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Mark Behrens

Publications and source records attributed to Mark Behrens.

At least 19 recordsLinked to original sources
arXiv2026

Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Greene and the first author, we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.

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arXiv2024

Periodic phenomena in equivariant stable homotopy theory

Building off of many recent advances in the subject by many different researchers, we describe a picture of A-equivariant chromatic homotopy theory which mirrors the now classical non-equivariant picture of Morava, Miller-Ravenel-Wilson, and Devinatz-Hopkins-Smith, where A is a finite abelian p-group. Specifically, we review the structure of the Balmer spectrum of the category of A-spectra, and the work of Hausmann-Meier connecting this to MU_A and equivariant formal group laws. Generalizing work of Bhattacharya-Guillou-Li, we introduce equivariant analogs of v_n-self maps, and generalizing work of Carrick and Balderrama, we introduce equivariant analogs of the chromatic tower, and give equivariant analogs of the smash product and chromatic convergence theorems. The equivariant monochromatic theory is also discussed. We explore computational examples of this theory in the case of A = C_2, where we connect equivariant chromatic theory with redshift phenomena in Mahowald invariants.

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arXiv2023

A deformation of Borel equivariant homotopy

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.

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arXiv2020

The 2-primary Hurewicz image of tmf

We determine the image of the 2-primary tmf-Hurewicz homomorphism, where tmf is the spectrum of topological modular forms. We do this by lifting elements of tmf_* to the homotopy groups of the generalized Moore spectrum M(8,v_1^8) using a modified form of the Adams spectral sequence and the tmf-resolution, and then proving the existence of a v_2^32-self map on M(8,v_1^8) to generate 192-periodic families in the stable homotopy groups of spheres.

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arXiv2019

The telescope conjecture at height 2 and the tmf resolution

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end we compute the structure of the tmf-resolution for a specifc type 2 complex Z. We find that, analogous to the height 1 case, the E1-page of the tmf-resolution possesses a decomposition into a v2-periodic summand, and an Eilenberg-MacLane summand which consists of bounded v2-torsion. However, unlike the height 1 case, the E2-page of the tmf-resolution exhibits unbounded v2-torsion. We compare this to the work of Mahowald-Ravenel-Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v2-torsion: either the unbounded v2-torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2-parabolas and the telescope conjecture is false. We also study how to use the tmf-resolution to effectively give low dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)-local Adams-Novikov spectral sequence for Z collapses.

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arXiv2019

C_2-equivariant stable homotopy from real motivic stable homotopy

We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.

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arXiv2019

Topological modular and automorphic forms

This article is a brief survey of the theory of topological modular forms (TMF) and the theory of topological automorphic forms (TAF). It will be a chapter in forthcoming "Handbook of Homotopy Theory" edited by Haynes Miller.

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arXiv2017

The Bousfield-Kuhn functor and Topological Andre-Quillen cohomology

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.

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arXiv2017

Detecting exotic spheres in low dimensions using coker J

Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which S^n has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which S^n has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.

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arXiv2017

Spectral algebra models of unstable v_n-periodic homotopy theory

We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological Andr\'e-Quillen cohomology, Koszul duality, and Goodwillie calculus.

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arXiv2017

On the E2-term of the bo-Adams spectral sequence

The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

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arXiv2015

On the ring of cooperations for 2-primary connective topological modular forms

We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.

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arXiv2012

On the homotopy of Q(3) and Q(5) at the prime 2

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenel's computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomura's 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

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arXiv2010

The Goodwillie tower for S^1 and Kuhn's theorem

We analyze the homological behavior of the attaching maps in the 2-local Goodwillie tower of the identity evaluated at S^1. We show that they exhibit the same homological behavior as the James-Hopf maps used by N. Kuhn to prove the 2-primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of S^1 at the prime 2.

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arXiv2010

The Goodwillie tower and the EHP sequence

We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to re-compute the 2-primary unstable stems through the Toda range (up to the 19-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod 2 stable homology of the Goodwillie layers of any functor from spaces to spaces.

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