  We introduce the notion of a local torus action modeled on the standard
representation (for simplicity, we call it a local torus action). It is a
generalization of a locally standard torus action and also an underlying
structure of a locally toric Lagrangian fibration. For a local torus action, we
define two invariants called a characteristic pair and an Euler class of the
orbit map, and prove that local torus actions are classified topologically by
them. As a corollary, we obtain a topological classification of locally
standard torus actions, which is a generalization of the topological
classification of quasi-toric manifolds by Davis and Januszkiewicz and of
effective two-dimensional torus actions on four-dimensional manifolds without
nontrivial finite stabilizers by Orlik and Raymond. We investigate locally
toric Lagrangian fibrations from the viewpoint of local torus actions. We give
a necessary and sufficient condition in order that a local torus action becomes
a locally toric Lagrangian fibration. Locally toric Lagrangian fibrations are
classified by Boucetta and Molino up to fiber-preserving symplectomorphisms. We
shall reprove the classification theorem of locally toric Lagrangian fibrations
by refining the proof of the classification theorem of local torus actions. We
also investigate the topology of a manifold equipped with a local torus action
when the Euler class of the orbit map vanishes.
