Torus actions, combinatorial topology, and homological algebra

Russian Mathematical Surveys

Volumn 55, Issue 5, 2000, Pages 825-921

  Buchstaber, Victor M ^a   Panov, Taras E ^a  

^a NONE

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[No Author keywords available]

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EID: 0034555865     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2000v055n05ABEH000320     Document Type: Article

References (107)

1. J. F. Adams, "On the cobar construction", Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409-412.
2. J. F. Adams, "On the non-existence of elements of Hopf invariant one", Ann. of Math. (2) 72 (1960), 20-104.
3. S. Akbulut and J. D. McCarthy, Casson's invariant for oriented homology 3-spheres, Princeton Univ. Press, Princeton, NJ 1990.
4. V. I. Arnol'd, "The cohomology ring of the colored braid group", Mat. Zametki 5 (1969), 227-231; English transl., Math. Notes 5 (1969), 138-140.
5. V. I. Arnol'd, "The cohomology ring of the colored braid group", Mat. Zametki 5 (1969), 227-231; English transl., Math. Notes 5 (1969), 138-140.
6. M. F. Atiyah and R. Bott, "A Lefschetz fixed point formula for elliptic complexes I", Ann. of Math. (2) 86 (1967), 374-407.
7. M. F. Atiyah and F. Hirzebruch, "Spin-manifolds and group actions", Essays in topology and related topics, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. 18-28.
8. M. Audin, The topology of torus actions on symplectic manifolds, Birkhäuser, Boston 1991.
9. A. Bahri and M. Bendersky, "The KO-theory of toric manifolds", Trans. Amer. Math. Soc. 352 (2000), 1191-1202; http://xxx.lanl.gov/abs/math.AT/9904087.
10. D. Barnette, "The triangulations of the 3-sphere with up to 8 vertices", J. Combin. Theory Ser. A 14 (1973), 37-42.
11. D. Barnette, "A proof of the lower bound conjecture for convex polytopes", Pacific J. Math. 46 (1973), 349-354.
12. V. V. Batyrev, "Quantum cohomology rings of toric manifolds", Journées de Géométrie Algébrique d'Orsay (July 1992), Astérisque 218 (1993), 9-34; http://xxx.lanl.gov/abs/alggeom/9310004.
13. M. Bayer and L. Billera, "Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets", Invent. Math. 79 (1985), 143-157.
14. L. Billera and C. Lee, "Sufficiency of McMullen's conditions for f-vectors of simplicial polytopes", Bull. Amer. Math. Soc. 2 (1980), 181-185.
15. A. Björner, "Subspace arrangements", Proc. of the First European Congress of Mathematics (Paris, 1992), vol. 1 (A. Joseph et al., eds.), Birkhäuser, Basel, 1994, pp. 321-370.
16. A. Björner and F. Lutz, "Simplicial manifolds, bistellar flips and 16-vertex triangulation of the Poincaré homology 3-sphere", Preprint 1999; http://www.math.kth.se/ ~bjorner/papers.html.
17. C. Boyer, K. Galicki, B. Mann, and E. Rees, "Compact 3-Sasakian 7-manifolds with arbitrary second Betti number", Invent. Math. 131 (1998), 321-344.
18. E. Brieskorn, "Sur les groupes de tresses (d'après V. I. Arnol'd)", Lecture Notes in Math. 317 (1973), 21-44.
19. A. Brønsted, An introduction to convex polytopes, Springer-Verlag, New York 1983; Russian transl., Mir, Moscow 1988.
20. V. M. Buchstaber and T. E. Panov, "Algebraic topology of manifolds defined by simple polytopes", Uspekhi Mat. Nauk 53:3 (1998), 195-196; English transl., Russian Math. Surveys 53 (1998), 623-625.
21. V. M. Buchstaber and T. E. Panov, "Algebraic topology of manifolds defined by simple polytopes", Uspekhi Mat. Nauk 53:3 (1998), 195-196; English transl., Russian Math. Surveys 53 (1998), 623-625.
22. V. M. Buchstaber and T. E. Panov, "Torus actions and combinatorics of polytopes", Trudy Mat. Inst. Steklov. 225 (1999), 96-131; English transl., Proc. Steklov Inst. Math. 225 (1999), 87-120.
23. V. M. Buchstaber and T. E. Panov, "Torus actions and combinatorics of polytopes", Trudy Mat. Inst. Steklov. 225 (1999), 96-131; English transl., Proc. Steklov Inst. Math. 225 (1999), 87-120.
24. V. M. Buchstaber and T. E. Panov, "Torus actions, equivariant moment-angle complexes and coordinate subspace arrangements", Zap. Nauchn. Sem. St. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 266 (2000), 29-50; English transl. in J. Math. Sciences (to appear); http://xxx.lanl.gov/abs/math.AT/9912199.
25. V. M. Buchstaber and T. E. Panov, "Torus actions, equivariant moment-angle complexes and coordinate subspace arrangements", Zap. Nauchn. Sem. St. Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 266 (2000), 29-50; English transl. in J. Math. Sciences (to appear); http://xxx.lanl.gov/abs/math.AT/9912199.
26. V. M. Buchstaber and T. E. Panov, "Torus actions determined by simple polytopes", Contemp. Math. 258 (2000), 33-46.
27. V. M. Buchstaber and T. E. Panov, "Moment-angle complexes and combinatorics of simplicial manifolds", Uspekhi Mat. Nauk 55:3 (2000), 171-172; English transl., Russian Math. Surveys 55 (2000), 567-569.
28. V. M. Buchstaber and T. E. Panov, "Moment-angle complexes and combinatorics of simplicial manifolds", Uspekhi Mat. Nauk 55:3 (2000), 171-172; English transl., Russian Math. Surveys 55 (2000), 567-569.
29. V. M. Buchstaber and N. Ray, "Flag manifolds and the Landweber-Novikov algebra", Geom. Topol. 2 (1998), 79-101; http://www.maths.warwick.ac.uk/gt/.
30. V. M. Buchstaber and N. Ray, "Toric manifolds and complex cobordisms", Uspekhi Mat. Nauk 53:2 (1998), 139-140; English transl., Russian Math. Surveys 53 (1998), 371-373.
31. V. M. Buchstaber and N. Ray, "Toric manifolds and complex cobordisms", Uspekhi Mat. Nauk 53:2 (1998), 139-140; English transl., Russian Math. Surveys 53 (1998), 371-373.
32. V. M. Buchstaber and N. Ray, "Tangential structures on toric manifolds, and connected sums of polytopes", Preprint UMIST no. 2000/5, Manchester 2000; Internat. Math. Res. Notices 2001 (to appear); http://xxx.lanl.gov/abs/math.AT/0010025.
33. J. W. Cannon, "Shrinking cell-like decompositions of manifolds. Codimension three", Ann. of Math. (2) 110 (1979), 83-112.
34. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ 1956; Russian transl., Inostr. Lit., Moscow 1960.
35. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ 1956; Russian transl., Inostr. Lit., Moscow 1960.
36. R. Charney and M. Davis, "The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold", Pacific J. Math. 171 (1995), 117-137.
37. B. Chen and M. Yan, "Singularity from Eulerian viewpoint", Trudy Mat. Inst. Steklov. 221 (1998), 305-319; English transl., Proc. Steklov Inst. Math. 221 (1998), 305-319.
38. B. Chen and M. Yan, "Singularity from Eulerian viewpoint", Trudy Mat. Inst. Steklov. 221 (1998), 305-319; English transl., Proc. Steklov Inst. Math. 221 (1998), 305-319.
39. D. A. Cox, "Recent developments in toric geometry", Proc. Symp. Pure Math. 62 (pt. 2) (1997), 389-436; http://xxx.lanl.gov/abs/alg-geom/9606016.
40. V. Danilov, "The geometry of toric varieties", Uspekhi Mat. Nauk 33:2 (1978), 85-134; English transl., Russian Math. Surveys 33:2 (1978), 97-154.
41. V. Danilov, "The geometry of toric varieties", Uspekhi Mat. Nauk 33:2 (1978), 85-134; English transl., Russian Math. Surveys 33:2 (1978), 97-154.
42. M. Davis and T. Januszkiewicz, "Convex polytopes, Coxeter orbifolds and torus actions", Duke Math. J. 62 (1991), 417-451.
43. C. De Concini and C. Procesi, "Wonderful models of subspace arrangements", Selecta Math. N.S. 1 (1995), 459-494.
44. M. De Longueville, "The ring structure on the cohomology of coordinate subspace arrangements", Math. Z. 233 (2000), 553-577.
45. N. Dobrinskaya, "The classification problem for quasitoric manifolds over a given simple polytope", Funktsional. Anal. i Prilozhen. 34:4 (2000) (to appear); English transl. in Functional Anal. Appl. 34 (2000) (to appear).
46. N. Dobrinskaya, "The classification problem for quasitoric manifolds over a given simple polytope", Funktsional. Anal. i Prilozhen. 34:4 (2000) (to appear); English transl. in Functional Anal. Appl. 34 (2000) (to appear).
47. N. P. Dolbilin, M. A. Shtan'ko and M. I. Shtogrin, "Cubic subcomplexes in regular lattices", Dokl. Akad. Nauk SSSR 291 (1986), 277-279; English transl., Soviet Math. Dokl. 34 (1987), 467-469.
48. N. P. Dolbilin, M. A. Shtan'ko and M. I. Shtogrin, "Cubic subcomplexes in regular lattices", Dokl. Akad. Nauk SSSR 291 (1986), 277-279; English transl., Soviet Math. Dokl. 34 (1987), 467-469.
49. R. D. Edwards, "The topology of manifolds and cell-like maps", Proc. International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki 1980, pp. 111-127.
50. S. Eilenberg and J. C. Moore, "Homology and fibrations. I", Comment. Math. Helv. 40 (1966), 199-236.
51. W. Fulton, Introduction to toric varieties, Princeton Univ. Press, Princeton, NJ 1993.
52. V. Gasharov, I. Peeva, and V. Welker, "Coordinate subspace arrangements and monomial ideals", Preprint 1999; http://www.mathematik.uni-marburg.de/~welker/sections/preprints.html.
53. M. Goresky and R. MacPherson, Stratified Morse theory, Springer-Verlag, Berlin-New York 1988; Russian transl., Mir, Moscow 1991.
54. M. Gromov, "Hyperbolic groups", Essays in group theory (S. M. Gersten, ed.), Springer, New York-Berlin 1987, pp. 75-264.
55. A. Hattori, "Almost complex toric manifolds and positive line bundles", Banach Center Publ. 45 (1998), 95-114.
56. F. Hirzebruch, T. Berger, and R. Jung, Manifolds and modular forms, 2nd ed., Max-Planck Inst., Bonn 1994.
57. M. Hochster, "Cohen-Macaulay rings, combinatorics, and simplicial complexes", Ring Theory II, Proc. 2nd Oklahoma Conference (1975) (B. R. McDonald and R. Morris, eds.), Dekker, New York 1977, pp. 171-223.
58. I. V. Izmest'ev, "Manifolds defined by simple polyhedra as configuration spaces of hinge mechanisms", Uspekhi Mat. Nauk 55:1 (2000), 185-186; English transl., Russian Math. Surveys 55 (2000), 176-177.
59. I. V. Izmest'ev, "Manifolds defined by simple polyhedra as configuration spaces of hinge mechanisms", Uspekhi Mat. Nauk 55:1 (2000), 185-186; English transl., Russian Math. Surveys 55 (2000), 176-177.
60. I. V. Izmest'ev, "3-dimensional manifolds defined by simple polytopes with coloured facets", Mat. Zametki 69:3 (2001) (to appear); English transl. in Math. Notes 69 (2001) (to appear).
61. I. V. Izmest'ev, "3-dimensional manifolds defined by simple polytopes with coloured facets", Mat. Zametki 69:3 (2001) (to appear); English transl. in Math. Notes 69 (2001) (to appear).
62. K. Jänich, "On the classification of O(n)-manifolds", Math. Ann. 176 (1968), 53-76.
63. A. G. Khovanskii, "Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevskii space", Funktsional. Anal. i Prilozhen. 20:1 (1986), 50-61; English transl., Functional Anal. Appl. 20 (1986), 41-50.
64. A. G. Khovanskii, "Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevskii space", Funktsional. Anal. i Prilozhen. 20:1 (1986), 50-61; English transl., Functional Anal. Appl. 20 (1986), 41-50.
65. R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton Univ. Press, Princeton, NJ 1977.
66. V. Klee, "A combinatorial analogue of Poincaré's duality theorem", Canad. J. Math. 16 (1964), 517-531.
67. P. Kleinschmidt, "A classification of toric varieties with few generators", Aequationes Math. 35 (1988), 254-266.
68. *(BP)", Amer. J. Math. 98 (1976), 591-610.
69. N. C. Leung and V. Reiner, "The signature of a toric variety", Preprint, 1999; http://www.math.umn.edu/~reiner/Papers/papers.html.
70. S. Maclane, Homology, Springer-Verlag, Berlin 1963; Russian transl., Mir, Moscow 1966.
71. M. Masuda, "Unitary toric manifolds, multi-fans and equivariant index", Tohoku Math. J. (2) 51 (1999), 237-265.
72. P. McMullen, "The numbers of faces of simplicial polytopes", Israel J. Math. 9 (1971), 559-570.
73. P. McMullen, "On simple polytopes", Invent. Math. 113 (1993), 419-444.
74. E. E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York 1977.
75. S. P. Novikov (ed.), Topology I (Current Problems in Mathematics, Fundamental Directions. vol. 12), VINITI, Moscow 1986; English transl., Encyclopaedia Math. Sci. vol. 12, Springer-Verlag, Berlin 1996.
76. S. P. Novikov (ed.), Topology I (Current Problems in Mathematics, Fundamental Directions. vol. 12), VINITI, Moscow 1986; English transl., Encyclopaedia Math. Sci. vol. 12, Springer-Verlag, Berlin 1996.
77. T. Oda, Convex bodies and algebraic geometry, Springer-Verlag, New York 1988.
78. P. Orlik and F. Raymond, "Actions of the torus on 4-manifolds", Trans. Amer. Math. Soc. 152 (1970), 531-559.
79. P. Orlik and L. Solomon, "Combinatorics and topology of complements of hyperplanes", Invent. Math. 56 (1980), 167-189.
80. P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, New York 1992.
81. U. Pachner, "Konstruktionsmethoden und das kombinatorische Homöomorphieproblem für Triangulationen kompakter semilinearer Mannigfaltigkeiten", Abh. Math. Sem. Univ. Hamburg 57 (1986), 69-85.
82. U. Pachner, "P.L. homeomorphic manifolds are equivalent by elementary shellings", European J. Combin. 12 (1991), 129-145.
83. χy-genus of a multioriented quasitoric manifold", Uspekhi Mat. Nauk 54:5 (1999), 169-170; English transl., Russian Math. Surveys 54 (1999), 1037-1039.
84. χy-genus of a multioriented quasitoric manifold", Uspekhi Mat. Nauk 54:5 (1999), 169-170; English transl., Russian Math. Surveys 54 (1999), 1037-1039.
85. T. E. Panov, "Hirzebruch genera of manifolds with torus actions", Preprint, 1999; http://xxx.lanl.gov/abs/math.AT/9910083.
86. I. Peeva, V. Reiner, and V. Welker, "Cohomology of real diagonal subspace arrangements via resolutions", Compositio Math. 117 (1999), 99-115.
87. L. S. Pontryagin, Foundations of combinatorial topology, 3rd ed., Nauka, Moscow 1986; English transl. of 1st ed., Graylock Press, Rochester, NY 1952.
88. A. A. Ranicki, "On the Hauptvermutung", The Hauptvermutung book, Kluwer, Dordrecht 1996, pp. 3-31.
89. G. Reisner, "Cohen-Macaulay quotients of polynomial rings", Adv. Math. 21 (1976), 30-49.
90. C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, reprint of 1972 ed., Springer-Verlag, Berlin 1982; Russian transl., Mir, Moscow 1974.
91. M. A. Shtan'ko and M. I. Shtogrin, "On the embedding of cubic manifolds and complexes in a cubic lattice", Uspekhi Mat. Nauk 47:1 (1992), 219-220; English transl., Russian Math. Surveys 47:1 (1992), 267-268.
92. M. A. Shtan'ko and M. I. Shtogrin, "On the embedding of cubic manifolds and complexes in a cubic lattice", Uspekhi Mat. Nauk 47:1 (1992), 219-220; English transl., Russian Math. Surveys 47:1 (1992), 267-268.
93. L. Smith, "Homological algebra and the Eilenberg-Moore spectral sequence", Trans. Amer. Math. Soc. 129 (1967), 58-93.
94. D. M. Y. Sommerville, "The relations connecting the angle-sums and volume of a polytope in a space of n dimensions", Proc. Roy. Soc. London Ser. A 115 (1927), 103-119.
95. R. Stanley, "Hubert functions of graded algebras", Adv. Math. 28 (1978), 57-83.
96. R. Stanley, "The number of faces of a simplicial convex polytope", Adv. Math. 35 (1980), 236-238.
97. *-vectors", European J. Combin. 14 (1993), 251-258.
98. R. Stanley, Combinatorics and commutative algebra, 2nd ed., Birkhäuser, Boston 1996.
99. R. Stanley, "Positivity problems and conjectures in algebraic combinatorics", Mathematics: frontiers and perspectives (V. Arnold et al., eds.), Amer. Math. Soc., Providence, RI 2000, pp. 295-319.
100. R. Stong, Notes on cobordism theory, Princeton Univ. Press, Princeton, NJ 1968; Russian transl., Mir, Moscow 1973.
101. T. Tay, N. White, and W. Whiteley, "Skeletal rigidity of simplicial complexes. I, II", European J. Combin. 16 (1995), 381-403, 503-523.
102. V. A. Timorin, "An analogue of the Hodge-Riemann relations for simple convex polytopes", Uspekhi Mat. Nauk 54:2 (1999), 113-162; English transl., Russian Math. Surveys 54 (1999), 381-426.
103. V. A. Timorin, "An analogue of the Hodge-Riemann relations for simple convex polytopes", Uspekhi Mat. Nauk 54:2 (1999), 113-162; English transl., Russian Math. Surveys 54 (1999), 381-426.
104. J. A. Todd (ed.), Proc. International Congress of Mathematicians (Edinburgh, 1958), Cambridge Univ. Press, Cambridge 1960.
105. V. Vassiliev, Topology of complements of discriminants, Phasis, Moscow 1997. (in Russian)
106. S. Yuzvinsky, "Small rational model of subspace complement", Preprint, 1999; http://xxx.lanl.gov/abs/math.CO/9806143.
107. G. Ziegler, Lectures on polytopes, Springer-Verlag, New York-Berlin 1995.