The mod 2 homology of the general linear goup of a 2-adic local field

Let F be a finite extension of Q2, of degree d. Our first main theorem gives an explicit computation of the mod two homology Hopf algebra of the infinite general linear group GLF . The answer is formulated in terms of the well-known homology algebras of the infinite unitary group U, its classifying space BU, and the classifying space BO of the infinite orthogonal group. Let P denote the subalgebra of H∗BO generated by the primitive elements; thus P is a polynomial algebra on generators of odd degree. Here and throughout and the paper, H∗(−) means H∗(−; F2). Let FP denote the subalgebra generated by the squares of the primitive elements; thus FP is a polynomial algebra on generators in degrees congruent to 2 mod 4. (F denotes the Frobenius operator on a bicommutative Hopf algebra.) Then there is a canonical embedding FP−→H∗BU with image the subalgebra P ′ generated by the primitives of H∗BU . Hence we can form the tensor products B = P ⊗FP H∗BU C = H∗BO ⊗FP B Theorem 1.1 Let F be a finite extension of Q2, of degree d. Then there are the following isomorphisms of Hopf algebras over the Steenrod algebra A: