Tobias Barthel, Nathaniel Stapleton
Let $A$ be a finite abelian $p$ group of rank at least $2$. We show that $E^0(BA)/I_{tr}$, the quotient of the Morava $E$-cohomology of $A$ by the ideal generated by the image of the transfers along all proper subgroups, contains $p$-torsion. The proof makes use of transchromatic character theory.
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The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava $E$-cohomology of finite abelian groups has played an important role in the study of power operations for Morava $E$-theory. The goal of this paper is to explore the relationship between level structures on the $p$-divisible group given by the trivial extension of the universal deformation by a constant $p$-divisible group and the Morava $E$-cohomology of the iterated free loop space of the classifying space of a finite abelian group.
We prove that the $p$-completed Brown-Peterson spectrum is a retract of a product of Morava $E$-theory spectra. As a consequence, we generalize results of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the notion of good group is determined by Brown-Peterson cohomology. Furthermore, we show that rational factorizations of the Morava $E$-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown-Peterson cohomology of such groups.
For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent…
We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is…
We give a short and much simplified proof of the main theorem of the recent study, by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton, of the Balmer spectrum for A-equivariant stable homotopy when A is a finite abelian p-group. This theorem says that if A is a finite abelian p-group of rank r, and X is a finite A-space that is acyclic in the (n+r)th Morava K-theory, then its space of fixed points, X^A, will be acyclic in the nth Morava K-theory. It is a chromatic homotopy version of P. A. Smith's classic theorem about the mod p homology of the fixed points of a fini…
We determine the Gross-Hopkins duals of certain higher real $K$-theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy fixed point spectra $E_n^{hG}$ are Gross-Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups $G$ which contain $p$-torsion. This generalizes previous results for $n=2$ and $p=3$.
We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonné correspondence between bicommutative Hopf algebras and Dieudonné modules. We extend P. Goerss's…
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and…
A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if A is a finite abelian p-group of rank r, then any finite A-space X which is acyclic in the nth Morava K-theory with n at least r will have its subspace F of fixed points acyclic in the (n-r)th Morava K-theory. This is a chromatic homotopy version of P.A.Smith's classical theorem that if X is acyclic in mod p homology, then so is F. The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoni…
We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses) for the automorphism group of V. We use Brauer characters and computer calculations. We construct and implement an algorithm for finding permutation submodules of maximal dimension inside modules for p-groups in characteristic p.
For a knot K in $S^3$ and a regular representation $ρ$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we construct according to Casson--or more precisely taking into account Lin's and Heusener's further works--a volume form on the SU(2)-representation space of $G_K$. Next, we compare these two apparently different points of view--the first by means of the Reidemeister torsion and the s…
We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.
In this paper we compute the total power operation for the Morava $E$-theory of any finite group up to torsion. Our formula is stated in terms of the $GL_n(Q_p)$-action on the Drinfeld ring of full level structures on the formal group associated to $E$-theory. It can be specialized to give explicit descriptions of many classical operations. Moreover, we show that the character map of Hopkins, Kuhn, and Ravenel from $E$-theory to $GL_n(Z_p)$-invariant generalized class functions is a natural transformation of global power functors on finite groups.
We show that the regulator, which is the difference between the homology torsion and the combinatorial Ray-Singer torsion, of fnite abelian coverings of a fixed complex has sub-exponential growth rate.
We give a brief survey of abelian torsions of 3-manifolds.
Let $X$ be a topological space upon which a compact connected Lie group $G$ acts. It is well-known that the equivariant cohomology $H_G^*(X;\Q)$ is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology $H_T^*(X;\Q)$, where $T$ is a maximal torus of $G$. This relationship breaks down for coefficient rings $\k$ other than $\Q$. Instead, we prove that under a mild condition on $\k$ the algebra $H_G^*(X,\k)$ is isomorphic to the subalgebra of $H_T^*(X,\k)$ annihilated by the divided difference operators.
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