Actual embedding

In mathematics , the actual embedding is an embedding term that is mainly used in topology when examining manifolds with a boundary .

definition

Be and manifolds with boundaries . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \partial M,\partial N}$

An image is called actual embedding if it ${\displaystyle f\colon M\rightarrow N}$

• is an embedding, and
• ${\displaystyle f(\partial M)\subset \partial N}$ such as
• ${\displaystyle f(M\setminus \partial M)\subset N\setminus \partial N}$

applies.

The definition is equivalent to the fact that the restriction is to an embedding and an actual mapping . ${\displaystyle M\setminus \partial M}$ ${\displaystyle M\setminus \partial M\to N\setminus \partial N}$

In the theory of differentiable manifolds one also demands that it is transversal to . ${\displaystyle f(M)}$ ${\displaystyle \partial N}$

literature

• Hempel, John 3-manifolds. Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004. ISBN 0-8218-3695-1