Barcelona striker Robert Lewandowski scored twice to set his side on course for a 5-0 UEFA Champions League thrashing of Swiss side Young Boys.
Jurgen Klopp stepped down as Liverpool manager in the summer citing exhaustion after a successful nine-year spell.
Rutter agreed a five-year contract with the Seagulls as he moves from the Championship to the Premier League.
The Warriors invited all their employees to be part of the welcoming committee for Thompson, who joined Dallas in July on a three-year, $50 million contract.
Xabi Alonso remains hugely popular with Liverpool fans and could have been in the home dugout had things gone differently.