The Adams–Novikov spectral sequence and the cohomology of Hopf algebras

In 1966, at the International Congress of Mathematicians in Moscow, Sergey Petrovich Novikov made a talk on the theory of cobordisms in problems of algebraic topology. The following year, his large article “Methods of algebraic topology from the point of view of cobordism theory” was published in the journal Izvestiya Akademii Nauk, mathematical series. It was a program for restructuring methods and approaches in algebraic topology based on the theory of cobordisms $U^*$. The focus of this paper was the construction of the Adams–Novikov spectral sequence (ANSS) and the algebra $A_U$ of all stable cohomology operations in the $U^*$-theory.
There are two types of operations in the algebra $A_U$: multiplication by an element of the ring of scalars $\Omega_U$ (isomorphic to the polynomial ring by the Milnor–Novikov theorem, 1960) and the action of the Landweber–Novikov algebra $S$, defined in terms of characteristic classes and the Thom isomorphism. The algebra of all stable operations is naturally described as a completed tensor product $\Omega_U\hat\otimes S$.
S.P.Novikov highlighted the fact that multiplication in this tensor product is not standard, but is determined by a nontrivial commutation rule for elements of the ring $\Omega_U$ and the algebra $S$.
The Adams–Novikov spectral sequence has become (and still remains) one of the main tools for computing homotopy groups of spectra. In the case of a spectrum of spheres, the second page this spectral sequence is a purely algebraic object, which is described in terms of the cohomology of the Landweber–Novikov algebra $S$ and its representation in the ring $\Omega_U$, which defines multiplication in the algebra $A_U$.
The fact that the Landweber–Novikov algebra $S$ is a Hopf algebra turned out to be fundamental.
To calculate second page of the ANSS, V.M.Buchstaber proposed a spectral sequence (1970), which connects the cohomology of the Hopf algebra S with its representation on the ring $\Omega_U$. The construction of this spectral sequence is based on the isomorphism of $S$-modules $\Omega_U\otimes Q\cong S_*\otimes Q$, where $S_*$ is a Hopf algebra dual to the algebra $S$.
The talk is focused on the construction of the Buchstaber spectral sequence for general Hopf algebras and a new structure in the cohomology of Hopf algebras determined by this spectral sequence (Buchstaber–Popelensky, 2024). We discuss problems in the theory of Hopf algebras in which this structure plays an important role. Connections with classical works in algebraic topology and homological algebra will be demonstrated.