The T-algebra spectral sequence: Comparisons and applications

In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence in detail. Under certain assumptions, we show that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence agree. Under further assumptions, we can apply a variation of an argument due to Jennifer French and show that these spectral sequences agree with the unstable Adams spectral sequence. From these equivalences we obtain information about filtration and differentials. Using these equivalences we construct the homological and cohomological Bockstein spectral sequences topologically. We apply these spectral sequences to show that Hirzebruch genera can be lifted to E_\infty-ring maps and that the forgetful functor from E_\infty-algebras in H\overline{F}_p-modules to H_\infty-algebras is neither full nor faithful.Comment: Minor revisions and more than a few typo corrections. To appear in Algebraic and Geometric Topolog

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends to a monad on the homotopy category and defines a category of homotopy $T$-algebras. In this setting there is a forgetful functor from the homotopy category of $T$-algebras to the category of homotopy $T$-algebras. Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy $T$-algebra map to a strict map of $T$-algebras. Once we have a map of $T$-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of $T$-algebra maps and the edge homomorphism on $\pi_0$ is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the $E_2$-term to be calculable in terms of Quillen cohomology groups. We provide worked examples in $G$-spaces, $G$-spectra, rational $E_\infty$ algebras, and $A_\infty$ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of $E_\infty$ ring spectra to the category of $H_\infty$ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space $E_\infty(\Sigma^\infty_+ \mathrm{Coker}\, J, L_{K(2)} R)$.Comment: 45 pages. Substantial revision. To appear in Advances in Mathematic

Derived induction and restriction theory

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for $G$-equivariant $E$-homology and cohomology, and generalizations of Quillen's $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, the classical bordism theories, connective real $K$-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.Comment: 63 pages. Many edits and some simplifications. Final version, to appear in Geometry and Topolog

On a nilpotence conjecture of J.P. May

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., $H_\infty$-ring spectra. Using an explicit nilpotence bound on the torsion elements in $K(n)$-local $H_\infty$-algebras over $E_n$, we reduce the conjecture to the nilpotence theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence results in various bordism rings including $M\mathit{Spin}_*$ and $M\mathit{String}_*$, results about the behavior of the Adams spectral sequence for $E_\infty$-ring spectra, and the non-existence of $E_\infty$-ring structures on certain complex oriented ring spectra.Comment: 17 pages. To appear in Journal of Topolog

Nilpotence and descent in equivariant stable homotopy theory

Let $G$ be a finite group and let $\mathscr{F}$ be a family of subgroups of $G$. We introduce a class of $G$-equivariant spectra that we call $\mathscr{F}$-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable $\infty$-category, with which we begin. We then develop some of the basic properties of $\mathscr{F}$-nilpotent $G$-spectra, which are explored further in the sequel to this paper. In the rest of the paper, we prove several general structure theorems for $\infty$-categories of module spectra over objects such as equivariant real and complex $K$-theory and Borel-equivariant $MU$. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex $K$-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.Comment: 63 pages. Revised version, to appear in Advances in Mathematic

In Vitro Consequences of Electronic-Cigarette Flavoring Exposure on the Immature Lung.

Background: The developing lung is uniquely susceptible and may be at increased risk of injury with exposure to e-cigarette constituents. We hypothesize that cellular toxicity and airway and vascular responses with exposure to flavored refill solutions may be altered in the immature lung. Methods: Fetal, neonatal, and adult ovine pulmonary artery smooth muscle cells (PASMC) were exposed to popular flavored nicotine-free e-cigarette refill solutions (menthol, strawberry, tobacco, and vanilla) and unflavored solvents: propylene glycol (PG) or vegetable glycerin (VG). Viability was assessed by lactate dehydrogenase assay. Brochodilation and vasoreactivity were determined on isolated ovine bronchial rings (BR) and pulmonary arteries (PA). Results: Neither PG or VG impacted viability of immature or adult cells; however, exposure to menthol and strawberry flavored solutions increased cell death. Neonatal cells were uniquely susceptible to menthol flavoring-induced toxicity, and all four flavorings demonstrated lower lethal doses (LD50) in immature PASMC. Exposure to flavored solutions induced bronchodilation of neonatal BR, while only menthol induced airway relaxation in adults. In contrast, PG/VG and flavored solutions did not impact vasoreactivity with the exception of menthol-induced relaxation of adult PAs. Conclusion: The immature lung is uniquely susceptible to cellular toxicity and altered airway responses with exposure to common flavored e-cigarette solutions