Après manger

Après manger -- after eating.
Après avoir mangé -- after having eaten.

Après mangé is wrong. It would be like saying "after eaten".

Après manger, nous sortons. -- After eating we go out.
Après avoir mangé cette pomme, j'en prends une autre. -- After having eaten this apple, I'll have another.

"Après manger" is a fixed expression which means the same as "après avoir mangé". "Après boire" also exists, but in general you can't use a present infinitive after "après".

This thread explains it:

https://forum.wordreference.com/threads/apr%C3%A8s-manger-boire-apr%C3%A8s-infinitif.1009468/