Odd primary Steenrod algebra, additive formal group laws, and modular invariants (Q2497082)

A central theorem in stable homotopy theory is \textit{J. W. Milnor's} description of the Hopf algebra of stable cooperations for ordinary mod \(p\) homology, \(H_*H\) [Ann. Math. 67, 150-171 (1958; Zbl 0080.38003)]. For \(p\) an odd prime this paper gives a new proof of this result, directly identifying the Hopf algebra in question with the Hopf algebra of quasi-strict automorphisms of additive formal group laws. The case \(p=2\) of this result was established by the author in an earlier paper [J. Math Kyoto Univ. 45-1, 39-55 (2005; Zbl 1081.55015)]. If \(Op (\enskip)\) is the functor associating to a nonnegatively graded commutative algebra \(R_*\) over \({\mathbb F}_p\) the set of multiplicative operations from \(H^*(\enskip)\) to \(H^* (\enskip) \otimes R_*\) and \(Aut_{{\mathbb F}_p}(\enskip)\) is the functor associating to \(R_*\) the group of quasi-strict automorphisms of formal group laws over \(R_* [\epsilon]/ (\epsilon^2)\) then these functors are representable, the first by \(H_* H\) and the second by \(E(\overline{\tau}_0,\overline{\tau}_1,\dots) \otimes {\mathbb F}_p [ \overline{\xi}_1,\overline{\xi}_2,\dots]\). The author defines a natural transformation between these functors and shows, using results of \textit{Hu\"nh Mùi} [Math. Z. 193, 151-163 (1986; Zbl 0593.55014)] on cohomology operations derived from modular invariants, that this induces an isomorphism of the representing Hopf algebras.