On Symmetric Products and the Steenrod Squares

In his paper Reduced Powers of Cohomology Classes (7) N. E. Steenrod defines a set of invariant cohomology operations. These "reduced powers" include as a special case the squares which Steenrod introduced earlier (6). As the definition of the reduced powers depends essentially on a transformation group in the product space K X K X ... X K (p factors) of a complex K it seemed likely that at least in certain cases a close relation between his operations and the Smith-Richardson theory of symmetric products would exist. (See (3), (4) and (5).) In this paper we will discuss the case p = 2, and obtain a definition of the Steenrod squares directly from the exact sequences which are associated with the symmetric product of a space. Our main result can be stated as follows: given a complex X, set K = X X X and let k be the space obtained from K by identifying points which correspond to each other under the involution which exchanges the factors of K. Let 1 be the image in k of the diagonal of K. The Smith theory gives rise to a homomorphism A*:Hf(K; I2) -* H'(k, 1; I2). Further, the cohomology analogue of the Smith-Richardson construction of a basis in Hn(k, 1; I2) leads to the following result: there exists a canonical isomorphism 4' of the direct sum Gn = Z7n' Hi(X; I2) into H2 (k, 1; I2) with A*H2n(K; I2) in its image. Under this isomorphism the element {at}, at = Sqiu; (i = 1, ... , n) u E H'(X), of Gn maps onto A*(u X u). It follows that the Steenrod squares can be defined as the components of the element A*(u X u) in the direct sum decomposition of A*Hf(K) defined by 4,.1 I wish to take this opportunity to thank Professor Steenrod for his helpful remarks and encouragement. 1. Let K be a finite cell complex, t:K -* K an involution (5) of K (i.e. a homeomorphism with period 2 of K onto itself with the following properties: (a) t maps cells onto cells. (b) If a cell is mapped onto itself by t it remains pointwise fixed.) It is a consequence of a, and b, that the set of fixed points under t form a subcomplex L of K. We will denote by k the quotient space K/t, formed by identifying points in K which are t images of each other. p shall denote the projection K -k and we set'l = p(L).