arXiv subjects
Publications and source records attributed to Kiran Luecke.
In the theory of spectral algebraic geometry, few objects receive as much study as the spectrum of topological modular forms. In this paper, we compute the strict units of topological modular forms, defined as the connective cover of the mapping spectrum from $\mathbb{Z}$ to the units spectrum.
We present a construction that manufactures $\E_\infty$ orientations of Tate fixed-point objects together with useful formulas for these maps, and then give a number of applications. For example, we produce a formula for the Frobenius homomorphisms of Thom spectra such as $\MU$ as well as certain lifts of Frobenius. We prove a rigidity property of $\MU$ as a \emph{cyclotomic} object. We construct a general obstruction theory for $\E_n$ complex orientations and establish various non-existence results for $p$-typical $\E_n$ orientations for low values of $p$ and $n$. We end with some miscellaneous further applications.
We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of $\mathbb E_\infty$ complex orientations to establish $p$-adic Kümmer congruences among iterated derivatives of the Weierstrass elliptic function. The machinery of Ando, Hopkins, and Rezk was developed with the intended application of taking congruence relations as input and producing $\mathbb E_\infty$-orientations as output. We run their machine in reverse, using as input the recent results of Carmeli and the first author on the existence of $\mathbb E_\infty$-orientations of Tate fixed-point objects.
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}.
It is well known that the $[0,1]$ and $[0,2]$ Postnikov truncations of the units of the topological $K$-theories $\glone \KO$ and $\glone \KU$, respectively, are split, and that the splitting is provided by the ($\Z/2$-graded) line bundles. In this paper we give a similar splitting for the $[0,1]$-truncation of the units of algebraic $K$-theory, considered as a sheaf on affine schemes. A crucial step is to produce the splitting for $\glone K(\Z)$. Along the way we also give a complete calculation of the connective spectrum of strict units of $K(\Z)$ and $K(\F_\ell)$ for a prime $\ell$. Finally, we show that the units of algebraic $K$-theory do not split as a presheaf. In fact we show they do not even split pointwise.
In this note I give a positive solution to Bullett's conjecture (posed in [1]) regarding a geometric presentation of the universal mod $p$ oriented ring spectrum.
We compute the first two k-invariants of the Picard spectra of $KU$ and $KO$ by analyzing their Picard groupoids and constructing their unit spectra as global sections of sheaves on the category of manifolds. This allows us to determine the E_\infty-structures of their truncations Pic(KU)[0,3] and Pic(KO)[0,2]. It follows that these truncated Picard spaces represent: the Brauer groups of Z/2-graded algebra bundles of Donovan-Karoubi, Moutuou and Maycock; the Brauer groups of super 2-lines; and the K-theory twists of Freed, Hopkins and Teleman. Our results also imply that that these spaces represent twists of String and Spin structures on manifolds and can be used to twist tmf-cohomology. Finally, we are able to identify pic(KU)[0,3] with a cotruncation of the Anderson dual of the sphere spectrum.
In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of [FT] and [FHT3]. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of [FT] as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which is a far more ubiquitous phenomenon than the varied techniques that were previously used to prove such formulas.
Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. Several authors (cf [ABG],[KM],[L]) have suggested that an equivariant version ought to be related to the work of Freed-Hopkins-Teleman ([FHT1],[FHT2],[FHT3]). However, a first attempt at this runs into apparent contradictions concerning twist, degree, and cup product. Several authors (cf. [BET],[G],[K]) have solved the problem over the complex numbers by interpreting the $S^1$-equivariant parameter as a complex variable and using holomorphicity as the technique for completion. This paper gives a solution that works integrally, by constructing a carefully completed model of $K$-theory for $S^1$-equivariant stacks which allows for certain ``convergent" infinite-dimensional cocycles.
Let $k$ be a perfect field of characteristic $p$. Associated to any (1-dimensional, commutative) formal group law of finite height $n$ over $k$ there is a complex oriented cohomology theory represented by a spectrum denoted $E(n)$ and commonly referred to as Morava $E$-theory. These spectra are known to admit $E_\infty$-structures, and the dependence of the $E_\infty$-structure on the choice of formal group law has been well studied (cf.\ [GH], [R], [L], Section 5, [PV]). In this note we show that the underlying homotopy type of $E(n)$ is independent of the choice of formal group law.
This two-page note gives a non-computational derivation of the dual Steenrod algebra as the automorphisms of the formal additive group. Instead of relying on computational tools like spectral sequences and Steenrod operations, the argument uses a few simple universal properties of certain cohomology theories.