arXiv subjects
Publications and source records attributed to Tilman Bauer.
We show that the topological elliptic genus from the cobordism ring of SU-manifolds to topological Jacobi forms lifts to connective topological Jacobi forms, and that this lift is surjective in homotopy.
There are two kinds of exotic spheres: bp spheres, which bound parallelizable manifolds, and non-bp spheres, or very exotic spheres, which do not. In the 1960s, W.-C. Hsiang showed that in each dimension where bp spheres exist, there is at least one which admits infinitely many inequivalent smooth free $S^1$-actions, and in each dimension congruent to $3$ modulo $4$, there is at least one bp sphere which admits infinitely many inequivalent smooth free $S^3$-actions. On the other hand, for each fixed prime $p$, smooth free $S^1$- and $S^3$-actions have only been recorded to exist for finitely many very exotic spheres with nontrivial $p$-local Kervaire--Milnor invariant, all in dimension less than approximately $p^3$. In this paper, we use topological modular forms to detect smooth free $S^1$- and $S^3$-actions on infinite families of very exotic spheres with nontrivial $2$- and $3$-local Kervaire--Milnor invariants.
As a generalization of the ring spectrum of topological modular forms, we construct a graded ring spectrum of topological Jacobi forms, $\operatorname{TJF}_*$. This is constructed as the global sections of a sheaf of $E_\infty$-ring spectra on the stacky universal elliptic curve using circle-equivariant $\operatorname{TMF}$. Complete calculations of its homotopy at odd primes and partial results at $p=2$ are given.
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^*(\Omega^n\Sigma^n X;\mathbf F_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf F_p)$ in a functorial way.
We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct sum of explicitly given indecomposables. This gives rise to a similar classification of not necessarily $p$-torsion objects that are either free as commutative algebras or cofree as cocommutative coalgebras. We also completely classify those objects that are indecomposable modulo $p$.
We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree commutative Hopf algebra on a commutative $k$-algebra $A$ if $P$ is the $p$-polar algebra underlying $A$; a dual result holds for free commutative Hopf algebras on finite $k$-coalgebras.
As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the latter is free on the $p$-polar affine line.
The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $\mathcal A$. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory ($2$-compact groups, topological modular forms) and may be of independent interest.
In this paper we study tensor products of affine abelian group schemes over a perfect field $k.$ We first prove that the tensor product $G_1 \otimes G_2$ of two affine abelian group schemes $G_1,G_2$ over a perfect field $k$ exists. We then describe the multiplicative and unipotent part of the group scheme $G_1 \otimes G_2$. The multiplicative part is described in terms of Galois modules over the absolute Galois group of $k.$ We describe the unipotent part of $G_1 \otimes G_2$ explicitly, using Dieudonné theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.
Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework to study the algebra of such functors, which I call formal plethories, in the case where $E_*$ is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.
Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A-infinity-monad and prove that it is an A-infinity-monad itself.
This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in each of the latter two cases, a sequence of algebraic Bockstein spectral sequences is used to compute the E_2 term of the elliptic Adams-Novikov spectral sequence from the elliptic curve Hopf algebroid. In a further step, all the differentials in the latter spectral sequence are determined. The result of this computation is originally due to Hopkins and Mahowald (unpublished).
We construct a combinatorial model of an A-infinity-operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R.
We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group.
For a compact Lie group $G$ with maximal torus $T$, Pittie and Smith showed that the flag variety $G/T$ is always a stably framed boundary. We generalize this to the category of $p$-compact groups, where the geometric argument is replaced by a homotopy theoretic argument showing that the class in the stable homotopy groups of spheres represented by $G/T$ is trivial, even $G$-equivariantly. As an application, we consider an unstable construction of a $G$-space mimicking the adjoint representation sphere of $G$ inspired by work of the second author and Kitchloo. This construction stably and $G$-equivariantly splits off its top cell, which is then shown to be a dualizing spectrum for $G$.
We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.