Embedding (simple definition)

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[edit] Definition

A smooth embedding of a smooth compact manifold N into a smooth manifold is a smooth injective map f:N\to M such that df 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 f:N\to M such that df is a monomorphism at each point. See an equivalent alternative definition.)

A map f:N\to\Rr^m of a polyhedron N is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of N. A PL embedding of a compact polyhedron N into \Rr^m is a PL injective map f:N\to\Rr^m.

A topological embedding of a compact subset N\subset\Rr^n into \Rr^m is a continuous injective map f:N\to\Rr^m.

Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c].

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