Minimax Problems Related to Cup Powers and Steenrod Squares

If F is a family of mod 2 k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp. This paper is about minimax estimates for the volumes of cycles in complicated families. The simplest example of a minimax problem is a classical result about curves in the unit disk. First consider the family of vertical lines in the unit disk. The longest line in the family has length 2. Then consider any other family of curves that sweeps out the unit disk. One of the curves in the other family must still have length at least 2. We illustrate the situation in Figure 1.