A locally flat embedding of a 
-dimensional manifold in an 
-dimensional manifold
is an embedding that, near each interior point
of its image, is homeomorphic to the standard inclusion 
. For a proper disk in a manifold
with boundary, boundary points have the corresponding
half-space local model. The condition concerns local topology, rather than differentiability.
A smooth embedded disk is locally flat, whereas a topological embedded disk can have wild points and fail to be locally flat. The disk embedding conjecture asks for disjoint locally flat disks from specified immersed disks and algebraic dual spheres in four-dimensional manifolds.
