No, the plutonium in the Fat Man device serves the same function as the highly enriched uranium in the Little Boy device. It is "fuel" not the trigger, which is made of [mumble], [mumble], [mumble] (I forget exactly what the trigger was). Conventional high explosives are used to force the nuclear material into a supercritcal configuration, whether it's plutonium or uranium.
The "trigger" was made of polonium and beryllium, and (I believe) some other fairly exotic materials. The guys at Los Alamos called the triggers "Urchins," for reasons that are nuclear. I mean unclear.
(steely)