I'm not sure what you are babbling about here. We are discussing the situation at the time Oswald was arrested (i.e. did the arresting officers have cause to suspect that the person who had killed Tippit was the same person who "killed the President") There was an obvious investigation going on at that moment into the death of JFK and Tippit. The WC had not even been formed.
A quote from the testimony of C.T. Walker, one of the officers who arrested Oswald;
Mr. BELIN. At that time, did anyone connect him with the assassination of the President?
Mr. WALKER. Not unless the crowd had assumed that is who we were after, I don't know.
Mr. BELIN. When you were after him, you were after him for what?
Mr. WALKER. For the killing of Officer Tippit.