The Topology of Involutions.

An extensive theory of the topology associated with finite groups of continuous transformations remains to be developed. Mr. M. Richardson has evaluated the Betti numbers of the "domain of discontinuity" in a case of considerable generality and his methods' yield equally well certain modulo p invariants. They seem to be inadequate, however, when p equals a, the order of the group. We shall briefly develop the theory in the simplest non-trivial exceptional case, namely, p = a. = 2. Here we have a topological involution T operating in S. If each point of S is identified with its transform, the topological invariants of the resulting ST are in a sense characteristic of T; for if T is deformed to the involution T' through a continuous family of involutions, then ST' is homeomorphic with ST. Our task is to study the mod 2 invariants of ST. As an application we shall evaluate the invariants of symmetric product spaces. 1. We shall take for S a connected n-complex and shall deal only with mod 2 topology.2 We assume that S admits a simplicial subdivision into Kn such that (a) T merely permutes the cells of K. (b) If a cell of K is invariant, it is pointwise invariant. The invariant cells' form a subcomplex KO and the non-invariant p-cells of K can be grouped in pairs such that the two cells of a pair are transforms of each other. Thus if TEP = EP, the p-cells of K are Epi (in KO) and Ep, Ep. If C = xjEP,4 we define C = TC to be the chain xSE'P.