I often hear the question: "Why only a front brake?" when I'm riding a fixed-gear. Font of wisdom Sheldon Brown, of course, has the answer:
You really should have a front brake. A front brake, all by itself, will stop a bicycle as fast as it is possible to stop. This is true because when you are applying the front brake to the maximum, there is no weight on the rear wheel, so it has no traction.One of the wonderful things about fixed-gear riding is that the direct feel you get for rear-wheel traction teaches you exactly how hard you can apply the front brake without quite lifting the wheel off of the ground.
This is a very valuable lesson for any cyclist who likes to go fast; it could save your life.
There is really no need for a rear brake on a fixed-gear bicycle. By applying back-pressure on the pedals, you can supply all the braking that the rear wheel really needs. In fact, it is fairly easy to lock up the rear wheel and make it skid, unless you are running a rather high gear.
Some fixed-gear fans make a point of not using their brake except in an emergency. I am not sure that this is a good idea. Heavy duty resisting is widely reputed to be bad for your legs, and to be counterproductive for building up muscles and coordination for forward pedaling.