Polytopes, Hopf algebras and Quasi-symmetric functions

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 page

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

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

Complex cobordism classes of homogeneous spaces

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 correcte