If cops can lie to you during an interrogation, and you ask for a lawyer, can a police officer pretend to be that lawyer?

I'm sorry if this is the wrong forum, but this is a question that I've had for a while.

I heard that, during an interrogation, the cops can lie to you. For instance, tell you that you failed a lie detector when you didn't, etc. So, if during questioning, you ask for a lawyer, can a police officer come into the room and pretend to be the requested lawyer? Are there any instances where the police CANNOT lie to you?

Thank you!