Tate-valued Characteristic Classes
Shachar Carmeli, Kiran Luecke
TL;DR
The paper develops a Tate-theoretic refinement of complex orientation theory, introducing the Euler-Tate framework that yields an $\mathbb{E}_\infty$-lift of the total Chern class via the Tate construction $R^{t\mathbb{T}}$. It proves the Euler-Tate characteristic class $c^{\mathfrak{et}}$ has the characteristic series $t^{-1}(x+_F t)$, situating the total Chern class inside a coherent $\mathbb{E}_\infty$-map, and thereby answering Segal's question affirmatively. A sharp, $\mathbb{E}_\infty$-lift construction generalizes Ando-French-Ganter to produce $(\omega)^{\#}:\mathrm{MU}\langle 2n-2\rangle\to R^{t\mathbb{T}}$ for $n=0,1,2,3$ with $R$ even periodic, yielding new $\mathbb{E}_\infty$-orientations and, in particular, an $\mathbb{E}_\infty$-lift of the Jacobi orientation for $\mathrm{tmf}^{t\mathbb{T}}$. The work also shows orientability results for the three forms of periodic complex bordism, propagating $\mathbb{E}_\infty$-orientations to a periodic integral form $\mathbb{Z}^{t\mathbb{T}}$, thereby enriching the landscape of highly structured orientations in stable homotopy theory.
Abstract
We define a projective variant of classical complex orientation theory. Using this, we construct a map of spectra which lifts the total Chern class, providing an alternative answer to an old question of Segal \cite{segal}, previously answered by Lawson et al \cite{lawsonetal}. We also lift and generalize the ``sharp'' construction of Ando-French-Ganter \cite{afg} to an operation on arbitrary $\EE_\infty$-complex orientations, thereby providing a rich source of new $\EE_\infty$-orientations for commutative ring spectra. In particular we give an $\EE_\infty$-lift of the Jacobi orientation, a generalization of the much-studied two variable elliptic genus. Finally, we construct some new complex orientations of periodic ring spectra as requested in \cite{hahnyuan}.