# 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