The last thing you want on slippery surfaces is a front brake, which should only be used in conditions with good traction. You absolutely need a rear brake of some kind for slippery conditions, and the reason you went down on the ice was that you most likely tried to use your front brake. If you are riding fixed, then use your legs to brake your rear wheel in slippery conditions and don't even touch your front brake. I used to ride on snow and ice covered roads, and rode a fixed gear with no brakes for that reason.