If you have ski poles or hiking poles with snow baskets, along with crampons, then you will travel up and down pretty well.
The ice axe is for your safety -- If you slip or trip, (and know how to self-arrest) it can save your life. Do NOT glissade (sit down and slide down feet-first) without an ice axe to use as a brake. If you get into an icy spot, you can take off like a rocket without a brake.
