GENERATING H*(BO(3), F2) AS A MODULE OVER THE STEENROD ALGEBRA

Abstract

This paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left ${\\cal A}$-module graded by degree, where ${\\cal A}$ denotes the Steenrod algebra over the field of two elements ${\\bb F}_2$. We recall that a homogeneous element f of grading d in a graded left ${\\cal A}$-module M is hit if there is a hit equation in the form of a finite sum $f=\\sum_{k>0}Sq^{k}(h_k)$, where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate ${\\cal A}$. We denote by Q = Q(M) = ${\\bb F}_2\\otimes_{\\cal A}$ M the quotient of the module M by the hit elements, where ${\\bb F}_2$ is here viewed as a right ${\\cal A}$-module concentrated in grading 0. Then Q is a graded vector space over ${\\bb F}_2$ and a basis for Q lifts to a minimal generating set for M as a module over ${\\cal A}$. The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = ${\\bb F}_{2}\\[\\sigma_1,\\ldots,\\sigma_n\\]$ is the polynomial subalgebra of P(n) = ${\\bb F}_{2}\\[x_1,\\ldots,x_n\\]$ generated by the elementary symmetric functions $\\sigma_i$ in the variables xj . In particular, $\\sigma_n=x_1\\cdots x_n$. The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over ${\\bb F}_2$, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by $\\sigma_n$, can be identified with the cohomology H*(MO(n), ${\\bb F}_2$) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by $\\sigma_n$. Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, ${\\cal A}$ is the algebra of universal stable operations in ordinary cohomology with ${\\bb F}_2$ coefficients and this explains the action of ${\\cal A}$ on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11].