-Reisne r fac e ring s o f simplicia l complexe s 3 5 3.2. Cohen-Macaula y ring s an d complexe s 3 8 3.3. Homologica l algebr a backgroun d 4 0 3.4. Homologica l propertie s o f fac e rings : Tor-algebra s an d Bett i number s 4 2 3.5. Gorenstei n complexe s an d Dehn-Sommervill e equation s 4 6

Chapter 4 . Cubica l complexe s 4 9 4.1. Definition s an d cubica l map s 4 9 4.2. Cubica l subdivision s o f simple polytope s an d simplicia l complexe s 5 0

Chapter 5 . Tori c an d quasitori c manifold s 5 7 5.1. Tori c varietie s 5 7 5.2. Quasitori c manifold s 6 3 5.3. Stabl y comple x structures , an d quasitori c representatives i n cobordis m

classes 6 9 5.4. Combinatoria l formula e fo r Hirzebruc h gener a o f quasitori c manifold s 7 4 5.5. Classificatio n problem s 8 2

Chapter 6 . Moment-angl e complexe s 8 5 6.1. Moment-angl e manifold s Zp define d b y simpl e polytope s 8 5 6.2. Genera l moment-angl e complexe s ZK 8 7 6.3. Cel l decompositions o f moment-angle complexe s 8 9 6.4. Moment-angl e complexe s correspondin g t o joins , connecte d sum s an d

bistellar move s 9 2

viii C O N T E N T S

6.5. Bore l construction s an d Davis-Januszkiewic z spac e 9 4 6.6. Walk aroun d th e constructio n o f ZK'- generalizations , analogue s an d

additional comment s 9 7

Chapter 7 . Cohomolog y o f moment-angl e complexe s an d combinatoric s o f triangulated manifold s 10 1

7.1. Th e Eilenberg-Moor e spectra l sequenc e 10 1 7.2. Cohomolog y algebr a o f ZK 10 2 7.3. Bigrade d Bett i number s o f ZK'- th e cas e o f general K 10 6 7.4. Bigrade d Bett i number s o f ZK'- th e cas e o f spherica l K 11 0 7.5. Partia l quotient s o f Zp 11 3 7.6. Bigrade d Poincar e dualit y an d Dehn-Sommervill e equation s 11 7

Chapter 8 . Cohomolog y ring s o f subspace arrangemen t complement s 12 5 8.1. Genera l arrangement s an d thei r complement s 12 5 8.2. Coordinat e subspac e arrangement s an d th e cohomolog y o f ZK- 12 7 8.3. Diagona l subspac e arrangement s an d th e cohomolog y o f SIZK- 13 3

Bibliography 13 5

Index 141

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Index

Affine equivalence , 7 Alexander duality , 2 6

simplicial, 2 7 Ample divisor , 6 1 Arithmetic genus , 8 0 Arnold relations , 12 6 Arrangement, 12 5

diagonal, 13 3 coordinate, 12 7 hyperplane, 12 5 /c-equal, 13 3 subspace (central) , 12 5

Artin group , 9 7

Bistellar equivalence, 3 2 moves (flips , operations) , 29 , 32 , 58 ,

Barycenter, 2 4 Barycentric subdivision , 2 4 Bigraded

Betti numbers , 4 2 differential module , 4 0 differential algebra , 41 , 42

Blow-up, blow-down , 31 , 58 Borel construction , 9 4 Boundary, 7 Bounded flag manifold , 7 2

Chain, 8 Characteristic map , 6 4

directed, 7 1 Characteristic pair , 6 5

directed, 7 1 Charney-Davis conjecture , 8 0 Chow ring , 5 9 Chromatic number , 11 4 Cobordisms

complex, 7 0 oriented, 7 5

Cohen-Macaulay complex, 39 , 49 , 10 6 ring (algebra) , 3 8

Colimit, 9 7 coloring, 99 , 11 4 Combinatorial

equivalence, 7 , 2 3 neighborhood, 2 6

Complement (o f a n arrangement) , 12 5 Cone, 2 3

convex polyhedral , 5 7 non-singular, 5 8 rational, 5 8 simplicial, 5 8 strongly convex , 5 8

Connected su m of simpl e polytopes , 1 0 of simplicia l complexes , 2 4

Convex polyhedron , 7 Core, 2 6 Coxeter group , 2 , 9 7 Coxeter complex , 2 Cross-polytope, 1 0 Cube, 8

standard, 8 topological, 4 9

Cubical complex, 4 9

abstract, 4 9 combinatorial-geometrical, 4 9 embeddable int o lattice , 5 0

subdivision, 5 4 of simpl e polytope , 5 1

Davis-Januszkiewicz space , 9 5 Dehn-Sommerville relations , 13 , 47, 62 , 11 2

for triangulate d manifolds , 12 0 Dehn twist , 9 9 Depth, 3 8 Dimension

homological, 4 0 Krull, 3 8 of cubica l complex , 4 9 of polytope , 7 of simplicia l complex , 2 1

Dolbeault complex , 7 5

Edge, 7 Edge vector , 7 6 Elementary shellings , 3 3 Eulerian complex , 4 7

Face missing, 2 5 of conve x polyhedra l cone , 5 8

141

142 INDEX

of cubica l complex , 4 9 of polytope , 7 of polyhedron , 2 2 proper, 7

Face poset , 8 Face rin g

of simpl e polytope , 2 0 of simplicia l complex , 3 5

Facet, 7 Facet vector , 6 4 Facial submanifold , 6 4 Factorization conjectur e (strong , weak) , 5 8 Fan, 5 7

complete, 5 8 non-singular, 5 8 normal, 6 0 polytopal, 6 7 simplicial, 5 8 strongly polytopal , 6 7 weakly polytopal , 6 7

Four Colo r Theorem , 99 , 11 4 Flag complex , 2 5 Flagification, 2 5 Flip, 5 8 /-vector

of cubica l complex , 4 9 of polytope , 1 2 of simplicia l complex , 2 2

^-conjecture, 2 9 ^-theorem, 16 , 47 , 61 , 11 3 g-vector

of polytope , 1 2 of simplicia l complex , 2 2

Geometrical realization , 2 2 Ghost vertex , 8 8 Gorenstein, Gorenstein *

complex, 46 , 80 , 89 , 11 3 ring (algebra) , 4 6

Graph product , 9 7

/i-vector of algebra , 3 8 of polytope , 1 2 of simplicia l complex , 2 2

Hard Lefschet z theorem , 61 , 11 3 Hauptvermutung de r Topologie , 3 0 Hilbert series , 3 7 Hinge mechanisms , 9 8 Hirzebruch genus , 7 5 Hirzebruch surface , 8 3 Hopf conjecture , 8 0 Homotopy colimit , 97 , 12 7 hsop (homogeneou s syste m o f parameters) ,

38

Ideal monomial, 3 6 Stanley-Reisner, 3 5

Index (o f a vertex) , 7 7 Intersection cohomology , 6 1 Intersection /i-vector , 6 2 Intersection pose t (o f a n arrangement) , 12 5

Join, 2 3

L-genus, 7 5 Link, 2 5 lsop (linea r syste m o f parameters) , 3 8 Lower Boun d Conjectur e (LBC) , 1 7

generalized (GLBC) , 20 , 29 , 62 , 11 3

Manifold PL (combinatorial) , 2 9 quasitoric, 63 , 64

non-toric, 6 8 stably complex , 6 9 toric, 5 8

unitary, 8 1 triangulated (simplicial) , 2 9 with corners , 6 3

Milnor hypersurfaces , 70 , 7 3 Milnor filtration , 9 9 Minimal

generator se t (basis) , 4 1 map, 4 1 resolution, 4 1

Mirroring construction , 9 8 Moment-angle

manifold, 8 7 complex 2 , 8 8

Moment curve , 1 1 Moment map , 63 , 13 0 Morse theory , 13 , 66, 11 3

stratified, 12 6 Multi-fan, 8 1 M- vector, 1 7

Noether normalizatio n lemma , 3 8

Omniorientation, 7 0 Orbifold, 58 , 9 8 Order complex , 2 5

of arrangement , 12 5 Oriented matroid , 12 6

Piecewise linea r (PL) homeomorphism, 2 3 manifold, 2 9

with boundary , 3 3 map, 2 3 sphere, 2 8

Poincare series , 3 7 Polar set , 9 Polyhedron, 2 1

convex, 7 Polytope

combinatorial, 8 convex, 7

INDEX 143

cyclic, 1 1 generic, 9 geometrical, 8 fc-neighborly, 1 1 lattice, 6 0 neighborly, 1 1 non-rational, 60 , 6 2 polar (dual) , 1 0 rational, 6 0 simple, 9 simplicial, 9 stacked, 1 9

Polytope algebra , 1 8 Poset, 8

Eulerian, 4 7 Poset category , 127 Positive cone , 8 Product

of simpl e polytopes , 1 0 of simplicia l complexes , 2 4

Pseudomanifold, 11 1 Pullback fro m th e linea r model , 9 8

Quadratic algebra , 3 6

Rank function , 12 5 Regular sequence , 3 8 Resolution

free, 4 0 Koszul, 4 1 minimal, 4 1 projective, 4 2

Ray, 5 8

Schlegel diagram , 2 8 Serre problem , 4 2 Sign (o f a vertex) , 7 6 Signature, 75 , 7 9 Simplex, 8

abstract, 2 1 geometrical, 2 1 standard, 8 regular, 8

Simplicial complex, 2 1

abstract, 2 1 dual, 2 6 geometrical, 2 1 /c-neighborly, 9 6 pure, 2 1 underlying (o f a fan) , 5 8

fan, 5 8 isomorphism, 2 3 manifold, 2 9 map, 2 3

non-degenerate, 2 3 sphere, 2 8 subdivision, 2 3

of cube , 5 1

stellar, 3 1 Skeletal rigidity , 2 9 Small cover , 9 8 Spectral sequenc e

Anderson, 12 7 Eilenberg-Moore, 101 , 103 , 116 , 13 4 Leray-Serre, 74 , 96 , 104 , 11 5

Sphere Barnette, 28 , 6 8 Bruckner, 2 8 homology, 2 8 non-PL, 28 , 3 0 non-polytopal, 2 8 PL, 2 8 Poincare, 3 0 polytopal, 2 8 simplicial, 2 8 stacked, 3 2

Stably comple x manifold, 6 9 structure, 6 9

canonical, 7 0 Stanley-Reisner rin g

of simpl e polytope , 2 0 of simplicia l complex , 3 5

Star, 2 5 Subcomplex

full, 2 6 cubical, 5 0 simplicial, 2 1

Support (o f a n arrangement) , 12 5 Supporting hyperplane , 7 Surgery, 3 1

equivariant, 9 3 Suspension, 2 3 Symplectic reduction , 13 0

Tangent bundle , 6 9 Tn-manifold, 6 3 Todd genus , 75 , 8 0 Tor-algebra, 4 4 Toric variety , 31 , 57 Torus, 5 7

algebraic, 5 7 Torus actio n

Hamiltonian, 13 0 locally standard , 6 4 standard, 6 3

Triangulation Conjecture , 3 1

Union (o f a n arrangement) , 12 5 Upper Boun d Conjectur e (UBC )

for polytopes , 1 8 for simplicia l spheres , 3 9

Vertex, 7 , 2 1 Vertex set , 2 1 Volume polynomial , 1 8

144 INDE X

2-arrangement, 12 6 Xy-genus, 7 5 f/j-equi variant, 63 , 6 5 -0-translation, 6 5