A History of the Arf-Kervaire Invariant Problem

Typing “Invariant theory” into Wikipedia yields the theory of functions like x1x2+x1x3+x2x3 which are unaltered by permuting the variables. In algebraic topology, particularly post-1950, a different notion of “invariant” emerged. This use of invariant (e.g. Hopf invariant, Arf-Kervaire invariant, λ-invariant) denotes an algebraic quantity that gives a partial answer to a topological question. Often invariants in this sense are very technical, both in their context and in their construction. However, a very simple invariant occurs in the game of Nim ([31], pp. 36–38). In the 1960s this game was popular among students due to its enigmatic appearance in Alain Resnais’s 1961 avant-garde movie L’Année Dernière à Marienbad. A set of matchsticks is divided arbitrarily into several heaps. Two players play alternately. A play consists of selecting a heap and removing from it any (nonzero) number of matchsticks. The winner is the player whose move leaves no remaining matchsticks. The question which the Nim invariant answers is, If my opponent and I play out of our skins, will I win? Suppose there are k heaps of matchsticks of sizes n1, . . . , nk. Recall that every positive integer n can be written in one and only one way as the sum of distinct powers of 2. This is called the binary