As any lawyer worth their salt will tell you, your case begins the moment you are read your rights and arrested. From that moment on, anything and everything you say and do is recorded and can be used against you in court. If you are being arrested for alleged assault and/or battery, you should exercise your right against self-incrimination and stay silent.
You are required to give them your basic information like your name, age, or your ID when asked, but you are not legally obligated to answer any questions, related to the case or not. If the police officers decide to ask you questions, you should politely inform them that you wish to remain silent and would like to speak to an attorney.
While you are not required to speak to the police officer, you are required to obey any and all lawful orders given to you by the arresting officers. Refusing to listen to lawful orders from the police can land you with additional charges, such as resisting arrest or assaulting an officer, and can provide behavioral evidence for the prosecution to use against you in court.
You should, however, remain alert and take in as much as you can. Any evidence you can remember or spot (such as cameras that may have picked up the incident or passerby's that may have witnessed the event) can help criminal defense attorneys build you a strong defense to fight your assault and battery charges.