Abstract:
We construct manifolds and orbifolds with quasitoric boundary. We show that these manifolds and orbifolds with boundary has a stable complex structure. These induce explicit (orbifold) complex cobordism relations among quasitoric manifolds and orbifolds. In particular, we show that a quasitoric orbifold is complex cobordant to some copies of fake weighted projective spaces. The famous problem of Hirzebruch is that which complex cobordism classes in $\Omega^U$ contain connected nonsingular algebraic varieties? We give some sufficient conditions to show when a complex cobordism class may contains an almost complex quasitoric manifold. Andrew Wilfong give some necessary condition of this problem up to dimension 8.

Abstract:
We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.

Abstract:
We construct toric manifolds of complex dimension $\geq 4$, whose orbit spaces by the action of the compact torus are not homeomorphic to simple polytopes (as manifolds with corners). These provide the first known examples of toric manifolds which are not quasitoric manifolds.

Abstract:
For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify small covers, the "real version" of quasitoric manifolds, over all dual cyclic polytopes.

Abstract:
Baker and Richter construct a remarkable $A_\infty$ ring-spectrum $M\Xi$ whose elements possess characteristic numbers associated to quasisymmetric functions; its relations, on one hand to the theory of noncommutative formal groups, and on the other to the theory of omnioriented (quasi)toric manifolds [in the sense of Buchstaber, Panov, and Ray], seem worth investigating.

Abstract:
We present some classification results for quasitoric manifolds (M) with (p_1(M)=-\sum a_i^2) for some (a_i\in H^2(M)) which admit an action of a compact connected Lie-group (G) such that (\dim M/G \leq 1). In contrast to Kuroki's work we do not require that the action of (G) extends the torus action on (M).

Abstract:
Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of the cobordism category with objects (d-1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincare duality, relating a space of submanifolds of M to a space of functions on M.

Abstract:
We construct small covers and quasitoric manifolds over a given $n$-colored simple polytope $P^n$ with interesting properties. Their Stiefel-Whitney classes are calculated and used as obstruction to immersions and embeddings into Euclidean spaces. In the case $n$ is a power of two we get the sharpest bounds.

Abstract:
The study of embeddings of smooth manifolds into Euclidean and projective spaces has been for a long time an important area in topology. In this paper we obtain improvements of classical results on embeddings of smooth manifolds, focusing on the case of quasitoric manifolds. We give explicit constructions of equivariant embeddings of a quasitoric manifold described by combinatorial data $(P,\Lambda)$ into Euclidean and complex projective space. This construction provides effective bounds on the dimension of the equivariant embedding.

Abstract:
We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.