  A fundamental result of toric geometry is that there is a bijection between
toric varieties and fans. More generally, it is known that some class of
manifolds having well-behaved torus actions, called topological toric manifolds
$M^{2n}$, can be classified in terms of combinatorial data containing
simplicial complexes with $m$ vertices. We remark that topological toric
manifolds are a generalization of smooth toric varieties. The number $m-n$ is
known as the Picard number when $M^{2n}$ is a {compact smooth} toric variety.
  In this paper, we investigate the relationship between the topological toric
manifolds over a simplicial complex $K$ and those over the complex obtained by
simplicial wedge operations from $K$. As applications, we do the following.
  1. We classify smooth toric varieties of Picard number 3. This is a reproving
of a result of Batyrev.
  2. We give a new and complete proof of projectivity of smooth toric varieties
of Picard number 3 originally proved by Kleinschmidt and Sturmfels.
  3. We find a criterion for a toric variety over the join of boundaries of
simplices to be projective. When the toric variety is smooth, it is known as a
generalized Bott manifold which is always projective.
  4. We classify and enumerate real topological toric manifolds when $m-n=3$.
  5. When $m-n \leq 3$, any real topological toric manifold is realizable as
fixed points of the conjugation of a topological toric manifold.
