Embedding (simple definition)
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Definition
A smooth embedding of a smooth compact manifold into a smooth manifold is a smooth injective map
such that
is a monomorphism at each point.
(See an equivalent alternative definition which works for non-compact manifolds and involves immersions.
A smooth immersion is a smooth map
such that
is a monomorphism at each point.
See an equivalent alternative definition.)
A map of a polyhedron
is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of
.
A PL embedding of a compact polyhedron
into
is a PL injective map
.
A topological embedding of a compact subset into
is a continuous injective map
.
Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c].