# zbMATH — the first resource for mathematics

Convex polytopes, Coxeter orbifolds and torus actions. (English) Zbl 0733.52006
This is a comprehensive survey with remarkable new results on the topic indicated in the title. The authors start with a convex n-polytope $$P^ n$$ that is called simple, if the number of codimension-one faces meeting at each vertex is n. They consider two types of group actions $$Z^ n_ 2$$ resp. $$T^ n$$ (torus action) on n-manifold $$M^ n$$ resp. 2n-manifold $$M^{2n}$$, such that the orbit space is just $$P^ n\cong M^ n/Z^ n_ 2$$ resp. $$\cong M^{2n}/T^ n$$. Up to an automorphism, the group action is required to be locally isomorphic to the standard representation of $$Z^ n_ 2$$ on $${\mathbb{R}}^ 2$$ resp. $$T^ n$$ on $${\mathbb{C}}^ n$$. In the first case $$M^ n$$ is called a “small cover” of $$P^ n$$, in the second $$M^{2n}$$ is a “toric manifold” over $$P^ n.$$
If m is the number of codimension-one faces of $$P^ n$$, then a homomorphism $$\lambda$$ : $$Z^ m_ 2\to Z^ n_ 2$$ resp. $$Z^ m\to Z^ n$$ specifies an isotropy subgroup for each codimension-one face. $$\lambda$$ is called “characteristic function” for $$M^ n$$ resp. $$M^{2n}$$. Let $$f_ i$$ denote the number of i-faces of $$P^ n$$, and let $$h_ j$$ denote the coefficient of $$t^{n-j}$$ in $$\sum f_ i(t- 1)^ i.$$ Then the f-vector $$(f_ 0,...,f_ n)$$ and the h-vector $$(h_ 0,...,h_ n)$$ determine one another. P. McMullen proved the inequality $$h_ i\leq \left( \begin{matrix} m-n+i-1\\ i\end{matrix} \right)$$ and conjectured simple combinatorial conditions on a sequence of integers to be the h- vector of a simple convex n-polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley [see A. Brøndsted, An introduction to convex polytopes, Springer, New-York (1983; Zbl 0509.52001)]. Such problems, particularly that of R. Stanley [Stud. Appl. Math. 54, 135-142 (1975; Zbl 0308.52009)] have led to injections of research aspects as follows
(A) The choice of a generic vector (never tangent to a proper face of $$P^ n)$$ allows to attach an integer-valued index to each vertex so that the number of vertices of index i is $$h_ i$$. This can be used to define here a cell structure which is perfect in the sense of Morse theory [see also A. G. Khovanskii, Funct. Anal. Appl. 20, 41-50 (1986); translation from Funkts. Anal. Prilozh. 20, No.1, 50-61 (1986; Zbl 0597.51014)].
(B) The “face ring“ of the simplicial complex K, dual to the boundary complex of $$P^ n$$, is characterized as a “Cohen-Macaulay complex” in G. Reisner, Adv. Math. 21, 30-49 (1976; Zbl 0345.13017). This topic is extensively studied in Sections 4 and 5 of the paper.
(C) Stanley established the necessity of McMullen’s condition by studying a certain quasi-smooth projective variety associated to $$P^ n$$, called a “toric variety”. Then the existence of a Kähler class yields the result. Nonsingular toric varieties are toric manifolds in the sense of this paper, however, the converse does not hold. In Section 6 the authors consider the tangent bundle of small covers and of toric manifolds. It is stably isomorphic to a sum of real resp. complex line bundles, and this leads to formulae for its characteristic classes.

##### MSC:
 52B70 Polyhedral manifolds 57Q91 Equivariant PL-topology 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry 14L30 Group actions on varieties or schemes (quotients) 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 55R50 Stable classes of vector space bundles in algebraic topology and relations to $$K$$-theory
Full Text:
##### References:
 [1] E. M. Andreev, On convex polyhedra in Lobačevskiĭ space , Math. USSR-Sb. 10 (1970), 413-440, English translation. · Zbl 0217.46801 · doi:10.1070/SM1970v010n03ABEH001677 [2] A. M. Bloch, H. Flaschka, and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra , preprint, 1989. · Zbl 0715.58004 [3] A. Brøndsted, An introduction to convex polytopes , Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York, 1983. · Zbl 0509.52001 [4] V. I. Danilov, The geometry of toric varieties , Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85-134, 247, Russian Math. Surveys 33 (1978), 97-154. · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305 [5] M. Davis, Smooth $$G$$-manifolds as collections of fiber bundles , Pacific J. Math. 77 (1978), no. 2, 315-363. · Zbl 0403.57002 · doi:10.2140/pjm.1978.77.315 [6] M. W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space , Ann. of Math. (2) 117 (1983), no. 2, 293-324. JSTOR: · Zbl 0531.57041 · doi:10.2307/2007079 · links.jstor.org [7] M. W. Davis, Some aspherical manifolds , Duke Math. J. 55 (1987), no. 1, 105-139. · Zbl 0631.57019 · doi:10.1215/S0012-7094-87-05507-4 [8] T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment , Bull. Soc. Math. France 116 (1988), no. 3, 315-339. · Zbl 0676.58029 · numdam:BSMF_1988__116_3_315_0 · eudml:87558 [9] D. Fried, The cohomology of an isospectral flow , Proc. Amer. Math. Soc. 98 (1986), no. 2, 363-368. JSTOR: · Zbl 0618.58030 · doi:10.2307/2045713 · links.jstor.org [10] M. Gromov, Hyperbolic groups , Essays in group theory ed. S. N. Gersten, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. · Zbl 0634.20015 [11] A. G. Khovanskiĭ, Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevskiĭ space , Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 50-61, 96. · Zbl 0597.51014 · doi:10.1007/BF01077314 [12] T. Oda, Convex bodies and algebraic geometry-toric varieties and applications. I , Algebraic Geometry Seminar (Singapore, 1987), World Sci. Publishing, Singapore, 1988, pp. 89-94. [13] P. Orlik and F. Raymond, Actions of the torus on $$4$$-manifolds. I , Trans. Amer. Math. Soc. 152 (1970), 531-559. JSTOR: · Zbl 0216.20202 · doi:10.2307/1995586 · links.jstor.org [14] G. Reisner, Cohen-Macaulay quotients of polynomial rings , Advances in Math. 21 (1976), no. 1, 30-49. · Zbl 0345.13017 · doi:10.1016/0001-8708(76)90114-6 [15] W. Thurston, The geometry and topology of $$3$$-manifolds , Princeton University, 1977, reproduced lecture notes. · Zbl 0393.57002 [16] C. Tomei, The topology of isospectral manifolds of tridiagonal matrices , Duke Math. J. 51 (1984), no. 4, 981-996. · Zbl 0558.57006 · doi:10.1215/S0012-7094-84-05144-5 [17] R. Stanley, The upper bound conjecture and Cohen-Macaulay rings , Studies in Appl. Math. 54 (1975), no. 2, 135-142. · Zbl 0308.52009 [18] R. Stanley, The number of faces of a simplicial convex polytope , Adv. in Math. 35 (1980), no. 3, 236-238. · Zbl 0427.52006 · doi:10.1016/0001-8708(80)90050-X [19] R. Stanley, Combinatorics and commutative algebra , Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1983. · Zbl 0537.13009 [20] W. van der Kallen, Homology stability for linear groups , Invent. Math. 60 (1980), no. 3, 269-295. · Zbl 0415.18012 · doi:10.1007/BF01390018 · eudml:142749
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.