Relations in the Homotopy of Simplicial Abelian Hopf Algebras

In this paper, we analyze the structure possessed by the homotopy groups of a simplicial abelian Hopf algebra over the field F2. Specifically, we review the higher-order structure that the homotopy groups of a simplicial commutative algebra and simplicial cocommutative coalgebra possess. We then demonstrate how these structures interact under the added conditions present in a Hopf algebra. Introduction The goal of this paper is to determine all the natural relations among primary operations that occur in the homotopy of a simplicial abelian Hopf algebra over F2, the field of two elements. Here abelian Hopf algebra means a unitary commutative algebra in the category of counitary cocommutative coalgebras. The motivation for this problem comes studying the second quadrant cohomology spectral sequence, over a finite field, associated to a cosimplicial space (see [3]). The E2-term of such a spectral sequence is the homotopy groups for the simplicial (unstable) algebra obtained from applying cohomology to this cosimplicial space. Given a simplicial commutative F2-algebra A. its homotopy groups possess operations δi : πnA. → πn+iA. as constructed in [4], [2], and [9]. As such they are called Cartan-Bousfield-Dwyer operations. In [10] it is shown that if A. arises from cohomology, as above, then, in the associated spectral sequence, these operations guarantee that Steenrod operations, which would violate instability at E∞, do not survive. Now if one started off with a cosimplicial iterated loop space, the associated cohomology is also a simplicial cocommutative coalgebra. The homotopy of such an object B. has a right action of Steenrod operations Sq : πnB. → πn−iB. which can be extracted from [7]. If B. arises from cohomology, as above, then these Steenrod operations determine Dyer-Lashof operations in the abuttment. This is shown in [19] and [20] examines a specific example. Now the cohomology of a cosimplicial iterated cosimplicial space is more than just a simplicial algebra and a simplicial coalgebra, it is a simplicial Hopf algebra. Given such an object H. the added Hopf condition guarantees certain “Nishida