# Generating H*(BO(3), F2) as a module over the Steenrod algebra

JANFADA, A. S. and WOOD, R.M.W. (2003) Generating H*(BO(3), F2) as a module over the Steenrod algebra. Mathematical Proceedings of the Cambridge Philosophical Society, 134 (2). pp. 239-257. ISSN 0305-0041

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].