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We relate the theory of moduli spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ of stable weighted curves of genus $0$ to the equivariant topology of complex Grassmann manifolds $G_{n,2}$, with the canonical action of the compact torus $T^n$. We prove that all spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ can be isomorphically or up to birational morphisms embedded in $G_{n,2}/T^n$ . The crucial role for proving this result play the chamber decomposition of the hypersimplex $\Delta _{n,2}$ which corresponds to $(\mathbb{C} ^{\ast})^{n}$-stratification of $G_{n,2}$ and the spaces of parameters over the chambers, which are subspaces in $G_{n,2}/T^n$. We show that the points of these moduli spaces $\overline{\mathcal{M}}_{0, \mathcal{A}}$ have the geometric realization as the points of the spaces of parameters over the chambers. We single out the characteristic categories among such moduli spaces. The morphisms in these categories correspond to the natural projections between the universal space of parameters and the spaces of parameters over the chambers. As a corollary, we obtain the realization of the orbit space $G_{n,2}/T^n$ as an universal object for the introduced categories. As one of our main results we describe the structure of the canonical projection from the Deligne-Mumford compactification to the Losev-Manin compactification of $\mathcal{M}_{0,n}$, using the embedding of $\mathcal{M}_{0, n}\subset \bar{L}_{0, n, 2}$ in $(\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$, the action of the algebraic torus $(\mathbb{C} ^{\ast})^{n-3}$ on $(\mathbb{C} P^{1})^{N}$ for which $\bar{L}_{0, n, 2}$ is invariant, and the realization of the Losev-Manin compactification as the corresponding permutohedral toric variety., Comment: 43 pages

In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the ``forgotten symmetric functions", which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers., Comment: Slightly revised version, to appear in the Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325

A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed., Comment: 67 pages; slightly amended English version of the paper published in Uspekhi Matematicheskikh Nauk (Russian Math Surveys)

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hyperelliptic functions associated with $V$ from the Weierstrass elliptic functions associated with $E_i$ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with $V$ to the appropriate subspaces in $\mathbb{C}^2$ are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with $E_i$. Further, we express the hyperelliptic functions associated with $V$ on $\mathbb{C}^2$ in terms of the Weierstrass elliptic functions associated with $E_i$. We derive these results by describing the homomorphisms between the Jacobian varieties of the curves $V$ and $E_i$ induced by the morphisms from $V$ to $E_i$ explicitly.

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero can be realized as the Chow quotient $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$. In our recent papers, the constructive description of the orbit space $G_{n,2}/T^n$ has been obtained. In getting this result our notions of the CW-complex of the admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for $T^n$-action on $G_{n,2}$ were of essential use. Using technique of the wonderful compactification, in this paper it is given an explicit construction of the space $\mathcal{F}_{n}$. Together with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled us to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. Thus, we showed that the space $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ can be realized as our universal space of parameters $\mathcal{F}_{n}$. In this way, we give description of the structure in $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$, that is $\overline{\mathcal{M}}(0,n)$ in terms of the CW-complex of the admissible polytopes for $G_{n,2}$ and their spaces of parameters., Comment: 32 pages

The problem of the description of the orbit space $X_{n} = G_{n,2}/T^n$ for the standard action of the torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ is widely known and it appears in diversity of mathematical questions. A point $x\in X_{n}$ is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of $X_n$ is introduced which opened the new approach to this problem. It is showed that for $n>4$ the set of critical points $\text{Crit}X_n$ belongs to our set of singular points $\text{Sing}X_{n}$, while the case $n=4$ is somewhat special for which $\text{Sing}X_4\subset \text{Crit}X_4$, but there are critical points which are not singular. The central result of this paper is the construction of the smooth manifold $U_n$ with corners, $\dim U_n = \dim X_n$ and an explicit description of the projection $p_{n} : U_{n}\to X_{n}$ which in the defined sense resolve all singular points of the space $X_n$. Thus, we obtain the description of the orbit space $G_{n,2}/T^n$ combinatorial structure. Moreover, the $T^n$-action on $G_{n,2}$ is a seminal example of complexity $(n-3)$ - action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity., Comment: 44 pages

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\bf u};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$. It is an analogue of the Weierstrass sigma function $\sigma(u;g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives of order 2 and higher of the function ${\sigma({\bf u};\lambda)}$ generate fields of hyperelliptic functions of ${\bf u} = (u_1,u_3)$ on the Jacobians of curves with a fixed parameter vector $\lambda$. We consider three Hurwitz series $\sigma({\bf u};\lambda)=\sum_{m,n\ge 0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\bf u};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ and $\sigma({\bf u};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ and $\mu_k(u_3;\lambda)$. It includes the latest results, which proofs use the fundamental fact that the function ${\sigma ({\bf u};\lambda)}$ is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space., Comment: 43 pages

In the focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of the model space $\mathfrak{E}$ with an action of the torus $T^k$ is given, such that there exists a $T^k$-equivariant homeomorphism $\mathfrak{E}\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak{E}/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of an $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where rk$G=$rk$H$, contains $(2n,k)$-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$., Comment: substantially revised, now 50 pages

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_{\sigma}$ and determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_\sigma$ such that $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$ and the notion of the spaces of parameters $F_{\sigma}$, which together describe $W_{\sigma}/T^{n}$ as the product $\stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$. To find the appropriate topology for the set $\cup _{\sigma} \stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$ we introduce the notions of the universal space of parameters $\tilde{\mathcal{F}}$ and the virtual spaces of parameters $\tilde{F}_{\sigma}\subset \tilde{\mathcal{F}}$ such that there exist the projections $\tilde{F}_{\sigma}\to F_{\sigma}$. Hence, we propose a method for the description of the orbit space $G_{n,k}/T^n$. Earlier we proved that the orbit space $G_{4,2}/T^4$, defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial \Delta _{4,2}\ast \mathbb{C} P^1$. We prove here that the orbit space $G_{5,2}/T^5$, defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space obtained by attaching the disc $D^8$ to the space $\Sigma ^{4}\mathbb{R} P^2$ by the generator of the group $\pi _{7}(\Sigma ^{4}\mathbb{R} P^2)=\mathbb{Z} _{4}$. In particular, $(G_{5,2}/G_{4,2})/T^5$ is homotopy equivalent to $\partial \Delta _{5,2}\ast \mathbb{C} P^2$. The methods and the results of this paper are fundaments for our theory of $(2l,q)$-manifolds., Comment: Section 12 improved, corrected the description of the homotopy type for G_{5,2}/T^5 and subsection 3.2 extended; final version, to appear in Moscow Mathematical Journal

We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene $C_{20}$). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -- eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge- and two edges-truncations., Comment: 10 pages, 23 figures

The lectures are devoted to a remarkable class of $3$-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -- fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Main notions are introduced during the lectures. The lecture notes are addressed to a wide audience., Comment: Lecture notes: 9 Lectures, bibliography, index, 117 pages, 83 figures

We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its restriction to real parameter domain $\lambda+\mu^2>0$ is a linearization of the family of nonlinear equations on two-torus modeling the Josephson effect in superconductivity. We show that for every $b,n\in\mathbb C$ satisfying a certain "non-resonance condition" and every $\lambda,\mu\in\mathbb C$, $\mu\neq0$ there exists an entire function $f_{\pm}:\mathbb C\to\mathbb C$ (unique up to constant factor) such that $z^{-b}\mathcal L(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb C$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those $\lambda$, $\mu$, $n$, $b$ for which the monodromy of the Heun equation has eigenvalue $e^{2\pi i b}$. It also describes those $\lambda$, $\mu$, $n$ for which the monodromy is parabolic: has multiple eigenvalue. We consider the rotation number $\rho$ of the dynamical system on two-torus as a function of parameters restricted to a surface $\lambda+\mu^2=const$. The phase-lock areas are its level sets having non-empty interiors. For general families of dynamical systems the problem to describe the boundaries of the phase-lock areas is known to be very complicated. Here we include the results in this direction obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). The result on parabolic monodromy implies the description of the union of their boundaries by an explicit functional equation. For every $\theta\notin\mathbb Z$ we get a description of the set $\{\rho\equiv\pm\theta(mod2\mathbb Z)\}$., Comment: 64 pages, 7 figures. To appear in Proc. Steklov Inst. of Math

In this work we explicitly construct polynomial vector fields $\mathcal{L}_k,\;k=0,1,2,3,4,6$ on the complex linear space $\mathbb{C}^6$ with coordinates $X=(x_2,x_3,x_4)$ and $Z=(z_4,z_5,z_6)$. The fields $\mathcal{L}_k$ are linearly independent outside their discriminant variety $\Delta \subset \mathbb{C}^6$ and tangent to this variety. We describe a polynomial Lie algebra of the fields $\mathcal{L}_k$ and the structure of the polynomial ring $\mathbb{C}[X, Z]$ as a graded module with two generators $x_2$ and $z_4$ over this algebra. The fields $\mathcal{L}_1$ and $\mathcal{L}_3$ commute. Any polynomial $P(X,Z) \in \mathbb{C}[X, Z]$ determines a hyperelliptic function $P(X,Z)(u_1, u_3)$ of genus $2$, where $u_1$ and $u_3$ are coordinates of trajectories of the fields $\mathcal{L}_1$ and $\mathcal{L}_3$. The function $2 x_2(u_1, u_3)$ is a 2-zone solution of the KdV hierarchy and $\frac{\partial}{\partial u_1}z_4(u_1, u_3)=\frac{\partial}{\partial u_3}x_2(u_1, u_3)$.

We present an infinite series of operations on fullerenes generalizing the Endo-Kroto operation, such that each combinatorial fullerene is obtained from the dodecahedron by a sequence of such operations. We prove that these operations are invertible in the proper sense, and are compositions of (1;4,5)-, (1;5,5)-, (2,6;4,5)-, (2,6;5,5)-, (2,6;5,6)-, (2,7;5,5)-, and (2,7;5,6)-truncations, where each truncation increases the number of hexagons by one., Comment: 20 pages, 56 figures

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations., Comment: 19 pages, 1 figure. To appear in Nonlinearity. Minor changes. The present version includes additional historical remarks and bibliography

The short article [V.M. Buchstaber, E.Yu. Netay, $\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology, Russian Math. Surveys, 69:4 (2014), 757-759] states results on $\mathbb{C}P(2)$-multiplicative Hirzebruch genera. The aim of the following text is to give a proof of Theorem 3. This proof uses only the technique of functional differential equations., Comment: 4 pages

We consider the canonical action of the compact torus $T^4$ on the Grassmann manifold $G_{4,2}$ and prove that the orbit space $G_{4,2}/T^4$ is homeomorphic to the sphere $S^5$. We prove that the induced differentiable structure on $S^5$ is not the smooth one and describe the smooth and the singular points. We also consider the action of $T^4$ on $CP^5$ induced by the composition of the second symmetric power $T^4\subset T^6$ and the standard action of $T^6$ on $CP^5$ and prove that the orbit space $CP^5/T^4$ is homeomorphic to the join $CP^2\ast S^2$. The Pl\"ucker embedding $G_{4,2}\subset CP^5$ is equivariant for these actions and induces embedding $CP^1\ast S^2 \subset CP^2 \ast S^2$ for the standard embedding $CP^1 \subset CP^2$. All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian $G_{4,2}(R)$ and the real projective space $RP^5$ for the action of the group $Z _{2}^{4}$. We prove that the orbit space $G_{4,2}(R)/Z _{2}^{4}$ is homeomorphic to the sphere $S^4$ and that the orbit space $RP^{5}/Z _{2}^{4}$ is homeomorphic to the join $RP^2\ast S^2$., Comment: Some typos corrected and few background information added, now 42 pages, to appear in Moscow Mathematical Journal

In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation., Comment: 11 pages, 6 figures

We consider in C^n the class of symmetric homogeneous quadratic dynamical systems. We introduce the notion of algebraic integrability for this class. We present a class of symmetric quadratic dynamical systems that are algebraically integrable by the set of functions h_1(t), ..., h_n(t) where h_1(t) is any solution of an ordinary differential equation of order n and h_k(t) are differential polynomials in h_1(t), k = 2, ..., n. We describe a method of constructing this ordinary differential equation. We give a classification of symmetric quadratic dynamical systems and describe the maximal subgroup in GL(n, C) that acts on this systems. We apply our results to analysis of classical systems of Lotka-Volterra type and Darboux-Halphen system and their modern generalizations., Comment: 13 pages

This paper is devoted to the one-dimensional heat equation and the non-linear ordinary differential equations associated to it. We consider homogeneous polynomial dynamical systems in the n-dimensional space, n = 0, 1, 2, .... For any such system our construction matches a non-linear ordinary differential equation. We describe the algorithm that brings the solution of such an equation to a solution of the heat equation. The classical fundamental solution of the heat equation corresponds to the case n=0 in terms of our construction. Solutions of the heat equation defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n=2. The group SL(2, C) acts on the space of solutions of the heat equation. We show this action for each n induces the action of SL(2, C) on the space of solutions of the corresponding ordinary differential equations. In the case n=2 this leads to the well-known action of this group on the space of solutions of the Chazy-3 equation. An explicit description of the family of ordinary differential equations arising in our approach is given., Comment: 20 pages

The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus $T^k$. As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous $SU$-spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any $T^k$-equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex $T^k$-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces., Comment: 67 pages; typos corrected and some explanations added; to appear in International Mathematics Research Notices - IMRN

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $\sigma$-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of $n$-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were "virtual", while in the present case the resulting bundles are effectively realizable., Comment: 42 pages

In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product $\times $ and a join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\mathcal{RP}$ has the structure of a graded Hopf algebra. It turns out that $\mathcal{P}$ has a natural Hopf comodule structure over $\mathcal{RP}$. Faces operators $d_k$ that send a polytope to the sum of all its $(n-k)$-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra $\mathcal{Z}$. This structure gives a ring homomorphism $\R\to\Qs\otimes\R$, where $\R$ is $\mathcal{P}$ or $\mathcal{RP}$. Composing this homomorphism with the characters $P^n\to\alpha^n$ of $\mathcal{P}$, $P^n\to\alpha^{n+1}$ of $\mathcal{RP}$, and with the counit we obtain the ring homomorphisms $f\colon\mathcal{P}\to\Qs[\alpha]$, $f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha]$, and $\F^*:\mathcal{RP}\to\Qs$, where $F$ is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over $\mathbb Q$, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism $f,\;f_{\mathcal{RP}}$, and $\F$ the images of two polytopes coincide if and only if they have equal flag $f$-vectors. Therefore algebraic structures on the images give the information about flag $f$-vectors of polytopes., Comment: 61 pages

We consider the geometrical addition law on the elliptic curve in Tate coordinates. It corresponds to the general formal group law over the ring of polynomials with integer coefficients of the parametra of the curve. We study the structure of this law and the differential equation that determines its exponent. We describe a 5-parametric family of Hirzebruch genera with integer values on stably complex manifolds. We introduce the general Krichever genus, which is given by a generalized Baker-Akhiezer function. This function has many of the fundamental properties of the Baker-Akhiezer function, but unlike it, it is not meromorphic, because it can have two branch points in the parallelogram of periods., Comment: 39 pages

From the paper of the first author it follows that upper and lower bounds for $\gamma$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and lower bounds for $\gamma$-vectors of flag nestohedra, particularly Gal's conjecture was proved for this case. In the present paper we obtain unimprovable upper and lower bounds for $\gamma$-vectors (consequently, for $g$-,$h$- and $f$-vectors) of graph-associahedra and some its important subclasses. We use the constructions that for an $(n-1)$-dimensional graph-associahedron $P_{\Gamma_n}$ give the $n$-dimensional graph-associahedron $P_{\Gamma_{n+1}}$ that is obtained from the cylinder $P_{\Gamma_n}\times I$ by sequential shaving some facets of its bases. We show that the well-known series of polytopes (associahedra, cyclohedra, permutohedra and stellohedra) can be derived by these constructions. As a corollary we obtain inductive formulas for $\gamma$- and $h$- vectors of the mentioned series. These formulas communicate the method of differential equations developed by the first author with the method of shavings developed by the second author.

In this paper we study the ring $\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its $(n-k)$-dimensional faces. It turns out that $\mathcal{D}$ is isomorphic to the universal Leibnitz-Hopf algebra with the antipode $\chi(d_k)=(-1)^kd_k$. Using the operators $d_k$ we build the generalized $f$-polynomial, which is a ring homomorphism from $\mathcal{P}$ to the ring $\Qsym[t_1,t_2,...][\alpha]$ of quasi-symmetric functions with coefficients in $\mathbb Z[\alpha]$. The images of two polytopes coincide if and only if their flag $f$-vectors are equal. We describe the image of this homomorphism over the integers and prove that over the rationals it is a free polynomial algebra with dimension of the $n$-th graded component equal to the $n$-th Fibonacci number. This gives a representation of the Fibonacci series as an infinite product. The homomorphism is an isomorphism on the graded group $BB$ generated by the polytopes introduced by Bayer and Billera to find the linear span of flag $f$-vectors of convex polytopes. This gives the group $BB$ a structure of the ring isomorphic to $f(\mathcal{P})$. We show that the ring of polytopes has a natural Hopf comodule structure over the Rota-Hopf algebra of posets. As a corollary we build a ring homomorphism $l_{\alpha}\colon\mathcal{P}\to\mathcal{R}[\alpha]$ such that $F(l_{\alpha}(P))=f(P)^*$, where $F$ is the Ehrenborg quasi-symmetric function., Comment: 42 pages

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus \Phi, which was introduced independently by Krichever and Loeffler in 1974, albeit from radically different viewpoints. In fact \Phi is a version of tom Dieck's bundling transformation of 1970, defined on T^k-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s., Comment: 35 pages, LaTeX. In v2 references made to the index theoretical approach to genera; rigidity and multiplicativity results improved; acknowledgements added

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the corresponding representations of T^k in the tangent spaces at the fixed points, as well as on the signs at these points. In that context, the effectiveness is also approached due to an explicit description of the relations between the weights and signs for an arbitrary couple of such structures. Special attention is devoted to the structures which are invariant under the canonical action of the group G. Using classical results, we obtain an explicit description of the weights and signs in this case. We consequently obtain an expression for the cobordism classes of such structures in terms of coefficients of the formal group law in cobordisms, as well as in terms of Chern numbers in cohomology. These computations require no information on the cohomology ring of the manifold G/H, but, on their own, give important relations in this ring. As an application we provide an explicit formula for the cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples., Comment: 30 pages; this is the extension of arXiv:0708.2054v2 [math.AT] with an application to some new problems

We consider compact homogeneous spaces G/H of positive Euler characteristic endowed with an invariant almost complex structure J and the canonical action \theta of the maximal torus T ^{k} on G/H. We obtain explicit formula for the cobordism class of such manifold through the weights of the action \theta at the identity fixed point eH by an action of the quotient group W_G/W_H of the Weyl groups for G and H. In this way we show that the cobordism class for such manifolds can be computed explicitly without information on their cohomology. We also show that formula for cobordism class provides an explicit way for computing the classical Chern numbers for (G/H, J). As a consequence we obtain that the Chern numbers for (G/H, J) can be computed without information on cohomology for G/H. As an application we provide an explicit formula for cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples., Comment: improvements in subsections 7.1 and 7.2; some small comments are added or revised and some typos corrected

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details., Comment: 23 pages, LaTeX2e; minor changes

We prove that the integral cohomology algebra of the moment-angle complex Z_K, or of the corresponding coordinate subspace arrangement complement U(K), is isomorphic to the Tor-algebra of the face ring Z[K] of simplicial complex K., Comment: 3 pages

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorial-geometrical objects in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some well-known topological problems., Comment: 87 pages, 11 figures, LaTeX2e, to appear in Russian Math. Surveys 55 (2000), no.5

We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the $B_{i,j}$ allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum $#$ for simple $n$-dimensional polytopes; when $P^n$ is a product of simplices, we describe $P^n# Q^n$ by applying an appropriate sequence of {\it pruning operators}, or hyperplane cuts, to $Q^n$., Comment: 22 pages, LaTeX2e, to appear in Internat. Math. Research Notices (2001)

Let $\rho:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a $T^m$-invariant bigraded cellular decomposition of the subset $\rho^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined by canonical cubical subdivisions of simplicial complexes. We describe relations between the combinatorics of simplicial complexes and the bigraded cohomology of corresponding moment-angle complexes. In the case when the simplicial complex is a simplicial manifold the corresponding moment-angle complex has an orbit consisting of singular points. The complement of an invariant neighbourhood of this orbit is a manifold with boundary. The relative Poincare duality for this manifold implies the generalized Dehn-Sommerville equations for the number of faces of simplicial manifolds., Comment: 28 pages, LaTeX2e, extended version of the paper published in Russian Math. Surveys 55 (2000), no. 3

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and the moment-angle complex defined by the simplicial complex. The moment-angle complex is a certain subset of a unit poly-disk in m-dimensional complex space invariant with respect to the action of an m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere, but otherwise has more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg-Moore spectral sequence. We also relate our results with well known facts in the theory of toric varieties and symplectic geometry., Comment: 16 pages, LaTeX 2e, to apper in Proceedings of the conference "Topology and Dynamics: Rokhlin Memorial", St. Petersburgh, Russia, 1999

An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T^{m-n}\subset T^m acting freely on Z_P such that Z_P/T^{m-n}=M^{2n}. We calculate the cohomology ring of Z_P and show that it is isomorphic to the cohomology ring of the face ring of P^n regarded as a module over the polynomial ring. In this way the cohomology of Z_P acquires a bigraded algebra structure, and the additional grading allows to catch the combinatorial invariants of the polytope. At the same time this gives an example of explicit calculation of the cohomology of the complement of an arrangement of planes, which is of independent interest., Comment: 36 pages, LaTeX 2e, to appear in Proceedings of the Steklov Institute of Mathematics vol. 225

We investigate geometrical interpretations of various structure maps associated with the Landweber-Novikov algebra S^* and its integral dual S_*. In particular, we study the coproduct and antipode in S_*, together with the left and right actions of S^* on S_* which underly the construction of the quantum (or Drinfeld) double D(S^*). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincare duality with respect to double cobordism theory; these lead directly to our main results for the Landweber-Novikov algebra., Comment: 23 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper5.abs.html