arXiv subjects

Nicholas J. Kuhn

Publications and source records attributed to Nicholas J. Kuhn.

At least 19 recordsLinked to original sources

arXiv2025

Type 2 complexes constructed from Brown-Gitler spectra

In a previous paper, one of us interpreted mod 2 Dyer-Lashof operations as explicit A-module extensions between Brown-Gitler modules, and showed these A-modules can be topologically realized by finite spectra occurring as fibers of maps between 2-local dual Brown-Gitler spectra. Starting from these constructions, in this paper, we show that infinite families of these finite spectra are of chromatic type 2, with mod 2 cohomology that is free over A(1). Applications include classifying the dual Brown-Gitler spectra after localization with respect to K-theory.

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arXiv2024

Dyer-Lashof operations as extensions of Brown-Gitler Modules

At the prime 2, let T(n) be the n dual of the nth Brown-Gitler spectrum with mod 2 homology G(n). Our previous work on computing the homology of an infinite loopspaces led us to observe that there are extensions between various of the right A-modules G(n) such that splicing with these gives an action of the Dyer-Lashof algebra on the sum over s and n of Ext_A^{s,s}(G(n),M). We give explicit constructions of these `Dyer-Lashof operation' extensions: one construction relates them to the cofiber sequence associated to the C_2-transfer. Another relates key `squaring' Dyer-Lashof operations to the Mahowald short exact sequences. Finally, properties of the spectra T(n) allow us to geometrically realize our extensions by cofibration sequences, with the implication that the sum over n of all the Adams spectral sequences computing [T(n),X] is a spectral sequence of modules over the Dyer-Lashof algebra.

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arXiv2024

Applications of the circle product with a right $Com$-module to the theory of commutative ring spectra

If Com is the reduced commutative operad, the category of Com-algebras in spectra is the category of nounital commutative ring spectra. The theme of this survey is that many important constructions on Com-algebras are given by taking the derived circle product with well chosen right Com-modules. Examples of constructions arising this way include the tensor product of a based space K with such an algebra I, and TQ(I), the Topological Andre-Quillen homology spectrum of I. We then show how filtrations of right Com-modules can be used to filter such constructions. A natural decreasing filtration on right Com-modules, when specialized to the Com-bimodule Com, defines the augmentation ideal tower of I, built out of the extended powers of TQ(I). A natural increasing filtration on right Com-modules, when specialized to the right Com-module used to define TQ(I), defines a filtration on TQ(I) built out of I and the spaces in the Lie cooperad. There are two versions of this in the literature -- by the author and by Behrens and Rezk -- and our setting here makes it easy to prove that these agree. Much of this applies with Com replaced by a more general reduced operad, and we make a few remarks about this.

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arXiv2023

An elementary proof of the chromatic Smith fixed point theorem

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 reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, A-CW complexes, which suggests some open problems.

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arXiv2022

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

In the early 1940's, P.A.Smith showed that if a finite p-group G acts on a finite complex X that is mod $p$ acyclic, then its space of fixed points, X^G, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study chromatic versions of this statement, with the question: given H<G and n, what is the smallest r such that if X^H is acyclic in the (n+r)th Morava K-theory, then X^G must be acyclic in the nth Morava K-theory? Barthel et.al. then answered this when G is abelian, by finding general lower and upper bounds for these `blue shift' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E.E.Floyd, which replaces acyclicity by bounds on dimensions of homology, and thus applies to all finite G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. In one direction, we are able to use classic constructions and representation theory to search for blue shift number lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that don't follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. As samples of new applications, we offer a new result about involutions on the 5-dimensional Wu manifold, and a calculation of the mod 2 K-theory of a 100 dimensional real Grassmanian that uses a C_4 chromatic Floyd theorem.

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arXiv2021

A short proof of the chromatic Smith Fixed Point Theorem

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 finite A-space.

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arXiv2021

Computing the Morava K-theory of real Grassmanians using chromatic fixed point theory

We study K(n)*(Gr(d,m)) for all n - the 2-local Morava K-theories of the real Grassmanian Gr(d,m) of d-planes in R^m, about which very little has been previously computed. We conjecture that the Atiyah-Hirzebruch Spectral Sequences computing these all collapse after the first possible non-zero differential, and give much evidence that this is the case. Computational patterns for all n seem similar to the known calculation of H*(Gr(d,m);Q), the n=0 case. We use a novel method to show that higher differentials can't occur: we get a lower bound on the size of K(n)*(Gr(d,m)) by constructing an action of C = the cyclic group of order 4, on our Grassmanians, and then applying the chromatic fixed point theory of the authors' previous paper. In essence, we bound the size of K(n)*(Gr(d,m)) from below by computing K(n-1)*(Gr(d,m)^C). Meanwhile, the AHSS after the first differential is determined by Q_n-homology, where Q_n is Milnor's nth primitive operation in mod 2 cohomology. Whenever we are able to calculate this, we have found that it agrees with our lower bound for the size of K(n)*(Gr(d,m)). We have two general families where we prove this: m at most 2^{n+1} and all d, and d=2 and all m and n. Computer calculations have allowed us to check many other examples with larger values of d.

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arXiv2019

Split Hopf algebras, quasi-shuffle algebras, and the cohomology of Omega Sigma X

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 as a morphism of graded k-vector spaces equipped with a Verschiebung map. Building on work on non-commutative Witt vectors by Goerss, Lannes, and Morel, we classify such free, `split' Hopf algebras. A topological consequence is that, if X is a based path connected space, then the Hopf algebra H^*(Omega Sigma X;k) is determined by the stable homotopy type of X. We also discuss the much easier analogous characteristic 0 results, and give a characterization of when our quasi--shuffle algebras are polynomial, generalizing the so-called Ditters conjecture.

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arXiv2018

Adams filtration and generalized Hurewicz maps for infinite loopspaces

We study the Hurewicz map h from the homotopy groups of a spectrum X to the R-homology of its 0th space X(0), where R is a connective commutative S-algebra. We prove that the decreasing filtration of the domain of h associated to an R-based Adams resolution is compatible with the augmentation ideal filtration of the range associated to the suspension spectrum of X(0)_+, an augmented commutative S-algebra. The proof makes use of the interplay of this filtration with Topological Andre Quillen Homology. An application is a Connectivity Theorem: Localize at a prime p and suppose X is (c-1)-connected for some positive c. If f in pi_*(X) has Adams filtration s and |f| < cp^s, then f maps to zero in R_*(X(0)). An application of that is a Finiteness Theorem: If the mod p cohomology of X is finitely presented as a module over the Steenrod algebra, then the image of the mod p Hurewicz map for X(0) is finite. We illustrate these theorems with calculations of the mod 2 Hurewicz image of BO, its connected covers, and tmf(0), and the mod p Hurewicz image of all the spaces in the BP and BP spectra. Enroute, we get new proofs of theorems of Milnor and Wilson. In the special case when X is a suspension spectrum and R = HZ/2, we recover results announced by Lannes and Zarati in the 1980s (with a totally different proof), and generalizations to all primes p. We get a chromatic version of this for the Hurewicz map for Morava E theory and all X.

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arXiv2016

Operad bimodules, and composition products on Andre-Quillen filtrations of algebras

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a canonical (and homotopically meaningful) decreasing O-algebra filtration I > I^2 > I^3 > ... satisfying various nice properties analogous to powers of an ideal in a ring. In this paper, we are explicit about these constructions. With R a commutative S-algebra, we study derived versions of the circle product M o_O I, where M is an O-bimodule, and I is an O-algebra in R-modules. Letting M run through a decreasing O-bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously studied structure. As a formal consequence, an O-algebra map from I to J^d induces compatible maps from I^n to J^{dn}, for all n. This is an essential tool in the first author's study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem.

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arXiv2014

The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A

In the early 1980's the author proved G.W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990's, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd dimensional spheres. In this paper we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.

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arXiv2014

Generic representation theory of finite fields in nondescribing characteristic

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,F)]-modules. As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if Char K = 0: the category Rep(F;K) will be semisimple. In a last section, we briefly discuss "q=1" analogues and consider representations of various categories of finite sets. The main result follows from a 1992 result by L.G.Kovacs about the semigroup ring K[M_n(\F)].

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arXiv2013

Nilpotence in Group Cohomology

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p--groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p--group P.

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arXiv2013

The Krull filtration of the category of unstable modules over the Steenrod algebra

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to better expose the algebraic properties of the Krull filtration.

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arXiv2013

The mod 2 homology of infinite loopspaces

We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1 term is the homology of the extended powers of X, and thus is a well known functor of H_*(X), including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra A, and a left module over the Dyer-Lashof operations. Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in E^1. We use an operad structure on the tower and the Z/2 Tate construction to show how Dyer-Lashof operations and differentials interact. These then determine differentials that hold for any spectrum X. These universal differentials then lead us to construct, for every A-module M, an algebraic spectral sequence depending functorially on M. The algebraic spectral sequence for H_*(X) agrees with the topological spectral sequence for X for many spectra, including suspension spectra and almost all generalized Eilenberg-MacLane spectra, and appears to give an upper bound in general. The E^infty term of the algebraic spectral sequence has form and structure similar to E^1, but now the right A-module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980's by W. Singer, J. Lannes and S. Zarati, and P. Goerss.

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arXiv2009

Correction to ""Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology"

Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not sufficiently proved, and, indeed, is likely incorrect as stated. This necessitates a little more argument to finish the proof of the main theorem of the original paper. The statement of this theorem, and all the examples, remain unchanged.

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arXiv2008

Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology

We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence.

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arXiv2008

A guide to telescopic functors

In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation.

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