We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.
Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of the finite sub-Hopf algebras $A(n)$ of the mod 2 Steenrod algebra that correspond to generators in the homotopy rings of certain periodic spectra. The purpose of this note is to present a proof of the conjecture.
Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k-vector spaces equipped with a Frobenius (pth-power) map. For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split a…
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Ger…
We show that the vector bundle on the moduli stack $M_\mathrm{ell}$ of elliptic curves associated to the $2$-cell complex $Cν$ is isomorphic to the de Rham cohomology sheaf $\mathrm{H}^1_\mathrm{dR}(\mathcal{E}/M_\mathrm{ell})$ of the universal elliptic curve $\mathcal{E}\to M_\mathrm{ell}$. We use this to calculate the homotopy groups of the $\mathbf{E}_{1}$-quotient $\mathrm{tmf} /\!\!/ ν$ of $\mathrm{tmf}$ by $ν$, called the spectrum of "topological quasimodular forms", by relating its Adams-Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitt…
Texas Christian University（德克萨斯基督教大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Louisiana at Lafayette（路易斯安那大学拉斐特分校）；University of Oregon（俄勒冈大学）；University of the Virgin Islands（维尔京群岛大学）
本文提出链环补空间的 bar 上同调作为同调球中链环的新不变量，它构成 Hopf 代数，可通过曲面及其交曲线计算，并定义了张量 Hopf 代数的典范子空间，编码并超越 Milnor 不变量。
We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.
